Conference Schedule

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• Summary
• Schedule Day 1
• Schedule Day 2
• Plenary Lectures
• Minisymposia
• Contributed Talks
• Posters

SUMMARY

4 plenary lectures, 19 minisymposia, 7 contributed talk sessions, and 1 poster session (11 posters).

• Plenary I — Guillaume Bal, University of Chicago — Sat., Sept 19 — 8:45–9:45 AM
• Plenary II — Jianfeng Lu, Duke University — Sat., Sept 19 — 1:45–2:45 PM
• Plenary III — Hailiang Liu, Iowa State University — Sun., Sept 20 — 8:00–9:00 AM
• Plenary IV — Selim Esedoglu, University of Michigan — Sun., Sept 20 — 9:15–10:15 AM

Daily Schedule

DAY 1 — Saturday, September 19, 2026

• 8:00 – 8:15 AM — Check in & Light Breakfast
  Lawson Computer Science Building, 305 N University St, West Lafayette, IN 47907
• 8:15 – 8:45 AM — Welcome and Opening Remarks
  LWSN 1142
• 8:45 – 9:45 AM — Plenary Talk I: Guillaume Bal (University of Chicago)
  LWSN 1142
  Title: Classification and scattering theory aspects of topological insulators
• 9:45 – 10:15 AM — Coffee Break
• 10:15 AM – 12:15 PM — Parallel Sessions:
  • LWSN 1106 — MS1: Computation and Analysis for HJE in Physics and Biology
  • LWSN 3102 — MS2: Advancements in Inverse Problems, Scattering and Spectral Theory — Part I
  • LWSN B151 — MS3: Recent Advancements in Learning for Complex Systems
  • LWSN B134 — MS4: Multifidelity Scientific Machine Learning for Reduced-Order Modeling of Complex Systems
  • LWSN B155 — MS5: Modeling and Data-Driven Approaches to Complex Biological Systems — Part I
  • LWSN 1142 — CT1: Contributed Talks I: Numerical Methods and Structure-Preserving Schemes
  • DSAI 1069 — CT2: Contributed Talks II: Analysis of Nonlinear PDEs and Spectral Theory
• 12:15 – 1:45 PM — Lunch & Poster Session
  Lobby (poster area)
  11 posters — see Poster Session listing
• 1:45 – 2:45 PM — Plenary Talk II: Jianfeng Lu (Duke University)
  LWSN 1142
  Title: Quantitative hypocoercive convergence estimates for underdamped Langevin equations
• 2:45 – 3:00 PM — Break
• 3:00 – 5:00 PM — Parallel Sessions:
  • LWSN 1106 — MS6: Mathematical Aspects of 2D Quantum Materials and Metamaterials
  • LWSN 3102 — MS7: Generative Models, Modern Optimization, and Autonomous Agents in Scientific Computing
  • LWSN B151 — MS8: Multiscale Dynamics in Nonlinear PDEs and Complex Systems
  • LWSN B134 — MS9: AI-Based Numerical PDE
  • LWSN B155 — MS10: Modeling and Data-Driven Approaches to Complex Biological Systems — Part II
  • LWSN 1142 — CT3: Contributed Talks III: Optimization, Inverse Problems, and Control
  • DSAI 1069 — CT4: Contributed Talks IV: Stochastic Modeling, Uncertainty Quantification, and Stochastic Control
• 5:00 – 5:30 PM — Coffee Break
• 5:30 – 6:30 PM — SIAM Great Lakes Section Business Meeting

DAY 2 — Sunday, September 20, 2026

• 7:30 – 8:00 AM — Light Breakfast
• 8:00 – 9:00 AM — Plenary Talk III: Hailiang Liu (Iowa State University)
  LWSN 1142
  Title: Deep Learning, PDEs, and Control: A Mean-Field Theory of Transformers
• 9:00 – 9:15 AM — Break
• 9:15 – 10:15 AM — Plenary Talk IV: Selim Esedoglu (University of Michigan)
  LWSN 1142
  Title: Algorithms for triple junction drag and surface grooving in thin polycrystalline films
• 10:15 – 10:30 AM — Break
• 10:30 AM – 12:30 PM — Parallel Sessions:
  • LWSN 1106 — MS11: Neural Surrogate Solvers for Fast PDE Simulation in Engineering Applications
  • LWSN 3102 — MS12: Advancements in Inverse Problems, Scattering and Spectral Theory — Part II
  • LWSN B151 — MS13: Nonlinear Waves and Coherent Structures
  • LWSN B134 — MS14: Modeling and Computational Methods for Applied Problems
  • LWSN B155 — MS15: Computational Mathematics for Time-Varying Systems and Control
  • LWSN 1142 — CT5: Contributed Talks V: Biomedical, Cellular, and Ecological Modeling
• 12:30 – 2:00 PM — Break (Lunch is not provided)
• 2:00 – 4:00 PM — Parallel Sessions:
  • LWSN 1106 — MS16: Discontinuous Galerkin and Structure-Preserving Methods for PDEs
  • LWSN 3102 — MS17: Advancements in Inverse Problems, Scattering and Spectral Theory — Part III
  • LWSN B151 — MS18: Analysis and Computation of Incompressible Fluid and Magnetohydrodynamic Systems
  • LWSN B134 — MS19: Advances in Numerical Optimization for Nonlinear Problems
  • LWSN 1142 — CT6: Contributed Talks VI: Mathematical Biology and Pattern Formation
  • LWSN B155 — CT7: Contributed Talks VII: Scientific Machine Learning and Data-Driven Methods
• 2:00 – 4:00 PM Parallel Sessions:

1. LWSN 1106 MS16: Discontinuous Galerkin and Structure-Preserving Methods for PDEs
2. LWSN 3102 MS17: Advancements in Inverse Problems, Scattering and Spectral Theory — Part III
3. LWSN B151 MS18: Analysis and Computation of Incompressible Fluid and Magnetohydrodynamic Systems
4. LWSN B134 MS19: Advances in Numerical Optimization for Nonlinear Problems
5. LWSN 1142 CT6: Contributed Talks VI: Mathematical Biology and Pattern Formation
6. LWSN B155 CT7: Contributed Talks VII: Scientific Machine Learning and Data-Driven Methods

• 4:00 – 4:15 PM Closing Remarks

Plenary Lectures

Plenary I. Guillaume Bal (University of Chicago)

Sat., Sept 19, 8:45 – 9:45 AM | LWSN 1142

Title: Classification and scattering theory aspects of topological insulators

Abstract: A surprising edge transport, displaying strong robustness to perturbations as an obstruction to Anderson localization, is guaranteed along interfaces separating two-dimensional insulators in different topological phases. We review the classification of several bulk and interface models of topological insulators and analyze their relation via a so-called bulk-edge correspondence. For suitable models, we present a spectral and scattering theory allowing for a quantitative description of the asymmetric edge transport along the waveguide in the presence of perturbations. We will also present recent results on the corresponding inverse scattering theory and observe that the non-trivial topology shows up again as a topological obstruction.

Plenary II. Jianfeng Lu (Duke University)

Sat., Sept 19, 1:45 – 2:45 PM | LWSN 1142

Title: Quantitative hypocoercive convergence estimates for underdamped Langevin equations

Abstract: The underdamped Langevin dynamics is perhaps one of the most familiar model used in sampling and non-equilibrium relaxation. Compared with its overdamped limit, the underdamped dynamics exhibits diffusive-to-ballistic acceleration of convergence. Nevertheless, quantifying such acceleration is challenging due to the degeneracy of diffusion. A large literature of hypocoercive estimates has been developed over the years to establish such quantitative rates.

In this talk, we will discuss some recent progress in sharp convergence rate estimates for underdamped Langevin dynamics, in chi-square divergence and in relative entropy, based on modified L^2 and modified entropy method respectively. Time permitting, we will also discuss application of these estimates to the design and analysis of sampling algorithms.

Plenary III. Hailiang Liu (Iowa State University)

Sun., Sept 20, 8:00 – 9:00 AM | LWSN 1142

Title: Deep Learning, PDEs, and Control: A Mean-Field Theory of Transformers

Abstract: Modern deep learning architectures can be studied through continuum limits, revealing connections with nonlinear PDEs, optimal transport, and control. In this talk, we develop a rigorous mean-field framework for transformer networks by taking simultaneous limits in the number of tokens and attention heads, followed by a continuous-depth limit. The resulting model couples a McKean–Vlasov transport equation for the token distribution with a Wasserstein gradient flow describing the evolution of the attention-parameter distribution during training.

We establish well-posedness, stability, and quantitative propagation-of-chaos estimates for the coupled dynamics, as well as the large-head limit of multi-head attention. We further connect the PDE formulation with optimization: entropy-regularized shallow attention models converge exponentially to global optima under a log-Sobolev condition, while genuinely deep transformers exhibit local linear convergence under a neural tangent kernel non-degeneracy condition.

The talk will highlight a broader PDE-for-AI perspective: continuum PDEs provide a new mathematical language for understanding learning dynamics and may also suggest principles for analyzing and designing future AI architectures.

Plenary IV. Selim Esedoglu (University of Michigan)

Sun., Sept 20, 9:15 – 10:15 AM | LWSN 1142

Title: Algorithms for triple junction drag and surface grooving in thin polycrystalline films

Abstract: Recent experimental results on the evolution of microstructure in polycrystalline materials during heat treatment have called into question the long established model in this subject, namely Mullins’ model based on multiphase mean curvature motion that has been influential in materials science as well as in mathematics. The quest to reconcile modeling and experiments, in thin films in particular, has identified several physical effects that may need to be accounted for. Among them are (1) triple junction drag and (2) surface grooving. I will describe new phase field and threshold dynamics methods that aim to approximate the evolution of microstructure in the presence of these effects. Surface grooving, which describes how the surface of the film may deform and interact with the microstructure, is especially challenging as it is modeled by surface diffusion — a high order geometric motion — coupled to motion by mean curvature at junctions.

Minisymposia

MS1. Computation and Analysis for HJE in Physics and Biology

Sat., Sept 19, 10:15 AM – 12:15 PM | LWSN 1106

Organizers:

• Yuan Gao (Purdue University) — gao662@purdue.edu

Description: This mini-symposium brings together analysts and computational mathematicians working on Hamilton–Jacobi and Fokker–Planck equations, which sit at the analytic core of many models in the physical and life sciences. A central theme is the interplay between the two: entropic Fokker–Planck equations arise as gradient flows in an optimal-transport geometry, while Hamilton–Jacobi equations enter as the dynamic-programming counterpart of the underlying action-minimization problems. The talks emphasize applications in physics and chemistry — where these equations appear as thermodynamic limits of chemical reaction systems, as models of aggregation and diffusion, and in the mean-field description of large interacting systems on continuous and graph state spaces — as well as related problems in biology such as reaction networks and collective behavior.

Talks:

1. Yuxi Han (Purdue University) — han891@purdue.edu
   Convergence of HJE with state constraint in chemical reactions
2. Emmanuel Gil Torres (Purdue University) — egiltorr@purdue.edu
   Dynamic Entropic Optimal Transport via Linearized ADMM
3. Monika Tomar (Purdue University)
   Linearly Solvable General-Sum Stochastic Differential Games
4. Maiia Makarova (Purdue University) — mmakarov@purdue.edu
   Diffusion Approximation and Transition Path for rare events

MS2. Advancements in Inverse Problems, Scattering and Spectral Theory — Part I

Sat., Sept 19, 10:15 AM – 12:15 PM | LWSN 3102

Organizers:

• Isaac Harris (Purdue University) — harri814@purdue.edu
• Plamen Stefanov (Purdue University) — Plamen-Stefanov@purdue.edu

Description: This minisymposium will showcase recent theoretical and computational developments across these closely related areas of mathematical physics and analysis. The program will bring together researchers working in the areas of inverse problems (and their applications), wave propagation, scattering phenomena, and spectral properties of differential operators. This will bring researchers in these areas together to facilitate useful discussions to hopefully develop fruitful collaborations.

Talks:

1. Isaac Harris (Purdue University) — harri814@purdue.edu
   On the Clamped Transmission Eigenvalue Problem
2. Pedro Morales (Purdue University) — pedromv9616@gmail.com
   Unconditional wave decay in dimension two
3. Jacob Shapiro (University of Dayton) — jshapiro1@udayton.edu
   Resolvent Estimates and wave decay for wavespeeds with Radial Regularity
4. Antonio Sa Barreto (Purdue University) — sabarre@purdue.edu
   The High Energy Distribution of Phase Shifts of Magnetic Schrodinger Operators on Nontrapping Asymptotically Hyperbolic Manifolds

MS3. Recent Advancements in Learning for Complex Systems

Sat., Sept 19, 10:15 AM – 12:15 PM | LWSN B151

Organizers:

• Rongjie Lai (Purdue University) — lairj@purdue.edu
• Shuhao Cao (University of Missouri–Kansas City) — scao@umkc.edu

Description: Modern machine learning provides efficient data-driven models for simulation, inverse problems, and prediction in complex physical systems. This minisymposium presents advances in structure-aware and physics-guided neural operators, nonlinear model reduction, and learning multiscale dynamical systems. The talks connect mathematical foundations with applications in materials, fluid dynamics, and PDE-based modeling, emphasizing physical consistency, generalization, and computational efficiency. Together, they demonstrate how numerical analysis, scientific machine learning, and domain knowledge can be integrated to build reliable and scalable models for scientific computing.

Talks:

1. Shuhao Cao (University of Missouri–Kansas City) — scao@umkc.edu
   Solver-in-the-Loop Joint Operator Learning
   Abstract: In this work, we propose a joint operator learning method for reconstructing images of conductivity coefficients from boundary data. Inspired by the idea of employing partial differential equation (PDE) solvers as preconditioners for this inverse problem, we investigate a "solver-in-the-loop" training mechanism. It allows the interaction between learnable parameters in a PDE solver module and those in neural networks for image reconstruction. Specifically, we employ a fractional Laplace-Beltrami operator with a learnable fractional order, which transforms boundary data into high-dimensional features. These features then serve as input to a neural network, significantly improving reconstruction accuracy. A Learning-Automated FEM module conveniently allows the auto-differentiation regarding an objective function to freely propagate through the PDE solver from the forward problem and the coupled neural networks for the inverse problem.
2. Naiyu Yin (Lehigh University) — nay224@lehigh.edu
   Physics-Guided Chemical Language Models for Polymer Property Prediction and Inverse Design
   Abstract: Chemical language models (CLMs) offer a data-driven route to polymer property prediction and design, but their promise is limited by the scarcity of experimental measurements: fine-tuning a pretrained CLM on a few dozen labeled polymers overfits and generalizes poorly. We present Poly4mer, a physics-guided CLM framework in which physical knowledge enters training at three distinct levels. First, the pretraining corpus is itself constructed from physics: a group-contribution model over roughly 200 functional groups generates structurally admissible hypothetical homopolymers, and Fire Dynamics Simulator runs supply the corresponding flammability labels, yielding roughly 120k structures with physically consistent structure–property relations. Second, property prediction acts as a regularizer within a joint encoder–decoder–predictor objective, so the latent polymer representation is shaped by physical response rather than by token reconstruction alone. Third, we impose known monotone relations between exposure conditions and fire response, namely incident heat flux versus peak heat release rate and time to ignition, as constraints on the predictor heads. After fine-tuning on limited experimental data, the resulting model reduces relative test error on all four flammability metrics compared to state-of-the-art CLMs, while retaining near-exact pSMILES reconstruction, which lets inverse design be posed as gradient-based optimization in latent space. Discovered candidates recover known flame-retardant chemistry, suggesting the model has learned genuine structure–property physics.
3. Suyi Gao (Purdue University) — gao757@purdue.edu
   Self-Supervised In-Context Operator Learning for Stochastic Mean-Field Control
   Abstract: Stochastic mean-field control coordinates large populations of interacting agents under noise, from swarm planning and systemic risk to Schrödinger bridges. Classical solvers discretize the Fokker–Planck equation on a mesh, so their cost grows exponentially with the state dimension and a tensor-product grid is already out of reach past two dimensions. Deep neural network solvers lift that ceiling but still handle one instance per run, re-optimizing whenever the initial law, the target, or a cost weight changes. This talk presents a mesh-free solution operator for a whole family of such problems, in which a task enters as a prompt and the optimal transport map comes out in one forward pass. We pass to the probability-flow ODE and parameterize its flow map by a prompt-conditioned coupling flow whose exact inverse and analytic log-determinant give the score at linear cost in the dimension, against the cubic cost of a generic map.
4. Kanad Sen (Purdue University) — rmaulik@purdue.edu
   Scale-Aware Learning of Chaotic Dynamics on Unstructured Meshes via Binned Spectral Losses
   Abstract: Surrogate modeling for high-dimensional nonlinear dynamical systems that exhibit chaos requires mechanisms that preserve not only pointwise accuracy but also the scale-dependent structure of physical fields. Bandwise spectral power losses, such as the binned spectral loss function, provide such supervision on structured grids, where Fourier modes define a standard frequency decomposition. On irregular meshes, however, no canonical Fourier basis exists, and spectral representations must be constructed from graph operators induced by mesh connectivity and geometry. In this study, we extend the binned spectral power loss for application to unstructured-mesh surrogate modeling of nonlinear dynamical systems. This is obtained by replacing Fourier bands with graph-Laplacian frequency bands, and we provide scalable Chebyshev and multilevel approximations for improving long-horizon rollout fidelity. In its full-spectrum form, our approach uses graph Laplacian eigenspaces to provide a graph analogue of Fourier band-power matching, but incurs the high cost of spectral decomposition. As a scalable approximation, we replace exact band projectors with sparse Chebyshev polynomial graph filters, avoiding explicit eigen-decomposition. When utilizing multilevel graph architectures, we introduce Graph Laplacian Energy Alignment for Meshes (GLEAM), which applies retained-subspace scale-aware supervision across graph hierarchies so that coarse and fine representations are regularized during autoregressive rollout. Our results show that the proposed spectral losses improve long-horizon rollout fidelity and preserve statistical invariants for the forecasting of turbulent flows on unstructured meshes, compared to deterministic baselines.

MS4. Multifidelity Scientific Machine Learning for Reduced-Order Modeling of Complex Systems

Sat., Sept 19, 10:15 AM – 12:15 PM | LWSN B134

Organizers:

• Di Qi (Purdue University) — qidi@purdue.edu
• Romit Maulik (Purdue University) — rmaulik@purdue.edu

Description: High-fidelity simulations of nonlinear PDEs and coupled multiphysics systems are often too expensive for repeated prediction, inference, optimization, and control. This minisymposium will highlight recent advances in scalable and multifidelity scientific machine learning for constructing efficient, accurate, and reliable reduced-order models of complex dynamical systems. Topics include projection-based and data-driven model reduction, nonlinear latent-space representations, operator learning and neural operators, differentiable and physics-constrained learning, learned closure models, and hybrid methods that combine classical reduced-order modeling with modern machine-learning components. The minisymposium will bring together theoretical, computational, and application-driven perspectives, including approximation and stability analysis, scalable algorithms, data assimilation, and robust generalization. Applications will include fluid dynamics, geophysical systems, transport processes, fusion and energy systems, and other nonlinear multiscale problems. The goal is to identify common mathematical and computational principles that connect classical model reduction, scientific machine learning, and high-performance simulation.

Talks:

1. Bahador Bahmani (Northwestern University) — bahador.bahmani@northwestern.edu
   A Multifidelity Stochastic Neural Operator: Applications to Mechanics of Materials
   Abstract: We present a multifidelity stochastic neural operator framework for learning complex mechanics models in the presence of both limited high-fidelity data and uncertainty. The approach combines information across multiple fidelity levels with low-dimensional representations of stochastic solution spaces, enabling efficient prediction of distributions of quantities of interest rather than only deterministic responses. We discuss connections to spectral stochastic operator learning and Bayesian multifidelity modeling, and demonstrate the framework on problems in mechanics of materials, including heterogeneous and architected material systems.
2. Alexander Hsu (Purdue University) — hsu297@purdue.edu
   Training-Free Universal Approximation by Prompting Random Transformers
   Abstract: How expressive is prompting a transformer? Answering this question is important for separating the roles of prompting, architecture, and pretraining in transformer models, and for determining whether task-specific behavior must be stored in model weights or can instead be induced at inference time through the prompt. We show, in an approximation-theoretic sense, that pretraining is not strictly necessary: a single-layer softmax attention network with random, untrained weights can approximate any Hölder function on a compact manifold when steered by an appropriate soft prompt. Guided by the connection between softmax attention and kernel methods, we construct explicit soft prompts—a prompt per target function, independent of the query—as solutions to linear systems matching attention logits to Gaussian kernel exponents, under which the frozen transformer emulates the classical Nadaraya-Watson kernel estimator. The construction requires only a mild rank condition on the weights, which we show holds almost surely under Gaussian initialization. The prompted network inherits the theoretical guarantees of kernel regression, leading to universal approximation theorems with minimax-optimal rates that depend on the intrinsic dimension. We further quantify the cost of prompting, exposing a tradeoff between the norm of the constructed soft prompt tokens, prompt length, and hidden dimension.
3. Yang Yang (Michigan Technological University) — yyang7@mtu.edu
   Numerical methods for the reinterpreted discrete fracture model
   Abstract: In this work, we propose discontinuous Galerkin methods and finite volume methods for the reinterpreted discrete fracture model (RDFM). This method characterizes fractures as low-dimensional Dirac-$\delta$ functions contained in the permeability tensor, making it suitable for conductive fractures on non-conforming grids without introducing additional degrees of freedom on fractures required by other models. Due to the application of the Dirac-$\delta$ functions and local mass conservation, we first discuss discontinuous Galerkin methods for the governing equations. To reduce the large degrees of freedom and significant computational cost, we further develop the finite volume methods on non-conforming meshes, assigning only one degree of freedom per cell and discretizing fluxes using the two-point flux approximation (TPFA). Numerical experiments show that direct applications of the properties of the Dirac-$\delta$ function to calculate the fracture flux may result in false numerical approximations, mainly because the single degree of freedom in each cell cannot effectively represent all the useful information of the fracture. As an alternative, we introduce a new virtual fracture treatment methodology, i.e. construct two projected fractures intersecting at the cell center in each cell. Then we derive the permeability of the virtual fractures, providing an equivalent representation of the fracture flux within each cell. The validity of the method and algorithm is demonstrated through several benchmark tests and a three-dimensional case, with comparative analysis showing significant advantages in computational efficiency over existing numerical methods within the RDFM framework.

MS5. Modeling and Data-Driven Approaches to Complex Biological Systems — Part I

Sat., Sept 19, 10:15 AM – 12:15 PM | LWSN B155

Organizers:

• Nour Khoudari (Purdue University) — nkhoudar@purdue.edu
• Tianna Burke (Purdue University) — burke230@purdue.edu
• Asini Konpola (Purdue University) — akonpola@purdue.edu
• Alexandria Volkening (Purdue University) — avolkening@purdue.edu

Description: Understanding complex biological systems requires approaches that can capture interactions across scales to reveal the mechanisms underlying emergent behavior. This mini-symposium will bring together researchers working at the intersection of mathematical biology, modeling, AI, machine learning, and topological data analysis with applications ranging from molecular systems and tissues to neuroscience and biophysics. By connecting theory, computation, modeling, and data analysis, the minisymposium aims to highlight new directions for uncovering biological mechanisms, integrating data, and making quantitative predictions.

Talks:

1. Mariya Savinov (University of Chicago) — savinov@uchicago.edu
   Constrained autocatalytic assembly drives a transition from coexistence to selection in protein network competition
   Abstract: The cytoskeleton—complex self-organized assemblies of protein filaments and auxiliary proteins—plays important roles in cell shape and mechanics through the dynamics of self-organized higher-order architectures such as networks. Multiple architectures coexist simultaneously in the cell, competing for common limited resources in a dynamic manner that is essential for the cell’s ability to quickly rearrange its cytoskeleton in response to external cues. How such networks coexist in the competitive cellular environment is still poorly understood, particularly in the relative importance of turnover, resource constraints, and degree of competition. Recently, our collaborators used a reconstituted system composed of protein filament-propelled beads in confinement (microwells) to demonstrate the importance of protein turnover for resource distribution between networks (Guerin et al. 2025, Curr. Biol.). Interestingly, they demonstrated that turnover permits such networks of different strengths to coexist, but if too many networks compete, a selection process occurs which favors assembly of strong networks over weak ones. In this work, they have further shown that the critical degree of competition where a transition from coexistence to selection occurs is highly sensitive to the turnover rate and amount of available protein, with slower turnover and smaller resource pools yielding selection at lower competition levels. To understand the mechanisms underlying this coexistence-selection transition in the competition, we developed a mathematical model for the network assembly dynamics as a system of coupled ODEs, incorporating parameters which correspond directly to the action of complementary proteins modulating turnover. Notably, we find that a Michaelis-Menten equation for the network assembly rate, indicative of autocatalytic growth which is constrained, is essential to capture the transition from coexistence to selection with increasing network competition. Our model reproduces key experimental results and explains the mechanism by which modulating turnover and the limiting monomer pool shifts the critical transition point. In conclusion, our findings reveal the precise mechanism through which turnover, resource constraints, and degree of competition determine the coexistence or selection of multiple competing protein filament networks, with broad implications of how cells can, in general, tune their various dynamic cytoskeletal architectures.
2. Kelsey Gasior (University of Notre Dame) — kgasior2@nd.edu
   The Molecular Dynamics Underlying Intracellular Phase Separation
   Abstract: An emerging mechanism for intracellular organization is liquid-liquid phase separation (LLPS). Found in both the nucleus and the cytoplasm, liquidlike droplets condense to create compartments that are thought to localize factors, such as RNAs and proteins, and promote biochemical interactions. Many RNA-binding proteins interact with different RNA species to create droplets necessary for cellular functions, such as polarity and nuclear division. Additionally, the proteins that promote phase separation are frequently coupled to multiple RNA binding domains and several RNAs can interact with a single protein, leading to a large number of potential multivalent interactions. This work focuses on a multiphase, Cahn-Hilliard diffuse interface model to examining the RNA-protein interactions driving LLPS. Using a ‘start simple, build up’ approach to model construction, these models explore how the molecular interactions underlying protein-RNA dynamics and RNA species competition control observable, droplet-scale phenomena. Numerical simulations reveal that RNA competition for free protein molecules contributes to intra-droplet patterning and the emergence of a heterogeneous droplet field. More in-depth analysis using combined sensitivity analysis techniques, such as Morris Method screening and Sobol’ method, highlights the complicated relationships underlying protein-RNA interactions and the results we can measure. Finally, droplet-level patterns are complicated when the initial conditions are considered. Under in vitro conditions, this model shows how experimental set up and initial conditions can produce complex droplet Turing patterns at phase separation. In-depth analysis using numerical simulations, shape analysis, and sensitivity analyses show that these systems are also susceptible to changes in the protein-RNA binding dynamics. The addition of a second RNA species competing for free protein introduces an element of asymmetry to the system and alters the emerging patterns at the onset of phase separation. But both systems show that certain initial conditions can produce sustained multi-droplet patterns as time goes to infinity. Ultimately, this targeted approach to intracellular LLPS begins to peel back the layers of complex molecular dynamics controlling observable LLPS phenomena that contribute to droplet regulation and, ultimately, cellular function.
3. Yue Sun (University of Wisconsin–Madison) — yue.sun@wisc.edu
   LBRMT: An Eulerian lattice Boltzmann framework for fluid–structure interaction
   Abstract: Computational approaches have become essential for complementing experimental and theoretical methods in the study of fluid–structure interaction (FSI). In this talk, I will present the lattice Boltzmann reference map technique (LBRMT), a fully Eulerian method for simulating FSI. The LBRMT builds on the reference map technique (RMT), which models large deformations of finite-strain solids on a fixed grid, and couples it with the lattice Boltzmann (LB) method for the fluid phase so that both solid and fluid are updated on the same computational grid. This Eulerian formulation, together with the parallelizable LB update and a single global velocity field for both phases, makes LBRMT well-suited to multi-body contact problems involving complex geometries. Through a range of LBRMT simulations, I will showcase its adaptability for various FSI scenarios, such as collective behavior in active and soft matter, with applications to biological systems. I will close by discussing our current work extending the LBRMT to be automatically differentiable, and its potential for inverse design in FSI-driven engineering and design.
4. Venkatesh Gopal (Elmhurst University) — vgopal@elmhurst.edu
   Octopus odor navigation - Extracting directional information from turbulent flow fields
   Abstract: Odor navigation is a primordial behavior exhibited by many animals in which they use odor plumes to locate food, find mates, navigate to their home, or avoid predators. Odor navigation is particularly challenging when the olfactory signal is transported by turbulent air or water currents because turbulence breaks up the odor plume into sparse patches that the animal encounters very infrequently. It is therefore an astonishing feat of animal sensation that many animals are capable of locating an odor source by tracking weak and intermittent olfactory signals over distances that are very large compared to their body length. In the first part of my talk, I will present our recent work on odor plume following by octopuses, which is the first direct laboratory observations of octopuses performing this behavior. One of our main findings was that the circular suckers on the arms of the octopus appear to be the primary olfactory organs, which led me to think about how the octopus might process olfactory information, and the role of sensor geometry in olfactory sensing. In the second part of my talk, which is more speculative, I will discuss interesting questions arising from our work on octopuses. In particular, I will focus on the broader question of how animals, or robots, can extract spatial and directional information from turbulent flow fields.

MS6. Mathematical Aspects of 2D Quantum Materials and Metamaterials

Sat., Sept 19, 3:00 – 5:00 PM | LWSN 1106

Organizers:

• Xuenan Li (Purdue University) — xuenan@purdue.edu

Description: This minisymposium will highlight recent mathematical advances in two-dimensional quantum materials and metamaterials, with particular emphasis on applications to complex materials arising in condensed matter physics, metamaterials, and photonics. Specifically, the session will feature theoretical and computational research exploring how wave-propagation phenomena, through the study of Schrödinger and Maxwell operators, provide insight into the properties of novel materials.

Talks:

1. Perry Kleinhenz (Illinois State University) — pbklein@IllinoisState.edu
   Observability of Schrodinger equations in Euclidean space
   Abstract: The Schrodinger equation is observable from a set if there is a uniform positive probability that a quantum particle will enter the set in finite time. On the flat torus, the Schrodinger equation is observable from any open set. In Euclidean space, these results have previously been extended to show periodic open sets can be used to observe the Schrodinger equation. In this talk we prove observability from non-periodic sets in Euclidean space. The proof uses a propagation of singularities approach to replace the observation set by its average along a direction.
2. Jia Shi (Indiana University Bloomington) — js289@iu.edu
   Constructing finite time singularities for some nonlinear PDEs
   Abstract: In this talk, I will introduce the implosion blow-up results for the compressible Euler, the compressible Navier-Stokes equations, and the defocusing nonlinear Schrödinger equation. We will discuss the existence of self-similar solutions and the stability near those solutions. During the talk, I will mention our work with Gonzalo Cao-Labora, Javier Gómez-Serrano, and Gigliola Staffilani on the first non-radial implosion result for those three equations. If there is time, I will also mention our work on self-similar solutions of a hydrodynamic equation that may formally arise as potential blow-up profiles of the focusing NLS equation.
3. Matthew Frazier (University of Chicago) — mjfrazier@uchicago.edu
   Topology and Bulk Edge Correspondence in Continuum Photonics
   Abstract: The bulk-edge correspondence (BEC), while ubiquitous in tight-binding and lattice models, has been shown to fail in some cases when applied to continuum models, which are characterized by a non-compact Brillouin zone that encompasses the entire 2d momentum space. In this talk, I will briefly review methods for defining topological invariants in the continuum setting and discuss recent advances in establishing the BEC for differential operators in continuum systems with continuous coefficients. These advances are illustrated by recent results from continuum photonic systems-macroscopic models of electromagnetic wave propagation in isotropic metals and semi-metals whose non-reciprocal material response results in non-trivial topology. When the material properties of such systems vary continuously, we find that the BEC holds robustly, while discontinuous transitions produce anomalous high-wavenumber effects which alter the BEC in a predictable way.
4. Xuenan Li (Purdue University) — xuenan@purdue.edu
   Pseudo-magnetism in a strained discrete honeycomb lattice
   Abstract: Slowly-varying nonuniform strain in honeycomb media can generate an effective pseudo-magnetic field, even though the underlying medium is non-magnetic. This phenomenon was first discovered in graphene and later in photonic crystals, and other physical settings. In this talk, I will start from a discrete nearest-neighbor tight-binding model for a nonuniformly strained honeycomb medium. I will then show how this model leads to a continuum magnetic Dirac Hamiltonian that describes wave packets concentrated near a Dirac point of the undeformed structure. I will focus on unidirectional deformations with bounded gradients that preserve translation invariance along the armchair direction. In this setting, we prove the existence of eigenmodes that behave like plane waves along the armchair direction and are strongly localized in the transverse direction, together with quantitative correction estimates. These results apply to deformations that produce an approximately constant pseudo-magnetic field perpendicular to the plane. They also lead to nearly flat bands and, as a consequence, a very high density of states. These analytical results are also supported by numerical simulations of the corresponding deformations. This is joint work with Michael I. Weinstein.

MS7. Generative Models, Modern Optimization, and Autonomous Agents in Scientific Computing

Sat., Sept 19, 3:00 – 5:00 PM | LWSN 3102

Organizers:

• Binghang Lu (Purdue University) — lu895@purdue.edu
• Changhong Mou (Utah State University) — changhong.mou@usu.edu

Description: Scientific computing is being reshaped by advances at the intersection of generative modeling, modern optimization, and autonomous agents. Diffusion-based generative models are opening new avenues for surrogate modeling, uncertainty quantification, and data-driven simulation of complex physical systems. Meanwhile, modern optimization methods are improving the training and reliability of the large-scale models that power these applications, and agentic frameworks are increasingly used to automate and accelerate scientific workflows, from experiment design to model discovery. This minisymposium brings together researchers working across these threads to share recent progress, highlight emerging challenges, and explore how generative modeling, optimization, and autonomous agents can be combined to advance scientific computing.

Talks:

1. Binghang Lu (Purdue University) — lu895@purdue.edu
   Muon-OGD: Muon-based Spectral Orthogonal Gradient Projection for LLM Continual Learning
2. Zheyuan Deng (Brown University) — zheyuan@brown.edu
   CONTRAMEM: Learning Self-Evolving Procedural Memory from Contrasting Multi-Model Trajectories
3. Runyu Zhang (Massachusetts Institute of Technology) — runyuzha@mit.edu
   Denoising as Projection: Understanding Gradient-Guided Diffusion as Constrained Optimization
4. Lang Cao (University of Illinois Urbana-Champaign) — langcao2@illinois.edu
   From RAG to Agentic Search: The Evolution of LLM-Based Information Collection and Organization
   Abstract: The transition from traditional Retrieval-Augmented Generation (RAG) to agentic search marks an important evolution in LLM-based information retrieval. While RAG primarily retrieves relevant documents and generates responses, agentic search enables autonomous task decomposition, search planning, multi-source information collection, source evaluation, evidence organization, and multi-step reasoning. This emerging trend reflects the development of LLMs from passive answer generators into active systems for intelligent information discovery, synthesis, and knowledge organization.

MS8. Multiscale Dynamics in Nonlinear PDEs and Complex Systems

Sat., Sept 19, 3:00 – 5:00 PM | LWSN B151

Organizers:

• Xiaofan Li (Illinois Institute of Technology) — lix@illinoistech.edu
• Ming Zhong (Illinois Institute of Technology) — mzhong3@central.uh.edu
• Baoli Hao (Illinois Institute of Technology) — bhao2@hawk.illinoistech.edu

Description: Multiscale problems arise naturally in nonlinear PDEs and complex systems, where dynamics at different spatial and temporal scales are often strongly coupled. This mini-symposium brings together recent developments in the analysis and computation of multiscale nonlinear systems, with emphasis on how different mathematical descriptions and methods can be connected across scales. Topics include kinetic and continuum models, reduced and statistical models for complex systems, and data-driven methods for identifying effective interactions and dynamics. The talks will address both analytical and computational questions, ranging from rigorous PDE analysis and asymptotic methods to model reduction, statistical modeling, and inverse problems. The goal is to highlight common mathematical ideas behind these problems and to encourage interactions between different approaches to multiscale dynamics.

Talks:

1. Baoli Hao (Illinois Institute of Technology) — bhao2@hawk.illinoistech.edu
   What Can Ionic Transport Data Reveal About Steric Interactions?
   Authors: Xiaofan Li (Illinois Institute of Technology), Chun Liu (Illinois Institute of Technology), Yiwei Wang (University of California, Riverside) Ming Zhong (University of Houston)
   Abstract: We develop a data-driven framework for coarse-graining multiscale transport systems from dynamical observations. Using ionic transport as a model problem, we learn a hierarchy of local, higher-order, and nonlocal constitutive descriptions within a common energy–dissipation structure. Our analysis connects interaction scales, information content, model stability, and predictive accuracy, providing a principled way to determine the simplest constitutive model justified by the observations. The framework also reveals how experimental resolution shapes what interaction physics can be learned and used for prediction.
2. Pei Liu (Penn State University) — pul21@psu.edu
   Energetic Variational Arbitrary Lagrangian-Eulerian Method for Gradient Flow: Discrete Gauge Symmetry Breaking
   Authors: Pei Liu (Penn State University)
3. Fanze Kong (University of Washington) — fzkong@uw.edu
   Turnpike stability in mean-field games with decreasing cost
   Authors: Fanze Kong(University of Washington)
   Abstract: Motivated by the theory of statistical physics, Huang et al. and Lasry et al. in 2007 independently proposed a class of strongly coupled PDEs named as Mean-field Games systems (MFGs) to describe the game among a huge number of rational players. Over the last decade, scholars extensively studied MFGs with the increasing cost by following techniques stated in the pioneering works of Lasry and Lions. Whereas, the properties of MFGs with the decreasing cost are not understood as well as the case of those with the increasing cost. In this talk, we focus on the turnpike stability of ergodic solutions to multi-population mean-field games with the decreasing cost. By constructing Lyapunov functional and imposing the matrix type spectral assumptions, we prove the local stability of ergodic steady states. Several examples including the stability of segregated steady states will be discussed.
4. Kunlun Qi (Michigan State University) — qikunlun@msu.edu
   On the Kinetic Description of Objective Molecular Dynamics (OMD): multiscale model reduction, numerics and data-driven applications
   Abstract: n the first part of this talk, a multiscale framework for objective molecular dynamics (OMD), a reduced molecular dynamics approach with inherent symmetries, will be presented as a typical example to do model reduction. This hierarchical framework bridges OMD with statistical kinetic equations and macroscopic hydrodynamic models. In the kinetic regime, we identify two distinct interaction scalings, leading to either a Mean-Field-type or Boltzmann-type equation. At the macroscopic level, we derive reduced Euler and Navier-Stokes systems through a detailed asymptotic analysis. The second part of the talk introduces a fast Fourier spectral method for numerically solving the derived kinetic equations, supported by convergence analysis and numerical tests to confirm its effectiveness. If time allows, I will also discuss our recent progress in applying data-driven and machine-learning methodologies to kinetic theory.

MS9. AI-Based Numerical PDE

Sat., Sept 19, 3:00 – 5:00 PM | LWSN B134

Organizers:

• Qingguo Hong (Missouri University of Science and Technology) — qingguohong@mst.edu
• Zhiliang Xu (University of Notre Dame) — zhiliangxu@nd.edu

Description: This mini-symposium explores emerging applications of artificial intelligence and machine learning in the numerical solution of partial differential equations (PDEs). Topics include physics-informed neural networks, neural operators, data-driven discretizations, training algorithms and so on. The symposium will emphasize both the opportunities and challenges of AI-based PDE solvers, including accuracy, generalization, efficiency, and the integration of physical and mathematical structure. It aims to bring together researchers from numerical analysis, scientific computing, and AI to exchange ideas and foster new collaborations.

Talks:

1. Mohammad Al-Saqqa (Central Michigan University) — alsaq1m@cmich.edu
   Physics-Informed Neural Networks (PINN) for the 2D Anisotropic Boussinesq System
   Abstract: Physics-informed neural networks (PINNs) solve PDEs by minimizing the equation residual rather than fitting data, but their accuracy typically comes with no guarantee. This talk develops rigorous error estimates for PINN approximations of the two-dimensional anisotropic Boussinesq system — buoyancy-driven flow in which dissipation acts only along selected directions. The central result is that the total solution error is controlled by the training residual, established in three steps: the residual can be made arbitrarily small within the class of tanh networks, an energy estimate bounds the total error by the residual, and numerical quadrature converts this into a fully a priori bound. Ensemble experiments confirm the theory — training and total error are tightly correlated and decay as the network and quadrature are refined. We close by contrasting this certified, mesh-free accuracy with a standard scheme that blows up at finite time.
2. Akram Moustafa (Central Michigan University)
   A neural-operator probe to long-time relaxation to a stratified shear in the two-dimensional anisotropic Boussinesq equations
3. Qingguo Hong (Missouri University of Science and Technology) — qingguohong@mst.edu
   Greedy algorithms for neural networks approximations for indefinite problems
4. Zhiliang Xu (University of Notre Dame) — zhiliangxu@nd.edu
   Energetic variational neural network discretizations of gradient flows

MS10. Modeling and Data-Driven Approaches to Complex Biological Systems — Part II

Sat., Sept 19, 3:00 – 5:00 PM | LWSN B155

Organizers:

• Nour Khoudari (Purdue University) — nkhoudar@purdue.edu
• Tianna Burke (Purdue University) — burke230@purdue.edu
• Asini Konpola (Purdue University) — akonpola@purdue.edu
• Alexandria Volkening (Purdue University) — avolkening@purdue.edu

Description: Understanding complex biological systems requires approaches that can capture interactions across scales to reveal the mechanisms underlying emergent behavior. This mini-symposium will bring together researchers working at the intersection of mathematical biology, modeling, AI, machine learning, and topological data analysis with applications ranging from molecular systems and tissues to neuroscience and biophysics. By connecting theory, computation, modeling, and data analysis, the minisymposium aims to highlight new directions for uncovering biological mechanisms, integrating data, and making quantitative predictions.

Talks:

1. Nour Khoudari (Purdue University) — nkhoudar@purdue.edu
   Characterizing the spatio-temporal dynamics of zebrafish skin pattern formation using topological data analysis
   Abstract: Zebrafish are known for their striped skin patterns which arise from the interactions between different types of pigment cells. We show interest in developing a mathematical framework, using topological data analysis, to study those patterns over time. We use different filtration types in persistent homology like the sweeping-plane filtration and cubical complexes of sublevel set persistence to classify those patterns based on their topological persistence and identify robust features such as the counts and widths of stripes, breaks, bridges, and time dynamics of pattern formation. By reinterpreting time as a filtration parameter in persistent homology, we propose new methods that can track the formation and degradation of stripes using density-like functions of pigment cells that are implicit functions of time. This allows us to estimate the time frames associated with normal developmental events and abnormalities in skin pattern development, which may indicate health, genetic, or environmental changes related to the fish, offering potential applications in biological research, genetics, and developmental biology.
2. Firas Khasawneh (Michigan State University) — khasawn3@msu.edu
   Classification of Epileptic iEEG using Topological Machine Learning
   Abstract: Epileptic seizure detection from EEG signals remains challenging due to the high dimensionality and nonlinear, potentially stochastic, dynamics of neural activity. In this work, we investigate whether features derived from topological data analysis (TDA) can improve the classification of brain states in preictal, ictal and interictal iEEG recordings from epilepsy patients using multichannel data. We analyze data from 55 patients, significantly larger than many previous studies that rely on patient-specific models. Persistence diagrams derived from iEEG signals are vectorized using several TDA representations, including Carlsson coordinates, persistence images, and template functions. To understand how topological representations interact with modern machine learning pipelines, we conduct a large-scale ablation study across multiple iEEG frequency bands, dimensionality reduction techniques, feature representations, and classifier architectures. Our experiments show that dimension-reduced topological representations achieve up to 80\% balanced accuracy for three-class classification. Interestingly, classical machine learning models perform comparably to deep learning models, achieving up to 79.17\% balanced accuracy, suggesting that carefully designed topological features can substantially reduce model complexity requirements. In contrast, pipelines preserving the full multichannel feature structure exhibit severe overfitting due to the high-dimensional feature space. These findings highlight the importance of structure-preserving dimensionality reduction when applying topology-based representations to multichannel neural data.
3. Dhananjay Bhaskar (University of Wisconsin–Madison) — dhananjay.bhaskar@wisc.edu
   Agentic AI Scientists for Autonomous Modeling and Discovery in Complex Biological Systems
   Abstract: A typical project in mathematical biology or biophysics begins with a biological phenomenon and proceeds through constructing a mathematical model, identifying parameters from the literature, selecting a simulation framework, writing and debugging code, generating hypotheses, and running parameter sweeps to test them. With the advent of agentic AI, substantial parts of this workflow can now be automated. In this talk, I will briefly review our work on agent-based modeling of epithelial–mesenchymal transition in cancer and embryonic pattern formation, and then describe an agentic AI framework for automating the construction and analysis of such models. The framework combines LangGraph-based orchestration, retrieval-augmented generation, MCP-based tool use, specialized code-writing and execution agents, and vector databases of scientific knowledge and reusable modeling components. I will demonstrate how it can translate a biological question into an executable model, iteratively debug and refine simulations, and conduct computational experiments to generate and test hypotheses.
4. Heber Lima da Rocha (Indiana University Bloomington) — hlimadar@iu.edu
   Agent-Based Modeling of Multicellular Systems with PhysiCell: Toward Uncertainty-Aware Predictions with UQ-PhysiCell
   Abstract: Agent-based models (ABMs) are a powerful framework for studying multicellular biological systems, particularly cancer, because they naturally capture how population-level dynamics emerge from the behaviors of individual cells and their interactions with a spatially heterogeneous environment. This talk presents PhysiCell, a well-established open-source framework for physics-based multicellular simulation, in which each cell is a discrete lattice-free agent coupled to reaction-diffusion equations describing oxygen, nutrients, drugs, and cytokines. Beyond this consolidated hybrid discrete-continuum core, recent developments have substantially expanded its expressive power and accessibility. A hypothesis grammar now allows cells to be modeled as signal processors: modelers declare, in a human-readable rule language, how microenvironmental and intercellular signals modulate cell behaviors such as cycling, death, motility, and secretion, disentangling signals from responses and enabling rapid construction and communication of mechanistic hypotheses without custom code. In addition, cloud-hosted deployments on nanoHUB and Galaxy make simulation and model exploration available through the browser, lowering the barrier to entry for experimental and computational researchers alike. These capabilities have been applied to tumor growth, immune-tumor interactions, and treatment response, producing models that reproduce experimentally observed behaviors. However, reproducing behaviors is not the same as predicting them: ABMs are stochastic, computationally expensive, and depend on high-dimensional, poorly constrained parameter spaces, which complicates calibration, uncertainty quantification, and comparison of competing mechanistic hypotheses. The second part of the talk addresses this challenge through UQ-PhysiCell, an extensible Python framework that orchestrates large ensembles of PhysiCell simulations with multiple levels of parallelism and integrates with established libraries for global sensitivity analysis, optimization, Bayesian inference, and surrogate modeling. By decoupling simulation execution from statistical analysis, UQ-PhysiCell enables reproducible, uncertainty-aware workflows that move ABMs beyond single best-fit simulations toward predictions with quantified confidence, illustrated with applications in computational oncology.

MS11. Neural Surrogate Solvers for Fast PDE Simulation in Engineering Applications

Sun., Sept 20, 10:30 AM – 12:30 PM | LWSN 1106

Organizers:

• Yueqi Wang (Purdue University) — wang7406@purdue.edu
• Shihao Wang (University of Wyoming) — Swang12@uwyo.edu

Description: Partial differential equations are fundamental to the modeling of complex physical and engineering systems, but their repeated high-fidelity simulation can be computationally expensive. This minisymposium focuses on neural surrogate solvers for accelerating PDE simulations while retaining important mathematical structures, physical information, and problem-specific solution features. The talks cover structure-preserving compositional neural operators, physics-guided few-step diffusion models, deep-learning surrogates for fractured-reservoir simulation, and connections between deep operator approximation and classical numerical methods. Together, the presentations highlight recent advances in efficient and reliable learning-based approaches for PDE simulation and engineering applications.

Talks:

1. Yueqi Wang (Purdue University) — wang7406@purdue.edu
   LegONet: Plug-and-Play Structure-Preserving Neural Operator Blocks for Compositional PDE Learning
   Abstract: Learned PDE solvers are often trained as monolithic surrogates for a specific equation, boundary condition and discretization. This makes them difficult to reuse when mechanisms change and it can limit stability under long-horizon rollout. We introduce Lego-like Operator Network (LegONet), a compositional framework that builds PDE solvers from plug-and-play, structure-preserving operator blocks defined on shared boundary-adapted spectral representations. LegONet separates boundary handling from mechanism learning, satisfying boundary conditions by construction. It also separates mechanism learning from time integration, enabling pretrained blocks to be assembled into new solvers without retraining. We also derive a finite-horizon error decomposition that separates block mismatch from splitting error and provides mechanism-level diagnostics for long-horizon predictions. Across ten time-dependent PDEs, LegONet delivers accurate closed-loop rollouts with improved stability under cross-PDE recombination and boundary reconfiguration. More broadly, this modular formulation suggests a path from task-specific neural solvers towards plug-and-play operator libraries for scientific computing.
2. Xiangrui Kong (Purdue University) — kong146@purdue.edu
   Ultra Fast PDE Solving via Physics Guided Few-step Diffusion
3. Shihao Wang (University of Wyoming) — Swang12@uwyo.edu
   Localization-Aware Deep Learning Surrogate for Depletion Delineation in Fractured Unconventional Reservoirs
4. Zecheng Zhang (University of Notre Dame) — zzhang48@nd.edu
   Deep Operator Approximation and Connection to the Numerical Methods

MS12. Advancements in Inverse Problems, Scattering and Spectral Theory — Part II

Sun., Sept 20, 10:30 AM – 12:30 PM | LWSN 3102

Organizers:

• Isaac Harris (Purdue University) — harri814@purdue.edu
• Plamen Stefanov (Purdue University) — Plamen-Stefanov@purdue.edu

Description: This minisymposium will showcase recent theoretical and computational developments across these closely related areas of mathematical physics and analysis. The program will bring together researchers working in the areas of inverse problems (and their applications), wave propagation, scattering phenomena, and spectral properties of differential operators. This will bring researchers in these areas together to facilitate useful discussions to hopefully develop fruitful collaborations.

Talks:

1. Yiran Wang (Emory University) — yiran.wang@emory.edu
   Inverse scattering of the relativistic Schrodinger operator at a fixed energy
2. Trung Truong (Marshall University) — truongt@marshall.edu
   Convexification for Mean Field Game Forecasting: Theory, Numerics, and Real Data Evidence
3. Thu Le (University of Wisconsin–Madison) — tle38@wisc.edu
   Direct imaging and polarization recovery for an elastic inverse source problem
4. General Ozochiawaeze (Purdue University) — oozochia@purdue.edu
   A Factorization Method for Clamped Obstacles from Near-Field Data

MS13. Nonlinear Waves and Coherent Structures

Sun., Sept 20, 10:30 AM – 12:30 PM | LWSN B151

Organizers:

• Xinyu Zhao (New Jersey Institute of Technology) — xz48@njit.edu
• Di Qi (Purdue University) — qidi@purdue.edu

Description: The study of nonlinear waves and coherent structures is of interest in fluid dynamics, plasma physics, optics, and materials science. These structures exhibit rich and varied behavior, ranging from solitary and traveling waves to dispersive shocks, wave turbulence, and complex multiscale interactions. Understanding their formation, evolution, stability, and interactions remains an active area of research in applied mathematics. The goal of this mini-symposium is to bring together researchers working on analytical, computational, and applied aspects of nonlinear wave dynamics and to provide a forum for discussing emerging ideas, state-of-the-art theoretical and computational tools, and applications.

Talks:

1. Vera Hur (University of Illinois Urbana-Champaign) — verahur@illinois.edu
   Stable undular bores: rigorous analysis and validated numerics
   Abstract: I will discuss the ‘global’ nonlinear asymptotic stability of the traveling front solutions to the Korteweg-de Vries–Burgers equation. Earlier works made strong use of the monotonicity of the profile, for relatively weak dispersion effects. We exploit the modulation of the translation parameter, establishing a new stability criterion that does not require monotonicity. Instead, a certain Schrodinger operator in one dimension must have exactly one negative eigenvalue, so that a rank-one perturbation of the operator can be made positive definite. We analytically verify that our stability criterion is met for an open set in the parameter regime including all monotone fronts. Our numerical experiments, revealing more stable fronts, suggest a computer-assisted proof. Joint with Blake Barker, Jared Bronski, and Zhao Yang.
2. Jia Shi (Indiana University Bloomington) — js289@iu.edu
   On the analyticity of the Muskat equation
   Abstract: The Muskat equation describes the evolution of the interface between two fluids in a porous medium. We consider the case of equal viscosities and different densities and present results on the analyticity of sufficiently smooth solutions. We show that such solutions are real analytic away from points where the interface turns over. Under additional assumptions, we also establish analyticity in a region whose width degenerates near the turnover points.
3. Chushan Wang (University of Chicago) — chushanwang@uchicago.edu
   Numerical methods for nonlinear Schrödinger equations with singular potential
   Abstract: Nonlinear Schrödinger equations with singular potentials, such as the Coulomb potential, arise widely in applications in quantum physics and chemistry. Such potentials, however, have very low regularity, and limit the regularity of the underline solutions even when the initial data are smooth, posing significant challenges in accurate numerical simulation. We propose an exponential integrator for the temporal discretization and rigorously establish its error estimate covering (almost) critically singular potentials. We further develop a novel Fourier-type spatial discretization that can efficiently handle the singularity of the potential.
4. Chris Vales (Dartmouth College) — Chris.Vales@dartmouth.edu
   Spatiotemporal pattern extraction by accelerated kernel methods
   Abstract: We consider the use of kernel methods for the extraction of spatiotemporal patterns from PDE simulation data. After a brief overview of the formulation of pattern extraction as the eigenvalue problem for a kernel integral operator, we consider the use of kernels that compare spatially local values of time snapshots of our simulation data, rather than full snapshots. The resulting kernel eigenfunctions are inherently spatiotemporal, with each one being able to encode complex spatiotemporal dynamics. We pay specific attention to the scalable implementation of the presented method via low rank approximation techniques and distributed GPU computing. Finally, we conclude with numerical results for the modified Hasegawa-Wakatani equations of plasma dynamics.

MS14. Modeling and Computational Methods for Applied Problems

Sun., Sept 20, 10:30 AM – 12:30 PM | LWSN B134

Organizers:

• Nurul Raihen (University of Toledo) — mdnurulislam.raihen@utoledo.edu
• Haridas K. Das — haridas@med.umich.edu

Description: Recent advances in modeling and computational methods have created new opportunities for understanding, predicting, and analyzing complex problems across the applied sciences. This minisymposium will bring together researchers developing mathematical, computational, and data-driven approaches for the study of real-world systems. The session will highlight recent developments in modeling, simulation, numerical and computational techniques, dynamical systems, data science, machine learning, and related analytical methods. Applications may arise from biological and biomedical systems, epidemiology, population and ecological dynamics, engineering, environmental systems, and other areas of science and technology. A central goal of the minisymposium is to demonstrate how modern computational and modeling approaches can be used to address challenging applied problems, particularly when complex dynamics, large datasets, uncertainty, or interacting processes are involved. The session will also provide a forum for discussing emerging methodologies, interdisciplinary applications, and future research directions, while encouraging collaboration among researchers working across complementary areas of applied mathematics, computation, and data science.

Talks:

1. Umar D. Islambekov (Bowling Green State University) — iumar@bgsu.edu
   A Novel Approach to Clustering Shape Data Using Iteratively Weighted Vectors with k-Means
2. Alessandro Selvitella (Purdue University Fort Wayne) — aselvite@pfw.edu
   On the Metastability of Learning Algorithms in Physics-Informed Neural Networks: Escape Routes and Regularity Barriers
3. Haridas K. Das — haridas@med.umich.edu
   Forecasting Challenge: Hybrid Data-Driven and Machine Learning Models for Short-Term and Long-Term Forecasting
4. Sultana Akter and Nurul Raihen (University of Toledo) — mdnurulislam.raihen@utoledo.edu
   Multi-Omics Integration of Mirna and Mrna Identifies Shared Neurodegenerative Genes and Their Pathways in Parkinson and Multiple Sclerosis Diseases

MS15. Computational Mathematics for Time-Varying Systems and Control

Sun., Sept 20, 10:30 AM – 12:30 PM | LWSN B155

Organizers:

• Firas Khasawneh (Michigan State University) — khasawn3@msu.edu
• Matthew Blake (Michigan State University) — blakem27@msu.edu

Description: Time-varying systems are ubiquitous in science and engineering. The analysis and control of these systems depend heavily on whether the underlying dynamics are deterministic or stochastic. In both cases, the geometry and topology of the input space and output data encode critical information that informs subsequent control actions. This minisymposium brings together researchers developing computational and mathematical approaches for the analysis, modeling, and control of time-varying systems. Topics include fundamental connections between the geometry and topology of 1D maps, parameterized and combinatorial persistent homology, and path-space frameworks for time-varying control. The minisymposium aims to foster interdisciplinary collaboration and identify emerging computational tools at the intersection of geometry, topology, and control.

Talks:

1. Goutam Das (Purdue University) — das347@purdue.edu
   Quantum Query Complexity of Model Predictive Path Integral Control
   Abstract: Model predictive path integral (MPPI) control simulates an ensemble of trajectories at each control step and forms a cost-weighted average of their control perturbations. The method is flexible for nonlinear and time-varying systems, but it can require many classical rollouts when accurate estimates are needed or when only a small fraction of the trajectories receive appreciable weight. In this talk, we ask whether quantum algorithms can reduce this sampling cost. We express each component of the finite-ensemble MPPI update as a ratio of bounded expectations and construct reversible rollout oracles that encode these expectations as quantum success probabilities, allowing them to be estimated by quantum amplitude estimation. The resulting query complexity improves quadratically over classical Monte Carlo in its dependence on both estimation accuracy and the average path weight, and below the exhaustive-evaluation threshold it matches known lower bounds for the associated scalar estimation problem. In the low-temperature limit, when the minimum-cost trajectory is unique, quantum minimum finding provides a quadratic reduction in oracle evaluations relative to exhaustive search. A guidance example confirms the predicted scalings, while an operation-count model shows why fewer oracle queries need not translate into an end-to-end computational advantage.
2. Andrew Haas (Purdue University) — haas60@purdue.edu
   The Conley-Morse persistence barcode: a homological signature of vector field evolution
   Abstract: A multivector field is a discrete object built from a simplicial complex that can be used to model aspects of continuous vector field dynamics. The Conley-Morse persistence barcode is a tool for tracking how important dynamical features change across a parameterized family of multivector fields. It combines ideas from Conley theory, Morse decompositions, and persistent homology to follow the births, deaths, and persistence of topological features associated with the evolving dynamics. Recent work provides efficient methods for computing this barcode, making it possible to construct homological signatures of evolving vector fields. This short talk will informally introduce the construction and illustrate how the barcode summarizes changing dynamical structure. This talk is based on work by myself, Tamal Dey, Michał Lipiński, and Manuel Soriano-Trigueros.
3. Matthew Blake (Michigan State University) — blakem27@msu.edu
   Unifying Total Persistence and Total Variation in 1D Continuous and Discrete Signals
   Abstract: 1-D signals are generated, continuously or discretely, by serial measurements of a single response variable over time or space, tracking physical phenomena like sound, electricity, pressure, or abstract values, like the cost of a stock or the exchange rate of a currency. Two metrics of such signals are total variation, which measures the total amount of vertical change in the signal, and total persistence, the sum of values of a process which isolates large and small “features” of the signal. Existing literature (Biswas 2023) connects these quantities in the case of continuous functions on compact domains with certain restrictions; we generalize this connection to discrete functions, (some) non-compact domains, and functions which do not have either of the restrictions in the existing literature, by deriving and generalizing a formula that computes these quantities directly. The extent of this generalization motivates a theorem which states the sufficient information needed from a signal to compute each of these quantities.
4. Elizabeth Munch (Michigan State University) — muncheli@msu.edu
   Canopies: A Generalization of Vines and Vineyards for Parameterized Persistent Homology
   Abstract: Static persistence, that is the persistent homology computed for a single filtration function $f:K \to \mathbf{R}$, has been highly utilized throughout topological data analysis. The first work looking at the persistent homology for a parameterized family of functions, $f_{p}:K \to \mathbf{R}$, came in the form of vineyards by Morozov et al, where the parameterization was over some interval, i.e. $p \in [a,b]$. More recently, there has been a growing interest in parameterized persistence where $p \in B$ for some more complex base space. For example, Turner et al introduced the Persistent Homology Transform (PHT), which studies the persistence of a family of functions on an embedded space $|K| \subset \mathbf{R}^n$ parameterized over $p \in \mathbf{S}^{n-1}$. The most general form of this came with the introduction of persistent homology bundles by Hickok, for arbitrary base space $B$. This parameterized input data holds interesting structure; of note is the discovery of monodromy in these bundles, a point in the persistence diagram might not come back to the same place while tracing out a closed loop in the base space. In this talk, we will introduce the concept of a *canopy*: a topological space with some additional structural data reminiscent of a bundle which stores the representatives of persistence classes over the parameterizing base space $B$. Canopies are particularly useful since they are well defined, even in the presence of monodromy. We show that for nice enough parameterized families of functions, we can construct such a canopy and give structure theorems for the types of issues that can give rise to monodromy.

MS16. Discontinuous Galerkin and Structure-Preserving Methods for PDEs

Sun., Sept 20, 2:00 – 4:00 PM | LWSN 1106

Organizers:

• Yang Yang (Michigan Technological University) — yyang7@mtu.edu
• Guosheng Fu (University of Notre Dame) — gfu@nd.edu

Description: This session brings together recent advances in discontinuous Galerkin and related numerical methods for partial differential equations, emphasizing accuracy, robustness, computational efficiency, and the preservation of essential mathematical and physical structures. Applications span flow in fractured porous media, compressible flow, relativistic hydrodynamics, and magnetohydrodynamics, highlighting challenges associated with lower-dimensional fractures, strong shocks, complex geometries, and physical constraints.

Talks:

1. Changhong Mou (Utah State University) — changhong.mou@usu.edu
   Neural-POD: A Plug-and-Play Neural Operator Framework for Infinite-Dimensional Functional Nonlinear Proper Orthogonal Decomposition
   Abstract: AI for science (AI4Science) models often suffer from discretization: learned representations remain tied to the training grid, limiting transfer across resolutions, solvers, and applications. We introduce Neural Proper Orthogonal Decomposition (Neural-POD), a plug-and-play neural operator that learns nonlinear, orthogonal basis functions directly in function space and can be integrated in both projection-based reduced order models and operator-learning frameworks such as DeepONet. Neural-POD replaces SVD-derived, resolution-dependent linear modes with continuous, resolution-invariant bases learned via sequential residual minimization, analogous to Gram-Schmidt orthogonalization. The framework supports training under task-specific norms, improves out-of-distribution generalization to unseen parameter regimes, and captures nonlinear structure in complex systems. Because the learned bases are interpretable and reusable, Neural-POD serves as a general representation module for AI4Science workflows. We demonstrate Neural-POD on Burgers' and Navier-Stokes equations.
2. Guosheng Fu (University of Notre Dame) — gfu@nd.edu
   Entropy-Stable and Physical-Constraint-Preserving DGSEM for Relativistic Hydrodynamics
   Abstract: High-order methods for relativistic hydrodynamics must preserve both nonlinear stability and the physical constraints of the solution, including positive density and pressure and subluminal velocity. This talk presents an entropy-stable discontinuous Galerkin spectral element framework combined with physical-constraint-preserving limiting. For special relativistic hydrodynamics, entropy-conservative two-point fluxes, entropy-stable interface dissipation, and a metric-independent admissible set lead to a robust high-order scheme with provable cell-average and nodal admissibility. Strong shocks and under-resolved flow features are treated using the oscillation-eliminating procedure which serves as high-order modal filtering. We also discuss extensions to general relativistic hydrodynamics on prescribed stationary spacetimes, where spatially varying metrics and geometric source terms require compatible flux–source discretizations. Numerical experiments demonstrate high-order accuracy, entropy stability, physical admissibility, and robustness for smooth and shock-dominated relativistic flows.
3. Siyuan Fan (University of Notre Dame) — sfan23@nd.edu
   A Structure-Preserving Operator-Splitting Scheme for Ideal MHD with DGSEM and Finite Elements
   Authors: Siyuan Fan (University of Notre Dame), Guosheng Fu (University of Notre Dame), Jielin Yang (University of Notre Dame)
   Abstract: We develop a structure-preserving high-order operator-splitting method for 2.5D ideal magnetohydrodynamics (MHD). The equations are decomposed into hydrodynamic and magnetic--velocity subproblems and coupled by second-order Strang splitting. The hydrodynamic subproblem is discretized by an entropy-stable discontinuous Galerkin spectral element method (DGSEM), supplemented with oscillation-eliminating (OE) stabilization and a positivity-preserving limiter. The magnetic--velocity subproblem employs compatible finite elements, with the current density and electric field reconstructed in H^1 and H(curl)-conforming spaces. This formulation advances the magnetic field through a compatible discrete curl, thereby preserving the global H(div) divergence-free property for compatible divergence-free initial data. Under periodic boundary conditions, the nonresistive magnetic semidiscretization conserves kinetic-plus-magnetic energy while leaving density and specific internal energy unchanged. Artificial resistivity is incorporated in curl form to regularize the magnetic field without compromising the divergence-preserving structure. The fully discrete scheme uses SSP Runge--Kutta time integration and preserves mass, positivity of density and pressure, and the discrete divergence-free constraint. Numerical experiments confirm theoretical spatial and temporal order of convergence, robust performance on standard MHD benchmarks, and magnetic-divergence errors near roundoff level.

MS17. Advancements in Inverse Problems, Scattering and Spectral Theory — Part III

Sun., Sept 20, 2:00 – 4:00 PM | LWSN 3102

Organizers:

• Isaac Harris (Purdue University) — harri814@purdue.edu
• Plamen Stefanov (Purdue University) — Plamen-Stefanov@purdue.edu

Description: This minisymposium will showcase recent theoretical and computational developments across these closely related areas of mathematical physics and analysis. The program will bring together researchers working in the areas of inverse problems (and their applications), wave propagation, scattering phenomena, and spectral properties of differential operators. This will bring researchers in these areas together to facilitate useful discussions to hopefully develop fruitful collaborations.

Talks:

1. Haoran Qi (Purdue University) — qi186@purdue.edu
   Microlocal Analysis of Wave Equations with Memory
2. Kiril Datchev (Purdue University) — kdatchev@purdue.edu
   Low frequency scattering of Dirichlet and Neumann obstacles
3. Joel Nathe (Purdue University) — jnathe@purdue.edu
   Recovery of a Null Form in the Wave Equation from Scattering Data
4. Plamen Stefanov (Purdue University) — Plamen-Stefanov@purdue.edu
   High-frequency wave propagation for the viscoelastic wave equation with singular memory

MS18. Analysis and Computation of Incompressible Fluid and Magnetohydrodynamic Systems

Sun., Sept 20, 2:00 – 4:00 PM | LWSN B151

Organizers:

• Xiaoming Zheng (Central Michigan University) — zheng1x@cmich.edu
• Jiahong Wu (University of Notre Dame) — jwu29@nd.edu
• M. Nader Alhomsi (Central Michigan University) — alhom1n@cmich.edu

Description: This mini-symposium brings together recent analytical and computational advances for incompressible fluid models and their buoyancy-driven and magnetohydrodynamic extensions. Two talks develop efficient time-stepping methods: a second-order generalized scalar auxiliary variable consistent-splitting scheme for the perturbed Boussinesq system, with unconditional stability and optimal error estimates; and a spectral-vanishing-viscosity stabilization that restores robustness of higher-order consistent splitting schemes for the Navier–Stokes equations at high Reynolds number. Two further talks address well-posedness and regularity: a hyperbolic Navier–Stokes model with acceleration convection, globally well-posed in 2D; and global regularity for 2D resistive MHD without velocity dissipation beyond small data. Together they highlight the interplay between structure-preserving discretization and rigorous analysis.

Talks:

1. Xiaoming Zheng (Central Michigan University) — zheng1x@cmich.edu
   A spectral-vanishing-viscosity stabilization of a higher-order consistent splitting scheme for the Navier--Stokes equations
   Authors: M Nader Alhomsi (Central Michigan University), Akram Moustafa (Central Michigan University), Mohammad Al-Saqqa (Central Michigan University), Jiahong Wu (University of Notre Dame)
   Abstract: Huang and Shen's high-order BDF-IMEX consistent-splitting schemes for incompressible Navier–Stokes are analyzed for arbitrary viscosity, revealing error bounds with inverse powers of viscosity and numerical breakdown at high Reynolds number. We stabilize the velocity update with a symmetric positive-semidefinite spectral vanishing viscosity operator built from directional Maday–Kaber–Tadmor kernels, proving viscosity-independent high-mode control. Three two-dimensional tests confirm robustness and design-order accuracy.
2. Mingyu Yu (University of Notre Dame) — myu4@nd.edu
   A hyperbolic Navier-Stokes model with acceleration convection
   Abstract: We propose a hyperbolic modification of the incompressible Navier-Stokes equation with an acceleration convection term $2u\cdot\nabla \partial_t u$. This model is motivated by expanding the second-order material derivative $D_t^2 u$. In contrast to the standard hyperbolic NSE model, the resulting system possesses a natural kinetic energy identity. We exploit this structure to establish global well-posedness for general initial data in 2D. We also investigate versions with anisotropic dissipation and prove global well-posedness for small initial data. Finally, we present numerical simulations comparing the introduced model with the classical hyperbolic NSE. The computations suggest possible finite-speed propagation and reveal substantial differences in their nonlinear dynamics. This is a joint work with Larios, Wu, and Yamazaki.
3. M. Nader Alhomsi (Central Michigan University) — alhom1n@cmich.edu
   A second-order consistent-splitting scheme, based on the generalized scalar auxiliary variable (GSAV) approach, for the two-dimensional perturbed Boussinesq system
   Abstract: We propose and analyze a second-order consistent-splitting scheme, based on the generalized scalar auxiliary variable (GSAV) approach, for the two-dimensional perturbed Boussinesq system. The system is obtained by subtracting a stable, linearly stratified hydrostatic equilibrium from the standard Boussinesq system. The time discretization extends the consistent-splitting generalized BDF2 framework of Huang and Shen [17] for the Navier-Stokes equations, treating the nonlinear convection and advection together with the linear buoyancy and stratification couplings explicitly, so that each time step reduces to a small number of decoupled linear systems. We prove an unconditional weak stability theorem for the GSAV scheme and derive optimal second-order error estimates for the velocity, pressure, and temperature. A careful tracing reveals that the error constant depends on the inverse viscosity and inverse thermal diffusivity through a quadruply-nested exponential, so the scheme is not robust as either tends to zero. Numerical experiments confirm the second-order convergence and reproduce the expected internal-wave dynamics and exponential relaxation toward hydrostatic balance in a long-time stratified-flow simulation.
4. Haoyue Teng (University of Notre Dame) — hteng@nd.edu
   Global Regularity for 2D Resistive MHD Beyond Small $H^4$ Data
   Authors: Chongsheng Cao (Florida International University), Shuang Liang (University of Notre Dame), Haoyue Teng (University of Notre Dame), Jiahong Wu (University of Notre Dame)
   Abstract: We consider the two-dimensional incompressible magnetohydrodynamic (MHD) equations on the torus with magnetic diffusion but no velocity dissipation. The global regularity problem for general smooth initial data remains open. Previous work established global well-posedness for initial data small in $H^4$, while our result allows an arbitrary prescribed $H^4$ bound. More precisely, global regularity holds provided that the initial vorticity is sufficiently small in $L^{\infty}$, the initial magnetic field is sufficiently small in $H^{\frac{3}/{2}}$ and has zero mean. The main difficulty is to control the vorticity in the absence of velocity dissipation. Our approach uses the magnetic stream function to obtain sufficiently strong control of the magnetic field. This allows us to control the vorticity and establish global regularity.

MS19. Advances in Numerical Optimization for Nonlinear Problems

Sun., Sept 20, 2:00 – 4:00 PM | LWSN B134

Organizers:

• Di Qi (Purdue University) — qidi@purdue.edu
• Xinyu Zhao (New Jersey Institute of Technology) — xz48@njit.edu

Description: Numerical optimization has broad applications in machine learning, data science, engineering design, and computational physics. The problems involved are often nonlinear, nonconvex, and high-dimensional, presenting severe computational challenges for algorithmic efficiency and reliability, as well as theoretical challenges for numerical analysis. This minisymposium brings together recent developments in numerical optimization and optimization-based computational methods for complex nonlinear problems and highlights the interplay among underlying mathematical structures, algorithmic design, and rigorous convergence and error analysis.

Talks:

1. Tianyu Kong (University of Chicago) — tianyuk@uchicago.edu
   Multiscale modeling of relaxation and electron dynamics for twisted bilayer graphene
   Abstract: Twisted bilayer graphene (TBG) has drawn significant interest due to recent experiments which show that TBG can exhibit strongly correlated behavior such as the superconducting and correlated insulator phases. We introduce a multiscale formulation to model the structural relaxation by coupling linear elasticity to a stacking energy that penalizes disregistry. We also prove the convergence of solutions of a time-dependent tight-binding model for the single-particle electronic dynamics of twisted bilayer graphene (TBG) to an equivalent multiscale continuum approximation. The tight-binding Hamiltonian depends explicitly on a small dimensionless parameter corresponding to the ratio of the atomic lattice constant to the moiré scale. We justify parameter scaling regimes near TBG’s magic angle at around 1 degrees where electronic tunneling and mechanical relaxation operate on comparable energy scales.
2. Xuda Ye (Purdue University) — ye481@purdue.edu
   Mean Square Error Analysis of Stochastic Runge–Kutta Integrators
   Abstract: We analyze the mean square error of stochastic Runge–Kutta integrators for overdamped Langevin dynamics whose potential is convex outside a bounded region. A decomposition splits the local error into a mean-zero term and a smaller remainder, and the discrete Poisson equation turns their moments into a bound on the error of a time average. We carry this out for two stochastic Runge–Kutta integrators proposed by Yang & Wang (2026), both of strong order 1.5 and weak order 2, which evaluate the gradient of the potential and no higher derivative. We show that their laws approach the law of the exact solution at second order in Wasserstein-1 distance, up to a logarithm, uniformly in the number of steps. For a test function with bounded derivatives up to third order, we prove that the mean square error over N steps with step size h is C[1/(Nh)+h^4], which is the optimal order in the discretization. Experiments measure the strong and weak orders and the sampling bias on a nonconvex potential in R^2, and compare the integrators on a diffusion model of CIFAR-10.
3. Jiaxing Li (Purdue University) — li4944@purdue.edu
   Global Convergence of an Efficient Splitting Method for the Defocusing Gross–Pitaevskii Ground State Problem
   Authors: Jiaxing Li (Purdue University), Xiangxiong Zhang (Purdue University), Shixin Zheng
   Abstract: For computing the ground state of the defocusing Gross–Pitaevskii energy, we propose and analyze two efficient schemes based on the Davis–Yin three-operator splitting, which treats the sphere constraint by normalization, the potential and interaction terms explicitly, and the kinetic energy by a resolvent. One iteration costs a single solve of I − γ∆ for the first scheme, and of I − γ∆ + γV1 for the second, with V1 denoting the separable part of the potential, and on structured meshes both operators can be inverted by fast GPU solvers. For monotone discrete Laplacians, including the second-order finite difference scheme and the lumped P1 finite element method on simplicial meshes with suitable angle conditions, we prove global convergence to the unique positive discrete ground state, for every positive normalized initial vector, for any constant step size below an explicit threshold. In contrast, the methods previously proven to converge globally to the ground state all invert a more difficult elliptic operator that depends on the current iterate. The proof combines a Lyapunov function, coupling the energy with a radial residual, with the positivity of the iterates preserved by the monotone discretization. In three-dimensional tests with up to 999^3 unknowns on one GPU, a simple variable step size rule makes the splitting schemes efficient in practice, comparable in wall-clock time to Riemannian conjugate gradient methods that also invert only a shifted Laplacian operator, and much more robust with respect to the choice of the initial guess.

Contributed Talks

CT1. Contributed Talks I: Numerical Methods and Structure-Preserving Schemes

Sat., Sept 19, 10:15 AM – 12:15 PM | LWSN 1142

Talks:

1. Rui Fang (The Ohio State University) — fang.1211@osu.edu
   Optimization-based structure-preserving methods for the Korteweg–de Vries (KdV) equation
   Authors: Rui Fang (The Ohio State University), Yulong Xing (The Ohio State University)
   Abstract: Structure-preserving numerical methods play a fundamental role in the long-time simulation of nonlinear dispersive waves by maintaining the physical invariants of the underlying system. Standard local discontinuous Galerkin (LDG) methods provide high-order accuracy and geometric flexibility, yet they generally do not preserve multiple invariants simultaneously. We develop optimization-based correction strategies that enforce discrete conservation properties for the generalized Korteweg--de Vries (KdV) equation. Specifically, we propose two optimization methods based on Davis--Yin splitting (DYS) that preserve mass and energy or mass and the Hamiltonian, respectively, together with an alternating direction method of multipliers (ADMM) formulation that simultaneously preserves the discrete mass, energy, and Hamiltonian. Numerical experiments include noidal-wave and solitary-wave benchmarks. Numerical tests show that the proposed corrections enforces the prescribed conservation laws and retain the convergence order of the underlying LDG discretization. Furthermore, the optimization-based methods significantly improve the long-time accuracy of the numerical solutions compared with the uncorrected LDG method. Numerical results indicate that applying the optimization only every few time steps is sufficient to maintain the prescribed conservation properties, resulting in negligible computational overhead. These results provide an efficient, accurate, and robust framework for constructing invariant-preserving discontinuous Galerkin methods for nonlinear dispersive equations with multiple conserved quantities.
2. Cesar Herrera (Purdue University) — herre125@purdue.edu
   Efficient Neural Network Methods for Numerical Hyperbolic PDEs
   Authors: César Herrera (Purdue University), Zhiqiang Cai (Purdue University)
   Abstract: Viewed as a class of approximating functions, neural networks provide continuous piecewise linear approximations capable of moving the mesh and generating irregular geometries. These features make neural networks appealing for approximating discontinuous functions with unknown interfaces. In the context of numerical PDEs, such functions arise in hyperbolic conservation laws, where traditional approaches like finite element methods still present numerical difficulties. However, to leverage the powerful approximating capabilities of neural networks, one must first establish a least squares formulation that preserves the physics of the original PDE and solve a computationally expensive high dimensional nonconvex optimization problem. I will present strategies for addressing these difficulties, focusing on the linear advection reaction equation.
3. BongSeok Kim (Purdue University) — kim4853@purdue.edu
   Hyperbolic Neural Moment Closure for Kinetic Equations: Boltzmann and Radiative Transfer
   Authors: Bongseok Kim (Purdue University), Jiahao Zhang (Purdue University), Johannes Krotz (University of Notre Dame), Dinshaw Balsara (University of Notre Dame), Ryan MacClarren (University of Notre Dame), Guang Lin (University of Notre Dame)
   Abstract: Moment methods provide computationally efficient approximations to kinetic transport equations by replacing the full distribution function with a finite set of moments. However, the resulting systems require closure models for unresolved higher-order moments. While machine learning (ML)-based closures can improve accuracy beyond classical analytic closures, unconstrained learned closures may violate hyperbolicity, producing non-real characteristic speeds and numerical instability. We propose a hyperbolic neural closure framework that guarantees real eigenvalues of the flux Jacobian associated with learned moment closures. Rather than directly predicting closure terms, we parameterize the Jacobian using two neural networks: a symmetric matrix network and a strictly convex entropy network whose Hessian defines a positive definite symmetrizer. These components yield a Jacobian that is similar to a symmetric matrix, thereby guaranteeing real eigenvalues. The closure is subsequently reconstructed by numerical integration of the learned Jacobian field along a prescribed path. We first demonstrate the framework for the M1 radiative transfer system. Numerical experiments show improved closure and solution accuracy compared with classical analytic closures, while maintaining stability in discontinuous Galerkin simulations. We further extend the framework to Hermite moment systems derived from the Boltzmann equation, demonstrating its applicability beyond radiative transfer. These results provide a general approach for constructing data-driven moment closures while preserving the hyperbolic structure required for stable numerical simulation.
4. Giselle Saylor (Oakland University) — gsaylor@oakland.edu
   An unconditionally stable embedded discontinuous Galerkin method for the phase field crystal equation
   Authors: Giselle Saylor (Oakland University), Tamas Horvath (Oakland University), Natasha S. Sharma (UT El Paso)
   Abstract: The phase field crystal (PFC) is a partial differential equation that models the growth of crystals in a liquid at the atomic scale in space, and diffusive scale in time. Since the PFC equation is sixth order and nonlinear, its numerical discretization often requires expensive algorithms. Moreover, proving existence and uniqueness of a solution involves the use of nonlinear functional analysis techniques. In this talk, we will use a class of finite element methods called embedded discontinuous Galerkin (EDG) for the PFC equation. We will show that the scheme is unconditionally energy stable and uniquely solvable.
5. Yuchuan Yang (University of Michigan) — yuchuan@umich.edu; Last speaker is the session chair
   Curvature motion of networks with triple junction drag
   Authors: Yuchuan Yang (University of Michigan), Selim Esedoglu (University of Michigan)
   Abstract: Curvature motion of networks is an important model for grain boundary evolution in polycrystalline materials. At triple junctions (free boundaries where three interfaces meet), one typically imposes the Herring angle condition. This force-balance condition requires the interfaces to meet at equilibrium angles throughout the evolution. However, recent experimental measurements have shown significant discrepancies with numerical simulations, raising questions about the validity of this assumption. One proposed alternative is triple junction drag, a boundary condition that allows triple junctions to relax toward force balance with finite speed rather than enforcing it instantaneously. In this talk, I will discuss recent work on this model, including its gradient flow structure, local well-posedness, and the topological changes that can occur during the evolution.

CT2. Contributed Talks II: Analysis of Nonlinear PDEs and Spectral Theory

Sat., Sept 19, 10:15 AM – 12:15 PM | DSAI 1069

Talks:

1. Trevor Leslie (Illinois Institute of Technology) — tleslie@illinoistech.edu
   The Euler Alignment System: Wellposedness and Limiting Configurations
   Authors: Trevor M. Leslie (Illinois Institute of Technology)
   Abstract: We consider the Euler Alignment system of collective behavior, which arises as a hydrodynamic limit of the Cucker--Smale system of ODEs. The primary interest in this system is in its compatibility with so-called "flocking" dynamics. We study the wellposedness of the system and long-time configurations, especially in the 1D and unidirectional settings.
2. Joshua Adeleke (Illinois Institute of Technology) — jadeleke@hawk.illinoistech.edu
   Unidirectional Entropic Solutions of the Pressureless Euler Alignment System
   Authors: Joshua O. Adeleke (Illinois Institute of Technology), Trevor M. Leslie (Illinois Institute of Technology)
   Abstract: We study the pressureless Euler alignment system in the setting of unidirectional velocity fields of the form u=(u,0,...,0). Our goal is to establish a well-posedness theory for this multidimensional class of solutions and to understand how nonlocal interactions between different spatial slices influence alignment and flocking behavior.
3. Joey Zou (Oakland University) — yzou@oakland.edu
   Gaussian-weighted normal operators on Euclidean space
   Authors: Joey Zou (Oakland University)
   Abstract: We consider the normal operator of the X-ray transform, weighted with Gaussian weights, in Euclidean space. We show the eigenfunctions of the normal operator are joint eigenfunctions of the harmonic oscillator and the spherical Laplacian, and we relate the spectrum to that of elliptic operators in the 1-cusp pseudodifferential calculus, considered recently by Jia, Vasy, Zachos et al. in considering integral geometry questions on asymptotically Euclidean spaces.
4. Yuliia Yershova (Michigan State University) — yershov1@msu.edu; Last speaker is the session chair
   Zero-range potential models for the thin structures with shrinking edges
   Authors: Yuliia Yershova (Michigan State University)
   Abstract: In this talk we will consider operator-theoretical models for the periodic thin structures (quantum waveguides) containing shrinking edges. We will discuss the internal structure of the limiting models and their connection with zero-range potentials, consider the dependence of the limiting properties on the frequency and study the structure of the spectrum.

CT3. Contributed Talks III: Optimization, Inverse Problems, and Control

Sat., Sept 19, 3:00 – 5:00 PM | LWSN 1142

Talks:

1. Huy Pham (Oakland University) — hnpham@oakland.edu
   Stable Recovery of Regularized Linear Inverse Problems
   Authors: Nghia Tran (Oakland University), Huy Pham (Oakland University), Nghia Vo (Oakland University)
   Abstract: Recovering a low-complexity signal from noisy observations by regularization methods is a fundamental problem in inverse problems and compressed sensing. A central question is whether the recovered solutions remain close to the original signal when the observed data are subject to perturbations. Stable recovery provides a quantitative answer to this question by ensuring that the original signal can be approximated linearly by optimal solutions of the corresponding Morozov or Tikhonov regularized optimization problems. In this work, we develop new characterizations of stable recovery in finite-dimensional spaces, with particular emphasis on the role of nonsmooth second-order information. These characterizations provide a deeper understanding of the variational structure underlying stability and offer new tools for analyzing when stable recovery can be guaranteed. Our approach connects stability properties of regularized inverse problems with second-order objects arising in nonsmooth and variational analysis, allowing us to formulate verifiable conditions for a broad class of structured regularization models. As an application of the proposed theory, we derive new sufficient conditions for stable recovery in analysis group sparsity problems. This framework includes important models such as group sparsity and isotropic total variation regularization, which are widely used to promote structured low-complexity solutions. We also present numerical experiments for these two classes of problems. The computational results provide favorable evidence for the practical use of our conditions in testing stable recovery and illustrate how the theoretical criteria can be implemented in concrete regularization models.
2. Nghia Vo (Oakland University) — nghiavo@oakland.edu
   Effective subspace Newton's method for Nonsmooth Optimization with Polyhedral Regularizers
   Authors: Nghia Tran (Oakland University), Nghia Vo (Oakland University), Khoa Vu (Wayne State University)
   Abstract: We propose several new nonsmooth Newton methods for solving convex composite optimization problems with polyhedral regularizers, while avoiding the computation of complicated second-order information associated with these functions. Under a tilt-stability condition at the optimal solution, the proposed methods achieve the quadratic convergence rates expected of Newton schemes. Numerical experiments on Lasso, generalized Lasso, OSCAR-regularized least-squares problems, and an image super-resolution task illustrate the applicability of the framework and demonstrate the acceleration obtained by the proposed Newton-type update over the corresponding first-order schemes, together with favorable performance relative to recently developed nonsmooth second-order methods on the tested instances.
3. Tuyen Tran (Loyola University Chicago) — ttran18@luc.edu
   New Globalized Newton-Type Methods for Nonconvex Optimization Problems
   Authors: Vo Thanh Phat (University of North Dakota), Tuyen Tran (Loyola University Chicago)
   Abstract: Newton's method is one of the most effective second-order algorithms for smooth optimization because of its fast local convergence. However, existing globally convergent Newton-type methods typically require convexity or strong convexity of the objective function, while approaches for nonconvex optimization often rely on Hessian regularization at every iteration. In this paper, we propose a general line-search Newton framework for unconstrained optimization that avoids repeated Hessian regularization by exploiting the Newton direction only when it is well-defined and suitable. The proposed framework encompasses several existing hybrid gradient-Newton methods as special cases and naturally yields a new extragradient Newton method. We establish global convergence under mild assumptions, including the Polyak-Lojasiewicz-Kurdyka (PLK) condition, allowing both isolated and nonisolated accumulation points. We further prove local superlinear and quadratic convergence under appropriate regularity assumptions. Finally, we apply the proposed framework to strongly quasiconvex optimization and provide, to the best of our knowledge, the first Newton-type algorithm together with a comprehensive convergence analysis for this important class of nonconvex optimization problems. Numerical experiments demonstrate the effectiveness of the proposed methods.
4. Kuangyu Ding (Purdue University) — ding433@purdue.edu
   On the Failure of KKT Convergence in Mirror Descent
   Authors: Kuangyu Ding (Purdue University), Kim-Chuan Toh (National Univeristy of Singapore)
   Abstract: For mirror descent generated by a Legendre kernel, perhaps one of the most basic questions in optimization is this: must every accumulation point of a bounded mirror descent sequence be Karush--Kuhn--Tucker (KKT) stationary under proper stepsizes? We show that the answer is no. A longstanding obstacle to resolving this question is the boundary blow-up of the Legendre gradient: it keeps every mirror step in the interior, while at a boundary limit, the inverse entropy metric vanishes on active coordinates and can erase the dual-feasibility in the KKT system. We construct $C^\infty$ objectives and bounded sequences generated by the Shannon-entropic mirror descent on the nonnegative orthant $\R_+^n$, for every $n\geq 3$, and on the probability simplex $\Delta_n$, for every $n\geq 4$, such that, in each case, the set of accumulation points is a smooth boundary closed curve of mirror-flow equilibria containing a nonempty relatively open arc of non-KKT points. The steps satisfy $\alpha_k\asymp k^{-\beta}$ with $\beta\in(1/2,1)$, the objective values are nonincreasing, and the objectives are entropy-relatively smooth. Hence the pathology stems from the degeneracy of the Bregman geometry at the boundary, rather than from failure of descent, or improper stepsizes. To the best of our knowledge, these provide the first counterexamples to KKT accumulation for bounded mirror descent sequences with nonincreasing objective values.
5. Yuezhu Xu (Purdue University) — xu1732@purdue.edu
   Learning Dynamical Systems with Guarantees
   Authors: Yuezhu Xu (Purdue University), S. Sivaranjani (Purdue University), Vijay Gupta (Purdue University)
   Abstract: Learning dynamical systems from data provides flexible models for complex nonlinear processes, but predictive accuracy alone does not ensure that the learned dynamics preserve properties important for analysis and decision-making. This talk considers how system-theoretic guarantees can be incorporated into learning-based dynamical models. We study neural ordinary differential equation models that directly represent continuous-time nonlinear dynamics and show how dissipativity can be enforced while retaining the predictive capability of the learned model. We also consider neural Koopman operator learning, where nonlinear dynamics are embedded into learned observable coordinates with approximately linear evolution. By carefully bounding errors arising from noisy data, Koopman operator truncation, and generalization to unseen data, dissipativity guarantees for the learned Koopman model can be transferred back to the underlying nonlinear system. The talk also briefly discusses scalable robustness certification through sensitivity and reachable-set analysis, providing a complementary perspective on uncertainty propagation through learned models.
6. Vipul Kumar Sharma (Purdue University) — sharm697@purdue.edu; Last speaker is the session chair
   Hard-Constrained Learning for Control with Optimality and Convergence Guarantees
   Authors: Vipul K. Sharma (Purdue University), Sivaranjani Seetharaman (Purdue University)
   Abstract: In autonomous control applications, guaranteeing real-world constraints (safety) is of critical importance. However, such guarantees are difficult to establish in a model-free, learning-based framework such as reinforcement learning (RL), particularly with infinite-horizon control objectives. Safe RL typically encourages rather than strictly enforces safety via penalties or regularization, whereas control approaches employ safety-filter projection methods that can guarantee safety during exploration but may lack theoretical guarantees. In this talk, we address these challenges by designing convergent and optimal policy gradient (PG) algorithms over a class of intelligent policies that guarantee safety by design. In our first work, we present a model-free safe RL framework for nonlinear dynamical systems by designing a class of truncated policies that are safe by construction. We then propose a convergent PG algorithm to directly search over this class of safe policies to find an optimal safe policy. In our second work, we consider RL with steady-state tracking constraints. We study PID policies that guarantee such tracking constraints by construction and propose both model-based and model-free PID PG algorithms by analytically deriving the first PID policy gradient expressions. We establish global convergence and optimality guarantees for our proposed RL-PID framework and show that it outperforms PPO and LQR in large-scale environments.

CT4. Contributed Talks IV: Stochastic Modeling, Uncertainty Quantification, and Stochastic Control

Sat., Sept 19, 3:00 – 5:00 PM | DSAI 1069

Talks:

1. Yisen Wang (Michigan State University) — wangyise@msu.edu
   Dynamics-Aware Sampling and Recovery for Tensor-Valued Dynamical Systems
   Authors: Longxiu Huang(Michigan State University) Yisen wang(Michigan State University) Seok-young Chung(Michigan State University)
   Abstract: We study dynamical sampling for third-order tensor-valued signals evolving under the t-product framework. The goal is to recover an unknown initial tensor from partial spatial observations collected over multiple time instances, thereby extending the classical time–space trade-off in dynamical sampling to structured tensor data. By exploiting the Fourier-domain representation of the t-product, we characterize how the spectral structure of the tensor evolution operator interacts with the spatial sampling pattern. We derive deterministic conditions for exact recovery and develop a reconstruction framework based on the resulting spectral decomposition. We also investigate randomized spatial sampling and obtain probabilistic recovery guarantees, including lower bounds on the sampling rate required for stable reconstruction with high probability. Our results show how temporal observations can compensate for missing spatial measurements while preserving the multilinear structure of the data. Numerical experiments on synthetic and real spatiotemporal data demonstrate the recovery behavior predicted by the theory and illustrate the effectiveness of the proposed tensor dynamical sampling framework.
2. Maxwell Bolt (Purdue University) — boltm@purdue.edu
   Modeling Diesel Output Particulate Matter as the Ornstein-Uhlenbeck Process
   Authors: Maxwell Bolt (Purdue University), Alex Alberts (Northwestern University), Akash S. Desai (Cummins Inc.), Peter Meckl (Purdue University), Ilias Bilionis (Purdue University)
   Abstract: Diesel engine particulate matter (PM) is one of the most challenging emission constituents to predict. As engines become cleaner and emissions levels drop, manufacturers need reliable methods to quantify the PM generated by production engines. Production engines typically do not have calibrated, time-resolved sensors for continuous engine-out PM mass estimation, so manufacturers rely on predictive models using available engine state measurements. In practice, this requires a computationally inexpensive model that provides PM estimates with calibrated uncertainty. Complex, multiscale physics make mechanistic models intractable and traditional data-driven methods struggle in transient drive cycles due to the stochastic nature of PM generation. Using high-frequency experimental PM measurements from transient engine tests, we introduce a novel PM model based on the Ornstein-Uhlenbeck (OU) process. The OU process is a mean-reverting stochastic process commonly used in financial modeling and is defined as the solution to a stochastic differential equation (SDE). We modify the OU process by parameterizing the terms of the SDE as functions of the engine state, which are then fit with a maximum likelihood estimate. In a synthetic example, we verify the ability of our model to learn a time-varying, parameterized OU process. We then train the model using real experimental data designed to dynamically cover the engine operating space and test the trained model on EPA-regulated drive cycles. For most drive cycles, we find the method accurately predicts cumulative PM mass output across time.
3. Shohreh Gholizadeh Siahmazgi (Wake Forest University) — gholizs@wfu.edu
   Rare Events in the Early Universe
   Authors: Shohreh Gholizadeh Siahmazgi (Wake Forest University), John Gemmer (Wake Forest University)
   Abstract: We study noise-induced transitions in stochastic cosmic inflation within the framework of Friedlin–Wentzell large deviation theory, with particular emphasis on the most probable transition path and the mean exit time. Our analysis reveals a slow-fast structure in the dynamics and shows that the most probable path admits an effective gradient description. This structure also allows us to reduce the mean-exit-time problem near the slow manifold and recover Kramers’ law for the escape time.
4. Yiming Chen (The Ohio State University) — chen.11042@osu.edu
   Stability and Error Estimates for Fully-Discrete Discontinuous Galerkin Methods for Stochastic Nonlinear Convection-Diffusion-Type Equations
   Authors: Yiming Chen (Ohio State University), Yunzhang Li (Fudan University), Yulong Xing (Ohio State University)
   Abstract: In this talk,  we consider nonlinear stochastic convection-diffusion-type equations driven by space-time mixed color noise.  We propose a high order local discontinuous Galerkin (LDG) method, coupled with Implicit-Explicit (IMEX) schemes for the equation of interest.  We establish the fully discrete high-moment stability and error estimates for the quasi-linear and fully nonlinear equations separately.  In particular,  the analytic estimates for the fully nonlinear equation are obtained on recursively defined subsets of the sample space whose probability converge to one.  Numerical experiments confirm the stability, convergence rates and the robustness of the proposed method for nonlinear stochastic models.
5. Haroun Meghaichi (The Ohio State University) — meghaichi.1@osu.edu
   Hyperbolicity-Preserving Stochastic Galerkin Methods for Conservation Laws Based on Associative Truncated Products on Polynomial Spaces
   Authors: Haroun Meghaichi (The Ohio State University), Yulong Xing (The Ohio State University)
   Abstract: Hyperbolicity-Preserving Stochastic Galerkin Methods for Conservation Laws Based on Associative Truncated Products on Polynomial Spaces Abstract: Stochastic Galerkin discretizations of nonlinear hyperbolic conservation laws can lose hyperbolicity because the standard pseudospectral product is generally nonassociative. We introduce a framework for hyperbolicity-preserving stochastic Galerkin systems based on associative truncated products on polynomial spaces. In one stochastic dimension, we characterize these products via a single polynomial datum, identify examples with favorable symmetry and spectral properties, and prove convergence to the classical product as polynomial degree grows. For rational-flux systems, we derive conditions ensuring hyperbolicity. Applications to the Euler equations demonstrate accurate, robust performance.
6. Temitope Iroko (University of Wisconsin-Milwaukee) — tciroko@uwm.edu; Last speaker is the session chair
   Optimal Threshold Policies for Asset Liquidation: An Impulse Control Approach
   Authors: Temitope Comfort Iroko (University of Wisconsin-Milwaukee)
   Abstract: This work studies an impulse control problem for optimal asset liquidation with a fixed transaction cost. The value of the asset is modeled as a geometric Brownian motion with known drift, and the objective is to maximize the expected discounted reward. We characterize a threshold policy for the liquidation problem and establish the existence and uniqueness of the optimal intervention thresholds. We also establish the optimality of the resulting threshold policy.

CT5. Contributed Talks V: Biomedical, Cellular, and Ecological Modeling

Sun., Sept 20, 10:30 AM – 12:30 PM | LWSN 1142

Talks:

1. Nitin Gopala (Grinnell College) — gopalani@grinnell.edu
   Mathematically Modeling the Impact of Phenotypic Plasticity On Pacific Treefrog Population Resilience Under Drought Stress
   Authors: Nitin Gopala (Grinnell College), Isaac Ho (Pepperdine University), Courtney Davis (Pepperdine University)
   Abstract: Amphibian populations around the world are declining due to drought and other stressors; however, Pacific treefrogs (Pseudacris regilla) remain common in Southern California. We aim to understand whether phenotypic plasticity in Pacific treefrogs enables their population to be resilient under drought stress. We develop a discrete stage-structured mathematical model of the Pacific treefrog life cycle. We model drought stress by reducing survivorship and decreasing breeding space, and we model plasticity by increasing the tadpole transition probability and varying body size (which alters fecundity). Numerical simulations suggest that treefrog populations are not at major risk of extinction in simulated drought conditions with or without plasticity. We predict that plasticity enables Pacific treefrog population persistence under conditions that cause extinction in the model without plasticity. Plasticity in developmental speed is most beneficial when stream drying threatens aquatic stages. However, plasticity has drawbacks; our model predicts that if plasticity delays maturation or reduces overall fecundity, population extinction may occur. By simulating the impact of features like phenotypic plasticity on population resilience, we learn when species are at greatest risk of decline and need protection.
2. Charuka Wickramasinghe (Northern Kentucky University) — wickramasc1@nku.edu
   Focused Ultrasound-Enhanced Drug Delivery to Brain Tumors: Quantitative and Translational Insights from CNS Physiologically Based Pharmacokinetic Modeling
   Authors: Charuka Wickramasinghe* (Northern Kentucky University), Andrew Wu* (Johns Hopkins University), Yuanyuan Jiang (Barbara Ann Karmanos Cancer Institute, Wayne State University), Xun Bao (Barbara Ann Karmanos Cancer Institute, Wayne State University), Jing Li (Barbara Ann Karmanos Cancer Institute, Wayne State University) * equal first authors
   Abstract: Purpose:
3. Qianyi Li (Iowa State University) — liqiany@iastate.edu
   A variational Neural Flow Matching for Aggregation-Diffusion Equations
   Authors: Qianyi Li (Iowa State University), Hailiang Liu (Iowa State University)
   Abstract: We propose a neural flow-matching method for nonlinear, nonlocal aggregation-diffusion equations arising in chemotaxis, polymer dynamics, and related applications. By leveraging the Onsager variational principle, the governing equation is reformulated as a constrained optimal control problem, in which a velocity field serves as the control and the density evolves under the associated continuity equation. We parameterize the velocity field by a neural network, and train it by adopting the objective function of the optimal control problem as the training loss, evaluated along particle trajectories generated by the network's own velocity field, with the associated score variable propagated through a coupled ODE and a multi-stage strategy extending the method to long time horizons. We provide a rigorous error analysis that decomposes the total error into approximation, optimization, and Monte Carlo sampling contributions. Numerical experiments confirm the effectiveness of the proposed method.
4. Luoding Zhu (IU Indianapolis) — Luozhu@iu.edu
   An implicit lattice Boltzmann immersed boundary method for FSI at low Re
   Authors: Jali Gill, Sanandan Ojha, Jared Barber
   Abstract: Cell migration plays a critical role in various biological processes, including cancer cell metastasis. This phenomenon is inherently complex and involves interactions between the deformable cell body and the surrounding fluid. Because cell migration occurs at extremely small flow speeds and spatial scales, the associated Reynolds numbers are very low. In that case, the standard explicit lattice Boltzmann–immersed boundary (LB-IB) method reaches steady state very slowly, making it inefficient and computationally expensive for modeling cell migration. To overcome this limitation, we propose an implicit lattice Boltzmann–immersed boundary method based on solving the stationary Stokes equations. The approach consists of three main steps: first, we linearize the original LB-IB mathematical formulation; second, we recast the linearized formulation into a large, sparse system of algebraic equations (Ax = b); and finally, we solve this system using iterative methods such as the Generalized Minimum Residual (GMRES) method with suitable preconditioners. We have applied this framework to simulate the relaxation of a perturbed cell toward its equilibrium state within a two-dimensional viscous incompressible fluid. Preliminary results indicate that the linearized LB-IB method maintains acceptable accuracy relative to the original explicit version while significantly reducing computation time. Our current work focuses on more efficient numerical solutions of the large sparse linear systems that arise in this framework.
5. Shakhnoza Takhirova (Bowling Green State University) — Takhirs@bgsu.edu; Last speaker is the session chair
   Left Censored Reduced-Rank Mixtures of Multivariate Regressions for County-Level Cancer Incidence and Pesticide Exposure
   Authors: —
   Abstract: County-level ecological studies of cancer incidence face correlated cancer sites, legally mandated suppression producing left-censored observations with known bounds when counts are small, and heteroskedastic counting noise dominating small-county variation. We develop a finite mixture of multivariate linear regressions to identify clusters of counties with similar cancer incidence patterns. Each latent county type has a reduced-rank coefficient matrix, so covariate effects share common directions. Suppressed rates are handled exactly in the likelihood as left-censored rather than imputed. An EM algorithm uses truncated multivariate normal moments in the E-step and closed-form rank-constrained coefficients in the M-step via residual-covariance-weighted SVD; rank is initialized from the fitted-signal spectrum and checked by BIC and cross-validated log-likelihood. In a county-level ecological study of 3,003 U.S. counties, eight cancer sites, 2018–2022, about 18% of site-by-county cells are censored; covariates include six agricultural pesticides (2002–2012), land use, environmental factors, and socioeconomic confounders. We use the method to test whether counties form latent clusters with shared incidence patterns.

CT6. Contributed Talks VI: Mathematical Biology and Pattern Formation

Sun., Sept 20, 2:00 – 4:00 PM | LWSN 1142

Talks:

1. Alexandria Volkening (Purdue University) — avolkening@purdue.edu
   Data-driven modeling of zebrafish patterns based on topological techniques
   Authors: Alexandria Volkening (Purdue University), Yue Liu (Duke University)
   Abstract: Many natural and social phenomena involve individual agents coming together to create group dynamics, whether the agents are drivers in a traffic jam, cells in a developing tissue, or locusts in a swarm. Here I will focus on the example of pattern formation in zebrafish, which are named for their dark and light stripes. Mutant zebrafish, on the other hand, feature different skin patterns, including spots and labyrinth curves. All of these patterns form as the fish grow due to the interactions of tens of thousands of pigment cells, making agent-based models a natural approach for describing cell behavior. However, stochastic, microscopic models are often not analytically tractable using traditional techniques, and parameter inference in biologically detailed, spatial agent-based models faces significant challenges. With this motivation, here I will describe how we are combining techniques from topological data analysis and approximate Bayesian inference to quantify structure in messy, cell-based patterns and infer parameter values from data.
2. Asini Konpola (Purdue University) — akonpola@purdue.edu
   Bayesian inference for an agent-based model of pattern formation in zebrafish skin
   Authors: Asini Konpola (Purdue University), Alexandria Volkening (Purdue University)
   Abstract: Complex systems in biology are those in which the interactions among individual agents give rise to diverse group dynamics. To better understand the factors driving these dynamics, we estimate parameters of the mathematical models that simulate such systems. But parameter estimation in agent-based models remains challenging due to many reasons, such as intractable likelihood functions, expensive model simulations, and quantifying the high-dimensional model outputs. Considering an existing agent-based model of pattern formation in zebrafish skin, we combine Bayesian methods for parameter estimation with pair-correlation functions of two different cell types to quantify patterns. Our results highlight which zebrafish mutants, cell types, and developmental time points provide the most informative insights into the mechanisms behind pattern formation.
3. Daecheol Kim (University of Illinois Urbana-Champaign) — dk43@illinois.edu
   Chemotaxis-Driven Pattern Formation in a Keller–Segel Model on Metapopulation Graphs and Graphons
   Authors: Daecheol Kim, Yefei Zhang, Yuxuan Zhao, and Daniel B. Cooney
   Abstract: We introduce and analyze a Keller–Segel-type model for chemotaxis on metapopulation networks. In our model, directed movement along network edges is modeled by a biased random walk towards higher chemoattractant concentrations, yielding a nonlinear graph dynamical system for population density and signal concentration. For dense graph sequences, we derive a nonlocal graphon equation and prove finite-time convergence of the discrete dynamics for deterministic weighted graphs and independent-edge dense graphs sampled on a fixed partition; we also obtain a relabeling-invariant limit in cut distance. By linearizing around the homogeneous equilibrium, we decouple the stability problem into a family of two-dimensional systems indexed by the graph Laplacian eigenvalues. This framework provides explicit chemotactic instability thresholds and identifies the dominant mode at the onset of pattern formation. For translation-invariant ring graphons, Fourier and weakly nonlinear analyses yield a cubic amplitude equation that distinguishes between supercritical and subcritical pattern-forming regimes. Finally, numerical simulations validate our theoretical predictions, illustrating how network structure drives the emerging spatial aggregation.
4. Bara Rababah and Amer Rawashdeh (Wayne State University) — baraa260@yahoo.com · hf1056@wayne.edu; Last speaker is the session chair
   On existence of chemotaxis compressible Navier-Stokes equations modeling vascular network formation
   Authors: Bara Rababah (Wayne State University), Tao Huang (Wayne State University), Amer Rawashdeh (Wayne State University)
   Abstract: We study a chemotaxis compressible Navier–Stokes system arising in the mathematical modeling of vascular network formation. The model couples the compressible Navier–Stokes equations, describing the density and velocity of endothelial cells, with a reaction–diffusion equation governing the concentration of a chemoattractant. The interaction between fluid motion and chemotactic effects introduces significant analytical challenges due to the nonlinear coupling between the density, velocity, and chemical concentration. In this talk, we investigate the existence of solutions to this coupled system in three spatial dimensions. We first discuss the mathematical structure of the model and derive the associated energy estimates, which provide the fundamental a priori bounds required for the analysis. Particular attention is given to controlling the nonlinear chemotactic interaction between the cell density and the chemoattractant concentration. Using these estimates together with compactness arguments and techniques from the theory of compressible Navier–Stokes equations, we establish the existence of global finite-energy weak solutions under suitable assumptions on the model parameters and the adiabatic exponent. The results provide a rigorous mathematical foundation for the chemotaxis compressible Navier–

CT7. Contributed Talks VII: Scientific Machine Learning and Data-Driven Methods

Sun., Sept 20, 2:00 – 4:00 PM | LWSN B155

Talks:

1. Daniele Schiavazzi (University of Notre Dame) — dschiavazzi@nd.edu
   Simulation-Based Probabilistic Reasoning: Nonparametric Belief Propagation with Arbitrary-Conditioning Probabilistic Flow
   Authors: Daniele Schiavazzi (University of Notre Dame), Sreejata Dey (University of Notre Dame)
   Abstract: Engineering decisions increasingly rely on chains of expensive computer simulations, and there is growing interest in delegating parts of this process to artificial AI agents. This requires an agent to reason under uncertainty across the whole chain: interpreting new evidence, tracking how uncertainty spreads, and inferring upstream causes from downstream observations. Current AI agents lack robustness in such a process. Belief propagation on factor graphs offers a natural remedy, since it breaks the problem into local pieces and keeps inference at the scale of a single simulation rather than the entire system. The difficulty is updating beliefs efficiently. A new downstream measurement must be traced back through a deterministic simulator, which conventionally calls for sampling methods (e.g.,. MCMC sampling) that are slow, and must be rerun from scratch each time new evidence is collected. I will discuss a new approach based on masked generative models. One model per simulation, trained offline on input-output pairs, captures every possible relation among its local variables; which one is used is decided at run time. Beliefs are then updated by reweighting stored samples, so new evidence is absorbed quickly, with no retraining. I will discuss some analytical examples, plus an application to the a three-simulator hypersonic vehicle.
2. Jichuan Tang (University of Notre Dame) — jtang4@nd.edu
   Reinforcement Learning for Active Solution Operators of Nonlinear Systems
   Authors: Jichuan Tang (University of Notre Dame), Patrick Brewick (University of Notre Dame), and Subhayan De (Northern Arizona University)
   Abstract: Using surrogate models to address real-world scientific problems in complex environments has attracted growing interest within the engineering community for a variety of nonlinear and dynamical applications. Surrogate models are typically created from numerous simulated realizations of a given computational or numerical model. Thus, one of the fundamental challenges for creating surrogates lies in sample efficiency, specifically, determining which model instances maximize surrogate performance. Active learning (AL) alleviates sample selection efforts by strategically selecting the most informative data samples when training a surrogate. This study reformulates AL from the perspective of sequential pool-based learning within a Markov decision process. It further introduces a versatile paradigm that leverages a reinforcement learning (RL) framework whose reward and state design transfers unchanged across partial differential equation (PDE) families to guide training sample selection for neural PDE surrogate modeling in high-dimensional parametric space. A Fourier neural operator (FNO) is adopted as the surrogate. We benchmark the proposed RL-based agent against baseline AL algorithms on time-dependent PDEs, namely Burgers' and compressible Navier--Stokes equations. Numerical experiments suggest that RL achieves the lowest error on three held-out metrics at the reported labeling budget compared with thirteen competing AL methods. The RL policy learns to balance stochastic and uncertainty-based methods by exploring high-amplitude extreme trajectories while maintaining diversity. It also avoids the collapse observed for deterministic uncertainty rules. The framework is therefore broadly applicable to scientific and engineering domains that demand accurate surrogate models of complex physical systems.
3. Sascha Ranftl (Purdue University) — sranftl@purdue.edu
   Deep Polynomial Chaos Expansion
   Authors: Johannes Exenberger (TU Vienna), Sascha Ranftl (Purdue University), Robert Peharz (TU Graz)
   Abstract: Polynomial chaos expansion (PCE) is a widely used surrogate modeling technique in uncertainty quantification and sensitivity analysis. By expressing the model response as a linear combination of basis polynomials [1] that are orthonormal with respect to the distribution of uncertain inputs, PCE enables tractable computation of key statistical quantities, including (conditional) means, covariances, and Sobol sensitivity indices [2]. These quantities are essential for understanding system behavior and identifying influential parameters. Despite substantial progress in higher-dimensional settings [3, 4, 5], classical PCE remains less scalable than modern neural networks. Conversely, standard neural networks typically do not allow exact and efficient computation of expectations and sensitivities, relying instead on costly, and often inaccurate, Monte Carlo approximations. We bridge this gap by combining PCE with ideas from tractable probabilistic circuits [6], yielding the deep polynomial chaos expansion (DeepPCE) [7]. Deep-PCE is a hierarchical generalization of PCE that scales gracefully to very high-dimensional inputs while preserving exact and efficient computation of Sobol sensitivity indices. Crucially, this is achieved without assuming sparsity, low effective dimensionality, latent embeddings, repeated re-orthogonalization, or sampling-based approximations. We demonstrate the method on the Sobol–G function as well as steady-state diffusion and Darcy flow with 100, 1024, and up to 4096 input dimensions. Thereat, DeepPCE achieves predictive performance comparable to that of multilayer perceptrons (MLPs), while retaining ordinary PCE’s ability to compute exact statistical inferences via simple forward passes at about three orders of magnitudes faster than the usual Monte Carlo integration of the MLP.
4. Yuchuan Zhang (Purdue University) — zhan5978@purdue.edu; Last speaker is the session chair
   Exact Enforcement of Dirichlet, Neumann, and Robin Boundary Conditions on General Curvilinear Polygonal Domains for Physics-Informed Machine Learning
   Authors: Yuchuan Zhang(Purdue University), Suchuan Dong(Purdue University)
   Abstract: We present a systematic method for exactly enforcing Dirichlet, Neumann, and Robin-type conditions on general polygonal domains with arbitrary curved boundaries. The method is built upon exact mappings between curved polygonal domains and corresponding standard domains, and combines TFC (theory of functional connections) constrained expressions with transfinite interpolations. When Neumann or Robin boundaries are present, especially when two such boundaries meet at a vertex, the induced compatibility constraints at the intersection must be incorporated to ensure exact enforcement of the imposed conditions on the adjoining boundaries. We present constructions for two situations: (i) a Neumann or Robin boundary intersecting only with Dirichlet boundaries, and (ii) two Neumann or Robin boundaries intersecting with each other. The method is implemented together with the extreme learning machine (ELM) technique for scientific machine learning, and the construction is further verified to be independent of the specific neural network architecture employed. Numerical experiments are presented for several linear and nonlinear, stationary and dynamic problems on two-dimensional domains with complex curved boundaries. The results demonstrate that the proposed method enforces the Dirichlet, Neumann, and Robin conditions exactly, with numerical boundary-condition errors at machine precision.

Poster Session

Sat., Sept 19, 12:15 – 1:45 PM (with Lunch) | Lobby (poster area)

Posters:

1. Laiba Gull (Purdue University) — lgull@purdue.edu
   Data-Driven Modeling of Cell Dynamics during Plant Growth
   Authors: Laiba Gull, Dinh Nhan Lai, Nour Khoudari, Yun Zhou, Alexandria Volkening
   Abstract: The development of plant tissue depends on cell behavior and signaling dynamics, as morphogens spread through connected cells. In the model fern Ceratopteris, a small group of cells undergoes division and grows in size to form a heart-shaped fern tissue with a meristem notch. Our work aims to determine what (unknown) signals affect cell behavior and drive notch formation through data-driven modeling. Working with time-lapse images of fern tissue, we construct cell–cell networks to study how signals may diffuse and decay across neighboring cells. In this poster, I will present our analysis of cell organization and preliminary models of morphogen dynamics.
2. Jiashu Han (University of Michigan) — shujhan@umich.edu
   A Semi-Lagrangian Particle Method for Plasma Simulations
   Authors: Jiashu Han(University of Michigan), Robert Krasny(University of Michigan), Alexander G.R. Thomas(University of Michigan)
   Abstract: We present a semi-Lagrangian particle method for two plasma models, one kinetic and one fluid. First, we solve the 1D1V two-species Vlasov–Poisson system, representing the electron and ion distribution functions on adaptively refined panels organized in a quadtree structure, with a 3×3 particle grid per panel. The electric field is computed by convolving the charge density with a regularized kernel, and the discrete N-body sums are evaluated with a GPU-accelerated treecode. The method is applied to ion acoustic waves, where we study long-time dynamics. Second, we extend the framework to 2D incompressible magnetohydrodynamics using a vorticity–current formulation, in which particles carry vorticity and current density, and the velocity and magnetic fields are recovered by Biot–Savart integrals with regularized kernels. The solver is verified on standard benchmarks including the Orszag–Tang vortex.
3. Afnan Hassan (Purdue University) — hassa129@purdue.edu
   A Multiscale Quantum/Classical Modeling of Metalloenzyme Redox Potentials: Application to Laccase Catalysis for Plastics Biodegradation
   Authors: Afnan Hassan (Purdue University), Lyudmila Slipchenko (Purdue University)
   Abstract: The accumulation of petroleum-based plastics is among the most pressing environmental challenges of our time, and their chemical inertness makes conventional recycling and degradation slow and incomplete. Enzymatic breakdown offers a sustainable alternative: laccases, copper-containing oxidases, can oxidize a broad range of substrates and are promising natural catalysts for converting recalcitrant polymers into recoverable products. However, laccase catalytic activity must be further optimized to make this plastics breakdown pathway industrially efficient and cost-effective. Computational modeling is an essential tool for rationally designing and optimizing new catalysts. The efficiency of a laccase is governed by its catalytic core, a cluster of four copper centers with distinct coordination environments. However, metalloenzymes containing transition-metal active sites are among the most challenging systems in computational chemistry. Their redox and catalytic properties depend on a delicate interplay between an electronically complex metal center which is often open-shell, near-degenerate, and strongly correlated, and a large, polarizable protein environment that reorganizes as the metal changes oxidation state. Predicting how the protein tunes the copper redox potential is therefore beyond experiment or intuition alone; it requires first-principles computation that couples a quantum-mechanical treatment of the metal site to a responsive, polarizable environment, capturing effects that can shift the potential by more than an electron-volt. Here, we develop and use the quantum-mechanical/effective-fragment-potential (QM/EFP) framework to model metalloenzyme redox potentials in the multicopper laccase from Trametes versicolor. The copper active site is treated by density functional theory within a polarizable classical environment, the two isoelectronic redox states are resolved by constrained DFT, and the redox free energy is obtained from a linear-response average of vertical energy gaps over molecular-dynamics ensembles, then referenced to the hydrogen electrode. Averaging over ten snapshots per state yields a stable free energy, a physically reasonable reorganization energy, and a reduction potential comparable to experiment. Future work will decompose this potential into per-residue contributions, guiding the rational engineering of laccase variants with enhanced activity toward synthetic polymers.
4. Jiyong Kwon (Purdue University) — kwon165@purdue.edu
   FAST-DeepONet: Factor-Augmented Branch Representations for High-Dimensional PDE Inputs in the Small-Sample Regime
   Authors: Jiyong Kwon (Purdue University), BongSeok Kim (Purdue University), Guang Lin (Purdue University)
   Abstract: Deep operator networks can become statistically unstable when partial differential equation inputs are observed at thousands of strongly correlated sensors but only a small number of operator samples is available. We introduce FAST-DeepONet, a branch representation combining a fixed spectral path with a regularized projection of the orthogonal residual, in which the directional penalty acts on the effective residual map after each of its rows is normalized. On Navier--Stokes flow a plain DeepONet degrades from $0.0394$ to $0.1556$ mean relative $L_2$ error as the branch grows from $129$ to $8193$ coordinates, while FAST-DeepONet stays near $0.04$, so the sensor grid can be refined without a statistical penalty. Across independent test sets for Navier--Stokes flow, Darcy flow, and signed terminal wavefield prediction it lowers mean relative $L_2$ error by $4.7\%$ to $37.0\%$ with three to seven times fewer trainable parameters. A spectral-only branch sharing the same basis separates the two paths: the fixed spectral path carries the improvement on Navier--Stokes and Darcy, while terminal wave prediction requires the residual path together with its directional penalty. FAST-DeepONet targets coordinate-query architectures and trains on solution values alone.
5. Jiaxing Li (Purdue University) — li4944@purdue.edu
   Global Convergence of an Efficient Splitting Method for the Defocusing Gross–Pitaevskii Ground State Problem
   Authors: Jiaxing Li (Purdue University), Xiangxiong Zhang (Purdue University), Shixin Zheng
   Abstract: For computing the ground state of the defocusing Gross–Pitaevskii energy, we propose and analyze two efficient schemes based on the Davis–Yin three-operator splitting, which treats the sphere constraint by normalization, the potential and interaction terms explicitly, and the kinetic energy by a resolvent. One iteration costs a single solve of I − γ∆ for the first scheme, and of I − γ∆ + γV1 for the second, with V1 denoting the separable part of the potential, and on structured meshes both operators can be inverted by fast GPU solvers. For monotone discrete Laplacians, including the second-order finite difference scheme and the lumped P1 finite element method on simplicial meshes with suitable angle conditions, we prove global convergence to the unique positive discrete ground state, for every positive normalized initial vector, for any constant step size below an explicit threshold. In contrast, the methods previously proven to converge globally to the ground state all invert a more difficult elliptic operator that depends on the current iterate. The proof combines a Lyapunov function, coupling the energy with a radial residual, with the positivity of the iterates preserved by the monotone discretization. In three-dimensional tests with up to 999^3 unknowns on one GPU, a simple variable step size rule makes the splitting schemes efficient in practice, comparable in wall-clock time to Riemannian conjugate gradient methods that also invert only a shifted Laplacian operator, and much more robust with respect to the choice of the initial guess.
6. Shin Lin (Colorado School of Mines) — shin_lin@mines.edu
   Neural Network Uncertainty Metrics on a Multilabel Cell Image Data
   Authors: Shin Lin (Colorado School of Mines), Cameron MacKenzie (Northwestern University), Luis Amaral, (Northwestern University)
   Abstract: Neural networks are notorious for making decisions that are overconfident. To address this problem, this research experiments with different techniques on quantifying neural network decision uncertainty on a weakly supervised, highly imbalanced, multi-label, multi-channel bioimage dataset. The data uses microscopy imaging on different types of human cells where specific subcellular structures and the protein of interest are labeled using target-specific fluorescent markers. The goal of the model is to identify which region(s) within the cell the protein of interest is located. The purpose of this project is to train a resnet18 (small convolution neural network) and evaluate its decision confidence using metrics including sigmoid, Mahalanobis distance, and the Shannon entropy methods, then identifying the most relevant metric for this type of dataset and model. Evaluations are based on per sample F1 score versus confidence values and the ability to identify out-of-distribution (OOD) and adversarial data. Metrics that are more able to represent per sample F1 scores with a positive correlation given any type of input data are better metrics to determine the confidence of a decision outcome.
7. Sontosh Kumar Sahani (Michigan Technological University) — sahani@mtu.edu
   High-order implicit bound-preserving discontinuous Galerkin methods with Local Lagrange Multipliers for wormhole propagation
   Authors: 1. Sontosh Kumar Sahani, PhD Candidate, Mathematical Sciences, Michigan Technological University 2. Yang Yang, Professor, Mathematical Sciences, Michigan Technological University
   Abstract: Wormhole propagation plays an important role in carbonate acidization, where injected acid reacts with the porous rock and forms preferential flow channels. Accurately simulating this process is challenging because the governing equations are strongly coupled and physically meaningful bounds on quantities such as porosity and acid concentration must be maintained. This poster presents a high-order discontinuous Galerkin framework equipped with a local bound-preserving strategy for simulating wormhole development. The proposed approach maintains the required physical constraints while retaining high-order accuracy. Numerical examples illustrate the method’s ability to resolve evolving dissolution patterns and complex wormhole structures and demonstrate the advantages of local bound preservation in challenging reactive-flow simulations.
8. Adriaan de Clercq (University of Chicago) — declercq@uchicago.edu
   Scattering Theory for the Dirac Equation with Weak Interface
   Authors: Adriaan de Clercq (University of Chicago), Guillaume Bal (University of Chicago)
   Abstract: The Dirac Equation with domain wall along an interface is a prototypical example of a topological insulator in Euclidean space. In this work, we develop a scattering theory for the Dirac operator in the regime where the domain wall is sufficiently weak to permit high-energy particles to propagate into the material bulk. In the presence of impurities, waves undergo scattering both along the interface and into the bulk, resulting in coupled one- and two-dimensional scattering behavior. We present results on developing an analytic scattering theory based on the limiting absorption principle, and characterize a class of "short-range" perturbations for which the limiting absorption principle remains valid. We study the relationship between the scattering matrix and the system's underlying topological invariant.
9. Cyril Cordor (University of Michigan, Ann Arbor) — ccordor@umich.edu
   Open-Quantum State Control with Control-Dependent Decoherence
   Authors: Cyril Morluyan Cordor (University of Michigan), Anthony M. Bloch (University of Michigan), Eitan Geva (University of Michigan)
   Abstract: The challenge in controlling open quantum systems is \textit{quantum decoherence} which arises from the entanglement of the system with the typically large number of environmental degrees of freedom. The true dynamics of an open quantum system are non-Markovian, but Markovian approximations such as the Lindblad and Redfield quantum master equations (QME) can be employed if the system is weakly coupled to the environment. Due to ease of use, the Lindblad master equation is a popular choice to model decoherence with the dissipation terms taken to be constant and/or independent of coherent, Hamiltonian controls. However, we argue that dissipation must be \textit{control-dependent}. In particular, for a thermal bath environment, the system Hamiltonian continuously "informs" the bath-induced, irreversible processes, because the system wants to relax to its thermal equilibrated state, \textit{i.e.} the canonical Gibbs state $e^{-\beta\Hhat_S}/\Tr[e^{-\beta\Hhat_S}]$, which is dependent on the system Hamiltonian and consequently the coherent controls. In this project, we investigate mathematically the differences in how the secular Redfield QME---an equivalent formulation to Lindblad---with control-dependent dissipation and the Lindblad QME with constant, control-independent dissipation model open system dynamics for the two-level system case in the presence of pure dephasing. We show that the secular Redfield dynamics are not just thermodynamically more consistent but reveal that Hamiltonian controls may wield a greater degree of influence on the dynamics than predicted by Lindblad with constant, control-independent dissipation.
10. Qiuyun Jin (Michigan Technological University) — qjin2@mtu.edu
    Stability and error estimates of local discontinuous Galerkin methods for incompressible miscible displacements with Darcy-Forchheimer model
    Authors: Qiuyun Jin (Michigan Technological University), Yang Yang (Michigan Technological University), Hui Guo (China University of Petroleum (East China)), Lulu Tian (China University of Petroleum (East China))
    Abstract: We develop and analyze local discontinuous Galerkin (LDG) methods for two-dimensional incompressible miscible displacements in porous media, where the flow is governed by the Darcy-Forchheimer model. The scheme couples a convection-diffusion equation for the concentration $c$ with a nonlinear elliptic equation for the pressure $p$ and velocity ${\bf v}$, giving rise to a strong nonlinear coupling that poses significant analytical challenges. We establish the stability of the semi-discrete scheme separately for the transport and flow variables, via a discrete energy argument and the inf-sup condition respectively. To handle the nonlinear Darcy-Forchheimer term, a cut-off operator is introduced for the velocity, and a modified coefficient is employed to control the coupling between velocity and concentration. Building on this framework, we derive optimal error estimates for $c$ and ${\bf v}$. The key analytical tools include specially designed projection operators whose compatibility with the alternating numerical fluxes ensures the cancellation of inter-element jump terms, together with a priori bounds on the numerical solution that enable the treatment of the nonlinear terms. Numerical experiments are presented to demonstrate the theoretical results.
11. Yeligay Segizbay (Purdue University Fort Wayne) — segiy01@purdue.edu
    Spring-Mass Models and Stride Timing in Human Locomotion
    Authors: Yeligay Segizbay (Purdue University Fort Wayne), Alessandro Selvitella (Purdue University Fort Wayne)
    Abstract: Human locomotion can be studied through low-dimensional dynamical models informed by biomechanical measurements. We investigate how the mass parameter in a spring-mass model influences a timing-based proxy for gait dynamics. Motivated by a three-dimensional motion-analysis dataset of healthy young adults walking and running overground and on a treadmill, we consider the vertical center-of-mass dynamics described by a second-order ordinary differential equation. For a range of mass values, we numerically simulate acceleration-time trajectories and estimate the time between successive vertical-acceleration peaks. We then compare the computed timing relation with the period-mass scaling predicted by the underlying oscillator model. The simulations show that increasing mass increases the peak-to-peak timing under fixed model parameters, consistent with the expected square-root dependence of oscillation period on mass. This study illustrates both the usefulness and the limitations of reductionist locomotion models: while a simple spring-mass system provides an interpretable baseline, realistic gait prediction will require parameter calibration and extensions incorporating damping, phase transitions, leg stiffness variation, and hybrid dynamics.