Project ISF: Data-driven Discovery of the Nonlinear Schrödinger Equation as a Governing Equation for Extreme Weather Events Program DUIRI - Discovery Undergraduate Interdisciplinary Research Internship Term Summer 2023 Status Accepted Research Area Global Sustainability, Data Science, Applied Math, Machine Learning, Earth Science, Atmospheric Sciences, Climate Sciences Description Faculty collaborators: Guang Lin Nonlinear dispersive waves play a fundamental role in nature, and in the 1970s Benney and collaborators proposed the celebrated nonlinear Schrodinger (NLS) equation as the canonical evolution equation governing the propagation of the slowly varying wave packet envelope. For atmospheric and climate systems, while we are obtaining more and more data through satellite and in-situ observations, whether the Nonlinear Schrödinger Equation can serve as a governing equation for extreme weather events remains an open question. Atmospheric blocking (the stagnation of the weather systems) often causes weather extremes in the mid- to high-latitudes, yet its first-order dynamics are still not well understood. While synoptic weather systems typically last 3-5 days, atmospheric blocking can last 10-20 days. Despite rich literature, atmospheric blocking’s first-order dynamics remain enigmatic. Inspired by growing evidence of the propagating Rossby wave packets as a critical aspect in block life cycles, it becomes natural to connect such observations with the solution of a particular type of nonlinear equation - the Nonlinear Schrödinger (NLS) Equation. A new data-driven approach of discovery equations for nonlinear dynamical systems has attracted increasing attention in recent years - spearheaded by Rudy et al (2017) and Purdue’s Zhang and Lin(2018). We will assess a key and classical hypothesis that views atmospheric blocking as governed by the NLS Equation, which was first proposed by Benny (1979) and Yamagata (1980) by the applied math community. Our goal is to fill the gap of the theory-driven approach and data-driven approach. We utilize a data-driven approach by using the machine learning approach to decipher the key underlying equation for an idealized atmospheric model, which can be compared against the theoretical derivations. By assessing this hypothesis, we will assess whether the evolution of coherent and propagating Rossby wave packets is instrumental in atmospheric block lifecycles. We will further learn the significance of confirming this NLS equation’s relation to atmospheric blocking and how it allows a physics-informed machine learning to further improve the predictability of blocking, and the implications in a warming climate. This project is in collaboration with Professor Guang Lin in Purdue Math/ME. The project would be supported by Institute for a Sustainable Future. Supervisor Lei Wang Mentor Type of work required Write codes to discover Partial Differential Equations, such as NLS, from an idealized model's output

More specifically, our goal is to fill the gap of the theory-driven approach and data-driven approach. We utilize a data-driven approach by using the machine learning approach to decipher the key underlying equation for an idealized atmospheric model, which can be compared against the theoretical derivations. By assessing this hypothesis, we will assess whether the evolution of coherent and propagating Rossby wave packets is instrumental in atmospheric block lifecycles. We will further learn the significance of confirming this NLS equation’s relation to atmospheric blocking and how it allows a physics-informed machine learning to further improve the predictability of blocking, and the implications in a warming climate.
Websites ApowerfultoolasafoundationfortheabovepaperisSparseIdentificationofNonlinearDynamical systems (SINDy). A Github link is here:
https://github.com/dynamicslab/pysindy

Install the SINDy package following the instructions above. If you are using Linux or macOS you can install PySINDy with pip: pip install pysindy. After you install, let’s do a simple test case to see if it works. We will use Lorenz 63 model to generate a large number of data, to provide these data to pysindy and finally ask pysindy to make a guess of the structure and associated parameters of the original equations. Use the attached sample code to do this exercise. Change the parameters from default values to a new set of values (your choice!) and show the results and discuss. Qualifications calculus, python, machine learning, team work, critical thinking Credit Hours 3 Weekly Hours 40 (estimated)