School of Physics Thesis Dissertation Defense

Presenter:        Matthew Golden

Title:                   Physics-Inspired Machine Learning of Partial Differential Equations

Date:                  Friday, July 14, 2023

Time:                  12:00 PM Eastern Time (US and Canada)

Location:          Howey, N201/202

Zoom Link:       https://gatech.zoom.us/j/93488424338

Committee:     Dr. Roman Grigoriev, Department of Physics, Georgia Institute of Technology (advisor)
                           Dr. Michael Schatz, Department of Physics, Georgia Institute of Technology

                           Dr. Kurt Wiesenfeld, Department of Physics, Georgia Institute of Technology

                           Dr. Elisabetta Matsumoto, Department of Physics, Georgia Institute of Technology

                           Dr. Alberto Fernandez-Nieves, Department of Physics, University of Barcelona

Summary:

The Sparse Physics-Informed Discovery of Empirical Relations (SPIDER) algorithm is a technique for data-driven discovery of partial differential equations (PDEs). SPIDER combines knowledge of symmetries, physical constraints like locality, the weak formulation of differential equations, and sparse regression to find new physical descriptions of data with spatiotemporal variation. SPIDER is a valuable tool in synthesizing scientific knowledge as demonstrated by its applications.

A novel feature of this algorithm is to not only learn physics in a symmetry-consistent way, but to learn only relations in irreducible representations. That is, relations are broken down into their indivisible parts, so that each PDE is learned truly independently. This prevents implicit bias and unknowingly using evidence from one relation for an independent one. A library of nonlinear functions is constructed for each irreducible representation of interest, and sparse linear combinations of these library terms are sought.

The weak formulation of differential equations is used: library terms are sampled not at individual points but integrated over spacetime domains with flexible weight functions. Integration by parts sidesteps numerical differentiation in many situations and increases robustness to noise by orders of magnitude. Clever weight functions can remove discontinuities and even entirely remove unobserved fields from analysis. Once these library terms have been sampled, sparse regression algorithms can find relations ranging from dominant balances to multi-scale quantitatively accurate PDEs.

Applications to numerical 3D fluid turbulence and 2D active nematic turbulence are presented. It is demonstrated that SPIDER can recover complete mathematical models of both systems and consistent redundancies across disjoint irreducible representations. In every representation analyzed, at least one relation is found. The active nematic system is of particular interest, as a new effective 2D description of the system is identified by SPIDER. While this system of equations holds only in regions of saturated microtubule density, it offers valuable physical insight.