Title:
Physics-Inspired Machine Learning of Partial Differential Equations
Physics-Inspired Machine Learning of Partial Differential Equations
Author(s)
Golden, Matthew Ryan
Advisor(s)
Grigoriev, Roman O.
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Abstract
This dissertation discusses the Sparse Physics-Informed Discovery of Empirical Relations (SPIDER) algorithm, which is a technique for data-driven discovery of governing equations of physical
systems. SPIDER combines knowledge of symmetries, physical constraints like locality, the weak
formulation of differential equations, and sparse regression to construct mathematical models of
spatially extended physical systems. SPIDER is a valuable tool in synthesizing scientific knowledge as demonstrated by its applications. First, libraries of terms are constructed using available
physical fields. The symmetries of a system allow libraries to be projected into independently transforming spaces, known as irreducible representations. This breaks relations down into their
indivisible parts; each minimal physical relation is learned independently to reduce implicit bias. A library of nonlinear functions is constructed for each irreducible representation of interest. Second, each library term is evaluated in the weak formulation. SPIDER is aimed at experimental systems with inherently noisy data making accurate estimation of derivatives difficult. The weak formulation solves this problem: library terms are integrated over spacetime domains with flexible weight functions. Integration by parts can avoid 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. Third, a sparse regression algorithm can
find parsimonious relations ranging from dominant balances to multi-scale quantitatively accurate relations. Applications to direct numerical simulation of 3D fluid turbulence and experimental 2D active nematic turbulence are presented. SPIDER recovered complete mathematical models of both systems. The active nematic system is of particular interest; SPIDER identified a 2D description contradicting widely accepted theoretical descriptions used for over a decade. SPIDER facilitated the discovery of a new physical constraint on the fluid flow.
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Date Issued
2023-07-30
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Resource Subtype
Dissertation