PDE Analysis via Inference: Treating Dynamical Systems as Data-Generating Processes

I gave the following talk at TTU’s Learning from Data seminar.

Abstract:

Due to their variety and potential complexity, a general treatment of the theory of partial differential equations is an open problem in analysis and mathematical physics. However, despite the theoretical challenges posed by finding, and solving, the equations governing realistic data, both biological and artificial learning processes do this all the time. In order to describe or predict the dynamics of an observed process, these algorithms select and solve models in almost arbitrarily complex scenarios; moreover, they do so with little error, and without ever taking a single integral. Inspired by both the variational calculus of inference and the statistical physics of noisy systems, we propose a way of viewing problems in PDE analysis as problems in deep and machine learning. A simple case of this formalism with analytic results—using maximum entropy to solve the heat equation—will be proven. Concrete realisations of further results will be suggested after that.

Talk: link to recording

Slides: TTU Oct 2021.pdf

Other details: link to website

Figures 1 and 2: see this python notebook