Toward Physical Generative Modeling

Explore advanced techniques for sampling physical measures in chaotic dynamical systems through this 42-minute conference talk from IPAM's Scientific Machine Learning Workshop. Discover three distinct approaches to effectively sampling observable invariant probability measures associated with chaotic systems, beginning with surrogate model development that preserves the physical measure of original systems. Learn about the statistical conditions under which neural network surrogate models can accurately reproduce statistical properties, including proven guarantees when incorporating Jacobian information into loss functions for systems satisfying linear response. Examine direct generative modeling approaches for physical measures, including flow-matching and score-based techniques that produce additional samples from target distributions. Understand the critical concept of "robustness of support" and how it ensures physical validity of generated samples despite inevitable learning errors in vector field or score function estimation. Investigate the mathematical conditions required for robust generative models, particularly the alignment of least stable finite-time Lyapunov vectors with unstable manifolds where physical measures are supported. Delve into computational methods for estimating scores of physical measures to enhance score-matching based generative model accuracy, and discover how precise score estimators enable Bayesian methodologies including parameter estimation and data assimilation for high-dimensional chaotic systems.