Discovering Reduced-Order Dynamical Models From Data

This work explores theoretical and computational principles for data-driven discovery of reduced-order models of physical phenomena. We begin by describing the theoretical underpinnings of multi-parameter models through the lens of information geometry. We then explore the behavior of paradigmatic models in statistical physics, including the diffusion equation and the Ising model. In particular, we explore how coarse-graining a system affects the local and global geometry of a “model manifold” which is the set of all models that could be fit using data from the system. We emphasize connections of this idea to ideas in machine learning. Finally, we employ coarse-graining techniques to discover partial differential equations from data. We extend the approach to modeling ensembles of microscopic observables, and attempt to learn the macroscopic dynamics underlying such systems. We conclude each analysis with a computational exploration of how the geometry of learned model manifolds changes as the observed data undergoes different levels of coarse-graining.