Bayesian Hidden Physics Models: Uncertainty Quantification for Discovery of Nonlinear Partial Differential Operators from Data

What do data tell us about physics-and what don't they tell us? There has\nbeen a surge of interest in using machine learning models to discover governing\nphysical laws such as differential equations from data, but current methods\nlack uncertainty quantification to communicate their credibility. This work\naddresses this shortcoming from a Bayesian perspective. We introduce a novel\nmodel comprising "leaf" modules that learn to represent distinct experiments'\nspatiotemporal functional data as neural networks and a single "root" module\nthat expresses a nonparametric distribution over their governing nonlinear\ndifferential operator as a Gaussian process. Automatic differentiation is used\nto compute the required partial derivatives from the leaf functions as inputs\nto the root. Our approach quantifies the reliability of the learned physics in\nterms of a posterior distribution over operators and propagates this\nuncertainty to solutions of novel initial-boundary value problem instances.\nNumerical experiments demonstrate the method on several nonlinear PDEs.\n