Bayesian Hidden Physics Models: Uncertainty Quantification for Discovery of Nonlinear Partial Differential Operators from Data
This work addresses the lack of uncertainty quantification in physics discovery from data, which is crucial for communicating credibility in scientific applications, though it is incremental by building on existing machine learning methods for differential equations.
The paper tackles the problem of discovering governing physical laws like differential equations from data by introducing a Bayesian model that quantifies uncertainty in learned nonlinear partial differential operators, demonstrating its effectiveness on several nonlinear PDEs with concrete uncertainty propagation to novel problem instances.
What do data tell us about physics-and what don't they tell us? There has been a surge of interest in using machine learning models to discover governing physical laws such as differential equations from data, but current methods lack uncertainty quantification to communicate their credibility. This work addresses this shortcoming from a Bayesian perspective. We introduce a novel model comprising "leaf" modules that learn to represent distinct experiments' spatiotemporal functional data as neural networks and a single "root" module that expresses a nonparametric distribution over their governing nonlinear differential operator as a Gaussian process. Automatic differentiation is used to compute the required partial derivatives from the leaf functions as inputs to the root. Our approach quantifies the reliability of the learned physics in terms of a posterior distribution over operators and propagates this uncertainty to solutions of novel initial-boundary value problem instances. Numerical experiments demonstrate the method on several nonlinear PDEs.