  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.
