Galerkin meets Laplace: Fast uncertainty estimation in neural PDEs

Christian Jimenez Beltran, Antonio Vergari, Aretha L Teckentrup, Konstantinos C. Zygalakis

Keywords: Deep Neural PDE, Deep Galerkin, PDE, PINNS, Laplace Approximation, Uncertainty, Bayesian

TL;DR: We propose DeepGALA as a fast and effective way to estimate uncertainty in deep neural PDE solvers that allows us to capture meaningful uncertainty in out of domain of the PDE solution and in low data regimes with little overhead.

Abstract:

The solution of partial differential equations (PDEs) by deep neural networks
trained to satisfy the differential operator has become increasingly popular. While
these approaches can lead to very accurate approximations, they tend to be over-
confident and fail to capture the uncertainty around the approximation. In this
work, we propose a Bayesian treatment to the deep Galerkin method (Sirignano &
Spiliopoulos, 2018), a popular neural approach for solving parametric PDEs. In
particular, we reinterpret the deep Galerkin method as the maximum a posteriori
estimator corresponding to a likelihood term over a fictitious dataset, leading thus
to a natural definition of a posterior. Then, we propose to model such posterior via
the Laplace approximation, a fast approximation that allows us to capture mean-
ingful uncertainty in out of domain interpolation of the PDE solution and in low
data regimes with little overhead, as shown in our preliminary experiments.

Submission Number: 66