Surrogate modeling for Bayesian inverse problems based on physics-informed neural networks

Abstract

The fundamental computing issues in Bayesian inverse problems (BIPs) stem from the need for repeated forward model evaluations, which are required by Markov chain Monte Carlo (MCMC) methods for posterior sampling. In this paper, we present an offline-online computational strategy for combining conventional MCMC with physics-informed neural networks (PINNs) for BIPs to achieve gains that cannot be reached with either component alone. In other words, we develop an offline learning technique that employs PINNs to construct a surrogate of the forward model, which we then utilize to build a new simulator for online sampling. During the online stage, the surrogate model is efficiently fine-tuned based on the current sample position, ensuring numerical accuracy in local regions. We specifically developed two types of online PINNs-based algorithms: transfer learning PINNs (TPINNs) and multi-fidelity PINNs (MPINNs). The proposed methods can obtain accurate posterior information with a small number of forward simulations, as demonstrated by numerical examples.