A Hybrid Two-level MCMC Framework to Accelerate Posterior Mean Estimation with Deep Learning Surrogates for Bayesian Inverse Problems

Bayesian inverse problems arise in various scientific and engineering domains, and solving them can be computationally demanding. This is especially the case for problems governed by partial differential equations, where the repeated evaluation of the forward operator is extremely expensive. Recent advances in Deep Learning (DL)-based surrogate models have shown promising potential to accelerate the solution of such problems. However, despite their ability to learn from complex data, DL-based surrogate models generally cannot match the accuracy of high-fidelity numerical models, which limits their practical applicability. We propose a novel hybrid two-level Markov Chain Monte Carlo (MCMC) method that combines the strengths of DL-based surrogate models and high-fidelity numerical solvers to {compute the posterior mean of Quantities of Interest (QoI) in} Bayesian inverse problems governed by partial differential equations. The intuition is to leverage the evaluation speed of a DL-based surrogate model as the base chain, and correct its errors using a limited number of high-fidelity numerical model evaluations in a correction chain; hence its name hybrid two-level MCMC method. Through a detailed theoretical analysis, we show that our approach can achieve the same accuracy as a pure numerical MCMC method while requiring only a small fraction of the computational cost. The theoretical analysis is further supported by several numerical experiments, namely a Poisson, a non-linear reaction-diffusion, and a Navier-Stokes equation. The proposed hybrid framework can be generalized to other approaches such as the ensemble Kalman filter and sequential Monte Carlo methods.