MCMC-Net: Accelerating Markov Chain Monte Carlo with Neural Networks for Inverse Problems

TL;DR

MCMC-Net introduces a neural network-based surrogate to accelerate Markov Chain Monte Carlo methods, maintaining accuracy while significantly reducing computational time in inverse PDE problems.

Contribution

The paper presents a novel neural surrogate for likelihood functions in MCMC, enabling faster sampling without sacrificing accuracy, and provides theoretical guarantees of approximation and convergence.

Findings

Achieves similar accuracy to classical MCMC with faster computation

Effective on three PDE-based inverse problems: electrical impedance, diffuse optical, photoacoustic tomography

Provides theoretical proof of approximation and convergence guarantees

Abstract

In many computational problems, using the Markov Chain Monte Carlo (MCMC) can be prohibitively time-consuming. We propose MCMC-Net, a simple yet efficient way to accelerate MCMC via neural networks. The key idea of our approach is to substitute the true likelihood function of the MCMC method with a neural operator based surrogate. We extensively evaluate the accuracy and speedup of our method on three different PDE-based inverse problems where likelihood computations are computationally expensive, namely electrical impedance tomography, diffuse optical tomography, and quantitative photoacoustic tomography. MCMC-Net performs similar to the classical likelihood counterpart but with a significant speedup. We conjecture that the method can be applied to any problem with a sufficiently expensive likelihood function. We also analyze MCMC-Net in a theoretical setting for the different use cases. We prove a universal approximation theorem-type result to show that the proposed network can approximate the mapping resulting from forward model evaluations to a desired accuracy. Furthermore, we establish convergence of the surrogate posterior to the true posterior under Hellinger distance.