Novel Deep neural networks for solving Bayesian statistical inverse

TL;DR

This paper introduces a fractional deep neural network method to efficiently perform forward solves in Bayesian inverse problems governed by PDEs, aiming to reduce computational costs in MCMC algorithms.

Contribution

The paper presents a novel neural network approach specifically designed for PDE-based Bayesian inverse problems, improving computational efficiency over traditional methods.

Findings

The neural network approach reduces the number of PDE solves needed.

Numerical examples demonstrate improved efficiency and accuracy.

Error estimates support the method's reliability.

Abstract

We consider the simulation of Bayesian statistical inverse problems governed by large-scale linear and nonlinear partial differential equations (PDEs). Markov chain Monte Carlo (MCMC) algorithms are standard techniques to solve such problems. However, MCMC techniques are computationally challenging as they require several thousands of forward PDE solves. The goal of this paper is to introduce a fractional deep neural network based approach for the forward solves within an MCMC routine. Moreover, we discuss some approximation error estimates and illustrate the efficiency of our approach via several numerical examples.