Bayesian deep learning framework for uncertainty quantification in high dimensions

TL;DR

This paper introduces a Bayesian deep learning approach combining neural networks and Hamiltonian Monte Carlo to efficiently quantify uncertainties in high-dimensional stochastic PDEs, effectively addressing the curse of dimensionality.

Contribution

The paper presents a novel Bayesian neural network framework with HMC sampling for uncertainty quantification in high-dimensional problems, showing near dimension-independent computational cost.

Findings

Effective uncertainty quantification in high dimensions

Computational cost nearly independent of problem dimension

Successful application to forward and inverse problems

Abstract

We develop a novel deep learning method for uncertainty quantification in stochastic partial differential equations based on Bayesian neural network (BNN) and Hamiltonian Monte Carlo (HMC). A BNN efficiently learns the posterior distribution of the parameters in deep neural networks by performing Bayesian inference on the network parameters. The posterior distribution is efficiently sampled using HMC to quantify uncertainties in the system. Several numerical examples are shown for both forward and inverse problems in high dimension to demonstrate the effectiveness of the proposed method for uncertainty quantification. These also show promising results that the computational cost is almost independent of the dimension of the problem demonstrating the potential of the method for tackling the so-called curse of dimensionality.