Dimension-reduced KRnet maps for high-dimensional Bayesian inverse problems

TL;DR

This paper introduces DR-KRnet, a dimension-reduction method combining VAE and invertible maps to efficiently solve high-dimensional Bayesian inverse problems, demonstrating improved accuracy and computational efficiency.

Contribution

The paper presents a novel DR-KRnet approach that integrates VAE priors with invertible transport maps for high-dimensional Bayesian inference, reducing computational costs.

Findings

Demonstrates high accuracy in numerical experiments

Shows improved efficiency over traditional methods

Validates effectiveness in high-dimensional settings

Abstract

We present a dimension-reduced KRnet map approach (DR-KRnet) for high-dimensional Bayesian inverse problems, which is based on an explicit construction of a map that pushes forward the prior measure to the posterior measure in the latent space. Our approach consists of two main components: data-driven VAE prior and density approximation of the posterior of the latent variable. In reality, it may not be trivial to initialize a prior distribution that is consistent with available prior data; in other words, the complex prior information is often beyond simple hand-crafted priors. We employ variational autoencoder (VAE) to approximate the underlying distribution of the prior dataset, which is achieved through a latent variable and a decoder. Using the decoder provided by the VAE prior, we reformulate the problem in a low-dimensional latent space. In particular, we seek an invertible transport map given by KRnet to approximate the posterior distribution of the latent variable. Moreover, an efficient physics-constrained surrogate model without any labeled data is constructed to reduce the computational cost of solving both forward and adjoint problems involved in likelihood computation. With numerical experiments, we demonstrate the accuracy and efficiency of DR-KRnet for high-dimensional Bayesian inverse problems.