Piecewise Deterministic Markov Processes for Bayesian Inference of PDE Coefficients

TL;DR

This paper introduces a flexible PDMP-based framework with surrogate-assisted thinning for efficient Bayesian inference in complex PDE inverse problems, significantly improving sampling accuracy and efficiency over traditional methods.

Contribution

It presents a novel surrogate-assisted thinning scheme for PDMP samplers, enhancing Bayesian inference in expensive PDE models with adaptive correction mechanisms.

Findings

GP-based PDMP samplers outperform traditional MCMC methods in accuracy and efficiency.

The Bouncy particle sampler shows the best overall performance and scalability.

The framework is versatile across different surrogate models and PDMP variants.

Abstract

We develop a general framework for piecewise deterministic Markov process (PDMP) samplers that enables efficient Bayesian inference in non-linear inverse problems with expensive likelihoods. The key ingredient is a surrogate-assisted thinning scheme in which a surrogate model provides a proposal event rate and a robust correction mechanism enforces an upper bound on the true rate by dynamically adjusting an additive offset whenever violations are detected. This construction is agnostic to the choice of surrogate and PDMP, and we demonstrate it for the Zig-Zag sampler and the Bouncy particle sampler with constant, Laplace, and Gaussian process (GP) surrogates, including gradient-informed and adaptively refined GP variants. As a representative application, we consider Bayesian inference of a spatially varying Young's modulus in a one-dimensional linear elasticity problem. Across dimensions, PDMP samplers equipped with GP-based surrogates achieve substantially higher accuracy and effective sample size per forward model evaluation than Random Walk Metropolis algorithm and the No-U-Turn sampler. The Bouncy particle sampler exhibits the most favorable overall efficiency and scaling, illustrating the potential of the proposed PDMP framework beyond this particular setting.