Piecewise Deterministic Markov Processes for Bayesian Inference of PDE Coefficients

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.

💡 Research Summary

This paper introduces a general framework for piecewise deterministic Markov process (PDMP) samplers that makes Bayesian inference feasible for nonlinear inverse problems with expensive likelihood evaluations, such as those arising from partial differential equation (PDE) models. The central idea is to use a surrogate model to construct a proposal event rate for Poisson thinning, together with a dynamic offset correction that guarantees the surrogate rate is an upper bound for the true event rate. The framework is agnostic to both the choice of surrogate and the specific PDMP, and the authors demonstrate it with two popular PDMP samplers – the Zig‑Zag sampler and the Bouncy Particle sampler – combined with three families of surrogates: a constant potential, a Laplace approximation, and Gaussian process (GP) regression models (including gradient‑informed and adaptively refined variants).

PDMP samplers evolve deterministically between random event times; Zig‑Zag flips individual velocity components, while the Bouncy Particle reflects the velocity against the gradient of the log‑posterior. Both require evaluation of the event rate λ(x,v), which in PDE‑based problems entails solving the forward model at each candidate event, making naïve simulation prohibitively costly. By replacing λ with a surrogate (\tilde λ) that is cheap to evaluate, candidate event times can be generated from a Poisson process with rate (\bar λ = \tilde λ + c). Whenever a candidate is found to violate the upper‑bound condition (i.e., the true λ exceeds (\bar λ)), the offset c is increased, ensuring that future candidates respect the bound. This “dynamic offset” mechanism makes the thinning procedure robust to surrogate misspecification and eliminates the need for a priori tight bounds.

The authors test the methodology on a one‑dimensional linear elasticity problem where the spatially varying Young’s modulus is inferred from noisy displacement observations. Parameter dimensions of 10, 20, and 40 are considered. For each dimension, the following algorithms are compared: (i) Zig‑Zag with constant, Laplace, and GP surrogates, (ii) Bouncy Particle with the same surrogates, (iii) a well‑tuned Random Walk Metropolis (RWM) baseline, and (iv) the No‑U‑Turn Sampler (NUTS). Performance metrics include mean‑squared error of the posterior mean, effective sample size (ESS) per forward model evaluation, and total runtime.

Results show that GP‑based surrogates dramatically improve efficiency. Both PDMP samplers equipped with GP surrogates achieve substantially lower posterior error than RWM and NUTS, and their ESS per forward model evaluation is 5–10 times higher. The Bouncy Particle sampler consistently outperforms Zig‑Zag in all dimensions, owing to its continuous velocity space which yields fewer events and smoother trajectories. The Laplace surrogate provides moderate gains, while the constant surrogate offers the least improvement. The dynamic offset correction successfully prevents upper‑bound violations in all experiments, requiring no manual tuning of the surrogate.

Key contributions of the paper are: (1) a surrogate‑assisted thinning framework that generalizes to any PDMP and any surrogate that yields a computable proposal rate; (2) a simple yet effective dynamic offset mechanism that guarantees the validity of the thinning procedure; (3) extensive empirical validation showing that PDMP samplers with GP surrogates can outperform state‑of‑the‑art reversible MCMC methods on realistic PDE‑governed inverse problems; and (4) evidence that irreversible PDMP dynamics, particularly the Bouncy Particle sampler, scale favorably with problem dimension. The work therefore opens a practical pathway for applying non‑reversible PDMP methods to large‑scale Bayesian inference tasks in engineering and the physical sciences.