Efficient Message Passing for Partial Differential Equation Priors

2026-09-26

AI Summary

This study addresses the challenge of efficiently incorporating partial differential equation (PDE) priors into probabilistic inference. To this end, it proposes a novel PDE-solving framework based on factor graph message passing, which approximates parametric posterior distributions via moment matching without requiring sampling or global optimization throughout the process. This approach enables precise quantification of uncertainty in structured predictions. Empirically, the proposed method achieves a tenfold improvement in training speed over existing techniques while attaining inference efficiency comparable to variational inference. Furthermore, its predictive accuracy matches that of mainstream baselines, and it yields more accurate posterior estimates.

Abstract

Prior information for real-world physical quantities is most elegantly expressed via partial differential equations (PDEs). In this paper, we propose a novel way to solve PDEs using probabilistic inference on a factor graph. In general, factor graphs provide a natural way to encode prior knowledge into a model as explicit factors; here, this knowledge is provided by a governing PDE, which narrows the solution space, while observed data further shape the posterior over the parameters. The approximate parameter posterior is inferred using message passing based on moment matching, without posterior sampling or global gradient-based optimization. We demonstrate our approach on the first-order advection and the second-order semi-linear Fisher-KPP equations, where it achieves predictive accuracy comparable to a standard baseline while providing structured predictive uncertainty. Moreover, the inferred posterior marginal means and uncertainty structure match more closely those obtained using Hamiltonian Monte Carlo than the evaluated mean-field variational inference baseline, while requiring up to 10x less training time in our experiments, with inference speed comparable to variational inference.

Problem

• Partial Differential Equations
• Probabilistic Inference
• Factor Graphs
• Message Passing
• Uncertainty Quantification

Innovation

• Partial Differential Equations
• Factor Graphs
• Message Passing
• Moment Matching
• Probabilistic Inference