Scalable Bayesian Inference for Nonlinear Conservation Laws

2026-05-29

AI Summary

This work addresses the ill-posed problem of inferring physical quantities in nonlinear conservation law systems from sparse or noisy observations by proposing the first Bayesian method that simultaneously enforces numerical conservation and provides structured uncertainty quantification. The approach embeds a conservative numerical scheme within a Gaussian process prior and leverages sparse approximation techniques to construct a scalable forward–inverse framework. In forward simulations, it preserves the accuracy of classical solvers while delivering principled uncertainty estimates; in inverse tasks, it reconstructs the full posterior distribution of nonparametric source terms with high precision in seconds—significantly outperforming neural network baselines that yield only point estimates and require several minutes of computation.

Abstract

Nonlinear conservation laws are at the heart of many of the most important dynamical systems in science and engineering. In practical applications, such systems are often subject to various sources of uncertainty, e.g. due to sparse or noisy measurements. Inferring physical quantities and fields of interest then becomes an ill-posed problem which both classical numerical methods and modern deep learning-based methods struggle to treat appropriately. Recent work has framed classical numerical methods as Bayesian inference under Gaussian process priors, resulting in a physics-aware treatment of uncertainties. Following this line of work, we develop a novel numerically conservative method for uncertainty-aware simulations of nonlinear conservation laws. We use recent sparse approximation techniques to scale up to large-scale forward and inverse problems. For forward simulation, we inherit the accuracy of classical solvers while providing structured uncertainty quantification. On inverse problems, we recover posteriors over nonparametric source fields in seconds -- outperforming neural baselines that take minutes to produce a less accurate point estimate.

Problem

• nonlinear conservation laws
• uncertainty quantification
• ill-posed inference
• Bayesian inference
• inverse problems

Innovation

• Bayesian inference
• nonlinear conservation laws
• uncertainty quantification
• sparse Gaussian processes
• inverse problems