Scalable Bayesian Inference for Nonlinear Conservation Laws

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.

Many important physical systems --- water flows in rivers and oceans, contaminants spreading through groundwater, fluid dynamics in engineering --- are governed by mathematical "conservation laws". In practice these models have to be combined with sparse and noisy data: a few sensor readings, a coarse measurement, a handful of monitoring wells. Classical numerical simulators give a single best-guess answer without telling us how reliable it is. Modern AI-based methods often perform poorly in this data-scarce regime and also lack a principled measure of uncertainty. We develop a method that combines the accuracy of classical numerical solvers with statistically calibrated uncertainty estimates: it produces not just a solution, but a quantitative confidence assessment of every point in that solution. Using recent advances in sparse linear algebra, the method scales to large problems --- for example, recovering an unknown contaminant source from sparse measurements in a 70,000-variable problem within minutes on a standard computer. This makes the approach useful wherever understanding model uncertainty is as important as the prediction itself: environmental monitoring, geophysical inversion, and scientific decision-making under uncertainty.