Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fields

TL;DR

This work tackles the scalability gap in probabilistic PDE solvers by recasting GP priors as Gaussian Markov random fields through SPDE discretizations, enabling sparse precision-based inference. It introduces physics-informed SPDE priors that closely mimic target PDE dynamics via linear proxies and embeds boundary conditions directly into the prior, while treating time as an additional dimension in a spatiotemporal FEM framework. Inference is performed with affine conditioning, efficient moment computation, and Gauss-Newton optimization for nonlinear PDEs, yielding a practical solver with principled uncertainty quantification. Experiments on Darcy flow and Burgers’ equation demonstrate accuracy on par with classical FEM with notable speedups and robust uncertainty handling, outperforming several state-of-the-art probabilistic solvers and offering scalable, physically consistent PDE solving for large-scale problems.

Abstract

Mechanistic knowledge about the physical world is virtually always expressed via partial differential equations (PDEs). Recently, there has been a surge of interest in probabilistic PDE solvers -- Bayesian statistical models mostly based on Gaussian process (GP) priors which seamlessly combine empirical measurements and mechanistic knowledge. As such, they quantify uncertainties arising from e.g. noisy or missing data, unknown PDE parameters or discretization error by design. Prior work has established connections to classical PDE solvers and provided solid theoretical guarantees. However, scaling such methods to large-scale problems remains a fundamental challenge primarily due to dense covariance matrices. Our approach addresses the scalability issues by leveraging the Markov property of many commonly used GP priors. It has been shown that such priors are solutions to stochastic PDEs (SPDEs) which when discretized allow for highly efficient GP regression through sparse linear algebra. In this work, we show how to leverage this prior class to make probabilistic PDE solvers practical, even for large-scale nonlinear PDEs, through greatly accelerated inference mechanisms. Additionally, our approach also allows for flexible and physically meaningful priors beyond what can be modeled with covariance functions. Experiments confirm substantial speedups and accelerated convergence of our physics-informed priors in nonlinear settings.

Figures (6)

• Figure 1: A flowchart of the high-level steps we propose for prior construction and inference. This conceptual sketch uses Burgers' equation as a concrete example for a simple nonlinear PDE.
• Figure 2: Advection-diffusion SPDE priors provide physical information. Starting from the same initial condition, the top row shows the analytic evolution of the advection-diffusion SPDE prior for Burgers' univariate equation at $t=1.5sec$ and $t=3.0sec$. For comparison the bottom row shows an analogous stationary product Matérn prior used in prior work. Both rows compare to the ground-truth dynamics of a nonlinear Burgers' equation in orange.
• Figure 3: Performance of our method on the nonlinear elliptic PDE from Chen2024. We evaluate the L2 error and runtime of GMRF-FEM with linear and quadratic basis elements for varying mesh resolutions $N_{xy}$.
• Figure 4: GMRF-based methods equal classic FEM solvers in accuracy at reasonable computational overheads. The plot shows the mean relative error across all problem instances for different discretization resolutions.
• Figure 5: Physics-informed priors yield faster convergence in terms of the discretization resolution. The plot shows the mean relative error across all problem instances for different discretization resolutions for the Burgers' equation IBVPs.