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Scaling up Probabilistic PDE Simulators with Structured Volumetric Information

Tim Weiland, Marvin Pförtner, Philipp Hennig

TL;DR

This work develops a scalable probabilistic PDE solver by placing a Gaussian process prior on the solution and conditioning on volumetric (finite-volume) observations rather than pointwise collocation. By combining tensor-product Matérn covariances with box-shaped volumes, the method reduces multidimensional integrals to one-dimensional components, enabling efficient exact updates for linear PDEs. The approach employs iterative, structure-exploiting linear algebra (IterGP with Kronecker-structured operators) and a Cholesky-based preconditioner to handle large Gram matrices, demonstrated on large-scale problems including tsunami propagation. Empirical results show GP-FVM achieves higher data efficiency and superior scaling compared to collocation, while providing a fully probabilistic posterior that captures both mathematical and computational uncertainty. This work advances practical probabilistic PDE simulation by uniting classical discretization ideas with modern Bayesian inference and scalable linear algebra.

Abstract

Modeling real-world problems with partial differential equations (PDEs) is a prominent topic in scientific machine learning. Classic solvers for this task continue to play a central role, e.g. to generate training data for deep learning analogues. Any such numerical solution is subject to multiple sources of uncertainty, both from limited computational resources and limited data (including unknown parameters). Gaussian process analogues to classic PDE simulation methods have recently emerged as a framework to construct fully probabilistic estimates of all these types of uncertainty. So far, much of this work focused on theoretical foundations, and as such is not particularly data efficient or scalable. Here we propose a framework combining a discretization scheme based on the popular Finite Volume Method with complementary numerical linear algebra techniques. Practical experiments, including a spatiotemporal tsunami simulation, demonstrate substantially improved scaling behavior of this approach over previous collocation-based techniques.

Scaling up Probabilistic PDE Simulators with Structured Volumetric Information

TL;DR

This work develops a scalable probabilistic PDE solver by placing a Gaussian process prior on the solution and conditioning on volumetric (finite-volume) observations rather than pointwise collocation. By combining tensor-product Matérn covariances with box-shaped volumes, the method reduces multidimensional integrals to one-dimensional components, enabling efficient exact updates for linear PDEs. The approach employs iterative, structure-exploiting linear algebra (IterGP with Kronecker-structured operators) and a Cholesky-based preconditioner to handle large Gram matrices, demonstrated on large-scale problems including tsunami propagation. Empirical results show GP-FVM achieves higher data efficiency and superior scaling compared to collocation, while providing a fully probabilistic posterior that captures both mathematical and computational uncertainty. This work advances practical probabilistic PDE simulation by uniting classical discretization ideas with modern Bayesian inference and scalable linear algebra.

Abstract

Modeling real-world problems with partial differential equations (PDEs) is a prominent topic in scientific machine learning. Classic solvers for this task continue to play a central role, e.g. to generate training data for deep learning analogues. Any such numerical solution is subject to multiple sources of uncertainty, both from limited computational resources and limited data (including unknown parameters). Gaussian process analogues to classic PDE simulation methods have recently emerged as a framework to construct fully probabilistic estimates of all these types of uncertainty. So far, much of this work focused on theoretical foundations, and as such is not particularly data efficient or scalable. Here we propose a framework combining a discretization scheme based on the popular Finite Volume Method with complementary numerical linear algebra techniques. Practical experiments, including a spatiotemporal tsunami simulation, demonstrate substantially improved scaling behavior of this approach over previous collocation-based techniques.
Paper Structure (30 sections, 1 theorem, 37 equations, 12 figures)

This paper contains 30 sections, 1 theorem, 37 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Affine Gaussian Process Inference
  4. PDE Solving as a Learning Problem
  5. Volumetric Information Operators
  6. One-dimensional domains
  7. Multi-dimensional domains
  8. Multi-dimensional codomains
  9. Large-Scale Inference
  10. Iterative techniques
  11. Cholesky preconditioners
  12. Related work
  13. Experiments
  14. Comparative Benchmarks
  15. Real-world example: Tsunami propagation
  16. ...and 15 more sections

Key Result

Theorem A.3

Let $\bm{\mathcal{L}}: \mathbb{B} \rightarrow \mathbb{R}^n$ be a bounded linear operator. Under assumption-rkbs-gaussian-randvar, the following results hold. Viewing $f$ as a $\mathbb{B}$-valued Gaussian random variable, the application of $\mathcal{L}$ yields Let $\bm{\varepsilon} \sim \mathcal{N}(\bm{\mu}, \bm{\Sigma})$ be an $\mathbb{R}^n$-valued Gaussian random variable such that $\bm{\vareps

Figures (12)

  • Figure 1: A physics-informed GP models the propagation of a tsunami near a coast. For visualization purposes, the seabed topography is not true to scale. The middle surface (in shades of blue) depicts the posterior mean over the deviation of the water from its mean height. The transparent surface at the top depicts the $95\%$ confidence upper bound, which includes computational uncertainty caused by early termination of the internal iterative solver. We recommend looking at an animated version of this simulation https://github.com/timweiland/gp-fvm
  • Figure 2: Learning to solve $\mathrm{d}u/\mathrm{d}x(x) = \cos(x)$ through collocation and FVM. Both approaches condition a GP prior on i) the boundary conditions $u(0) = u(2 \pi) = 0$ and ii) a discretization of the differential equation. Collocation discretizes via point observations , whereas FVM discretizes via integral observations . The plots depict the resulting posteriors over $u(x)$ and $\mathrm{d}u/\mathrm{d}x(x)$, as well as the corresponding ground truths. We also plot three samples for each posterior.
  • Figure 3: Scaling behavior of GP-FVM compared to collocation for various problem classes. The plots show the mean RMSE across all IBVPs in each problem class. For a), we also compare to neural network baselines from PDEBench.
  • Figure 4: IterGP actions show how the solver progresses through the time dimension. Depicted are the first $500$ actions of IterGP with a CG policy for the tsunami example. Each action is reduced to the FVM components, and then summed up over the spatial dimensions. The result is a vector containing the weight by which each time slice is targeted.
  • Figure 5: Scaling behavior of GP-FVM compared to collocation for 1D Advection with $\beta = 0.4$. The plot shows the mean maximum absolute error (MAE) across all IBVPs in each problem class.
  • ...and 7 more figures

Theorems & Definitions (4)

  • Definition 2.1
  • Definition 4.1
  • Definition A.1
  • Theorem A.3