Scaling up Probabilistic PDE Simulators with Structured Volumetric Information

TL;DR

This paper introduces a scalable probabilistic PDE simulation framework that combines finite volume discretization with advanced linear algebra, enabling efficient uncertainty quantification in complex real-world problems.

Contribution

It presents a novel scalable approach integrating finite volume methods with numerical linear algebra for probabilistic PDE simulation, improving over previous collocation-based techniques.

Findings

Enhanced scalability demonstrated in tsunami simulation

Improved uncertainty quantification over traditional methods

Effective combination of discretization and linear algebra techniques

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.