Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems

TL;DR

This paper introduces a scalable probabilistic PDE solver that combines stochastic dual descent and active learning to efficiently handle large-scale and high-dimensional problems with quantified uncertainty.

Contribution

It presents a novel $h$-adaptive probabilistic solver that significantly reduces computational complexity and adaptively selects collocation points for improved efficiency.

Findings

Successfully applied to 2D and 3D elliptic PDEs

Effective in time-dependent parabolic PDEs

Achieves linear complexity per iteration

Abstract

Solving partial differential equations (PDEs) within the framework of probabilistic numerics offers a principled approach to quantifying epistemic uncertainty arising from discretization. By leveraging Gaussian process regression and imposing the governing PDE as a constraint at a finite set of collocation points, probabilistic numerics delivers mesh-free solutions at arbitrary locations. However, the high computational cost, which scales cubically with the number of collocation points, remains a critical bottleneck, particularly for large-scale or high-dimensional problems. We propose a scalable enhancement to this paradigm through two key innovations. First, we develop a stochastic dual descent algorithm that reduces the per-iteration complexity from cubic to linear in the number of collocation points, enabling tractable inference. Second, we exploit a clustering-based active learning strategy that adaptively selects collocation points to maximize information gain while minimizing computational expense. Together, these contributions result in an $h$-adaptive probabilistic solver that can scale to a large number of collocation points. We demonstrate the efficacy of the proposed solver on benchmark PDEs, including two- and three-dimensional steady-state elliptic problems, as well as a time-dependent parabolic PDE formulated in a space-time setting.