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

arXiv: https://arxiv.org/abs/2508.09623 · HTML: https://mlsanity.com/paper/2508.09623/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 25 references that are in the map.
- First posted on arXiv 13 Aug 2025.

## Builds on (most cited first, 9 of 9)

- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Pathwise Conditioning of Gaussian Processes](https://mlsanity.com/paper/2011.04026/) (2020)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems](https://mlsanity.com/paper/1701.04006/) (2017)
- [Stable Implementation of Probabilistic ODE Solvers](https://mlsanity.com/paper/2012.10106/) (2020)
- [Stochastic Gradient Descent for Gaussian Processes Done Right](https://mlsanity.com/paper/2310.20581/) (2023)
- [Probabilistic Interpretation of Linear Solvers](https://mlsanity.com/paper/1402.2058/) (2014)
- [Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanics](https://mlsanity.com/paper/2409.10972/) (2024)

## Abstract (arXiv)

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.