# Scaling up Probabilistic PDE Simulators with Structured Volumetric Information

arXiv: https://arxiv.org/abs/2406.05020 · HTML: https://mlsanity.com/paper/2406.05020/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 40 references that are in the map.
- First posted on arXiv 7 Jun 2024.

## Builds on (most cited first, 23 of 23)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [Constant-Time Predictive Distributions for Gaussian Processes](https://mlsanity.com/paper/1803.06058/) (2018)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Thoughts on Massively Scalable Gaussian Processes](https://mlsanity.com/paper/1511.01870/) (2015)
- [Posterior and Computational Uncertainty in Gaussian Processes](https://mlsanity.com/paper/2205.15449/) (2022)
- [Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems](https://mlsanity.com/paper/1701.04006/) (2017)
- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [Solving High Frequency and Multi-Scale PDEs with Gaussian Processes](https://mlsanity.com/paper/2311.04465/) (2023)
- [Probabilistic ODE Solutions in Millions of Dimensions](https://mlsanity.com/paper/2110.11812/) (2021)
- [Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2110.11847/) (2021)
- [Sample Path Regularity of Gaussian Processes from the Covariance Kernel](https://mlsanity.com/paper/2312.14886/) (2023)
- [ProbNum: Probabilistic Numerics in Python](https://mlsanity.com/paper/2112.02100/) (2021)
- [Numerical methods for Mean field Games based on Gaussian Processes and Fourier Features](https://mlsanity.com/paper/2112.05414/) (2021)

## Built on by (most cited first, 2 of 2)

- [Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fields](https://mlsanity.com/paper/2503.08343/) (2025)
- [Scalable Bayesian Inference for Nonlinear Conservation Laws](https://mlsanity.com/paper/2605.31127/) (2026)

## Abstract (arXiv)

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.