Scaling up Probabilistic PDE Simulators with Structured Volumetric Information

Tim Weiland, Marvin Pförtner, Philipp Hennig
Proceedings of the 2nd International Conference on Probabilistic Numerics, PMLR 341:212-226, 2026.

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 well-known 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v341-weiland26a, title = {Scaling up Probabilistic {P}DE Simulators with Structured Volumetric Information}, author = {Weiland, Tim and Pf\"ortner, Marvin and Hennig, Philipp}, booktitle = {Proceedings of the 2nd International Conference on Probabilistic Numerics}, pages = {212--226}, year = {2026}, editor = {Karvonen, Toni and Bosch, Nathanael and Cockayne, Jon and Gessner, Alexandra and Hennig, Philipp and Kouw, Wouter}, volume = {341}, series = {Proceedings of Machine Learning Research}, month = {09--11 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v341/main/assets/weiland26a/weiland26a.pdf}, url = {https://proceedings.mlr.press/v341/weiland26a.html}, 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 well-known 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.} }
Endnote
%0 Conference Paper %T Scaling up Probabilistic PDE Simulators with Structured Volumetric Information %A Tim Weiland %A Marvin Pförtner %A Philipp Hennig %B Proceedings of the 2nd International Conference on Probabilistic Numerics %C Proceedings of Machine Learning Research %D 2026 %E Toni Karvonen %E Nathanael Bosch %E Jon Cockayne %E Alexandra Gessner %E Philipp Hennig %E Wouter Kouw %F pmlr-v341-weiland26a %I PMLR %P 212--226 %U https://proceedings.mlr.press/v341/weiland26a.html %V 341 %X 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 well-known 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.
APA
Weiland, T., Pförtner, M. & Hennig, P.. (2026). Scaling up Probabilistic PDE Simulators with Structured Volumetric Information. Proceedings of the 2nd International Conference on Probabilistic Numerics, in Proceedings of Machine Learning Research 341:212-226 Available from https://proceedings.mlr.press/v341/weiland26a.html.

Related Material