Mechanistic PDE Networks for Discovery of Governing Equations

Adeel Pervez, Efstratios Gavves, Francesco Locatello
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:48962-48973, 2025.

Abstract

We present Mechanistic PDE Networks – a model for discovery of governing partial differential equations from data. Mechanistic PDE Networks represent spatiotemporal data as space-time dependent linear partial differential equations in neural network hidden representations. The represented PDEs are then solved and decoded for specific tasks. The learned PDE representations naturally express the spatiotemporal dynamics in data in neural network hidden space, enabling increased modeling power. Solving the PDE representations in a compute and memory-efficient way, however, is a significant challenge. We develop a native, GPU-capable, parallel, sparse and differentiable multigrid solver specialized for linear partial differential equations that acts as a module in Mechanistic PDE Networks. Leveraging the PDE solver we propose a discovery architecture that can discovers nonlinear PDEs in complex settings, while being robust to noise. We validate PDE discovery on a number of PDEs including reaction-diffusion and Navier-Stokes equations.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-pervez25a, title = {Mechanistic {PDE} Networks for Discovery of Governing Equations}, author = {Pervez, Adeel and Gavves, Efstratios and Locatello, Francesco}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {48962--48973}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/pervez25a/pervez25a.pdf}, url = {https://proceedings.mlr.press/v267/pervez25a.html}, abstract = {We present Mechanistic PDE Networks – a model for discovery of governing partial differential equations from data. Mechanistic PDE Networks represent spatiotemporal data as space-time dependent linear partial differential equations in neural network hidden representations. The represented PDEs are then solved and decoded for specific tasks. The learned PDE representations naturally express the spatiotemporal dynamics in data in neural network hidden space, enabling increased modeling power. Solving the PDE representations in a compute and memory-efficient way, however, is a significant challenge. We develop a native, GPU-capable, parallel, sparse and differentiable multigrid solver specialized for linear partial differential equations that acts as a module in Mechanistic PDE Networks. Leveraging the PDE solver we propose a discovery architecture that can discovers nonlinear PDEs in complex settings, while being robust to noise. We validate PDE discovery on a number of PDEs including reaction-diffusion and Navier-Stokes equations.} }
Endnote
%0 Conference Paper %T Mechanistic PDE Networks for Discovery of Governing Equations %A Adeel Pervez %A Efstratios Gavves %A Francesco Locatello %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-pervez25a %I PMLR %P 48962--48973 %U https://proceedings.mlr.press/v267/pervez25a.html %V 267 %X We present Mechanistic PDE Networks – a model for discovery of governing partial differential equations from data. Mechanistic PDE Networks represent spatiotemporal data as space-time dependent linear partial differential equations in neural network hidden representations. The represented PDEs are then solved and decoded for specific tasks. The learned PDE representations naturally express the spatiotemporal dynamics in data in neural network hidden space, enabling increased modeling power. Solving the PDE representations in a compute and memory-efficient way, however, is a significant challenge. We develop a native, GPU-capable, parallel, sparse and differentiable multigrid solver specialized for linear partial differential equations that acts as a module in Mechanistic PDE Networks. Leveraging the PDE solver we propose a discovery architecture that can discovers nonlinear PDEs in complex settings, while being robust to noise. We validate PDE discovery on a number of PDEs including reaction-diffusion and Navier-Stokes equations.
APA
Pervez, A., Gavves, E. & Locatello, F.. (2025). Mechanistic PDE Networks for Discovery of Governing Equations. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:48962-48973 Available from https://proceedings.mlr.press/v267/pervez25a.html.

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