Learning Physical Operators using Neural Operators

Vignesh Gopakumar, Ander Gray, Daniel Giles, Lorenzo Zanisi, Matt J. Kusner, Timo Betcke, Stanislas Pamela, Marc Peter Deisenroth
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:3223-3231, 2026.

Abstract

Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work introduces a physics-informed training framework that addresses these limitations by decomposing PDEs using operator splitting methods, training separate neural operators to learn individual non-linear physical operators while approximating linear operators with fixed finite-difference convolutions. This modular mixture-of-experts architecture enables generalisation to novel physical regimes by explicitly encoding the underlying operator structure. We formulate the modelling task as a neural ordinary differential equation (ODE) where these learned operators constitute the right-hand side, enabling continuous-in-time predictions through standard ODE solvers and implicitly enforcing PDE constraints. Demonstrated on incompressible and compressible Navier–Stokes equations, our approach achieves better convergence and superior performance when generalising to unseen physics. The method remains parameter-efficient, enabling temporal extrapolation beyond training horizons, and provides interpretable components whose behaviour can be verified against known physics.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-gopakumar26a, title = { Learning Physical Operators using Neural Operators }, author = {Gopakumar, Vignesh and Gray, Ander and Giles, Daniel and Zanisi, Lorenzo and Kusner, Matt J. and Betcke, Timo and Pamela, Stanislas and Deisenroth, Marc Peter}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {3223--3231}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/gopakumar26a/gopakumar26a.pdf}, url = {https://proceedings.mlr.press/v300/gopakumar26a.html}, abstract = { Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work introduces a physics-informed training framework that addresses these limitations by decomposing PDEs using operator splitting methods, training separate neural operators to learn individual non-linear physical operators while approximating linear operators with fixed finite-difference convolutions. This modular mixture-of-experts architecture enables generalisation to novel physical regimes by explicitly encoding the underlying operator structure. We formulate the modelling task as a neural ordinary differential equation (ODE) where these learned operators constitute the right-hand side, enabling continuous-in-time predictions through standard ODE solvers and implicitly enforcing PDE constraints. Demonstrated on incompressible and compressible Navier–Stokes equations, our approach achieves better convergence and superior performance when generalising to unseen physics. The method remains parameter-efficient, enabling temporal extrapolation beyond training horizons, and provides interpretable components whose behaviour can be verified against known physics. } }
Endnote
%0 Conference Paper %T Learning Physical Operators using Neural Operators %A Vignesh Gopakumar %A Ander Gray %A Daniel Giles %A Lorenzo Zanisi %A Matt J. Kusner %A Timo Betcke %A Stanislas Pamela %A Marc Peter Deisenroth %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-gopakumar26a %I PMLR %P 3223--3231 %U https://proceedings.mlr.press/v300/gopakumar26a.html %V 300 %X Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work introduces a physics-informed training framework that addresses these limitations by decomposing PDEs using operator splitting methods, training separate neural operators to learn individual non-linear physical operators while approximating linear operators with fixed finite-difference convolutions. This modular mixture-of-experts architecture enables generalisation to novel physical regimes by explicitly encoding the underlying operator structure. We formulate the modelling task as a neural ordinary differential equation (ODE) where these learned operators constitute the right-hand side, enabling continuous-in-time predictions through standard ODE solvers and implicitly enforcing PDE constraints. Demonstrated on incompressible and compressible Navier–Stokes equations, our approach achieves better convergence and superior performance when generalising to unseen physics. The method remains parameter-efficient, enabling temporal extrapolation beyond training horizons, and provides interpretable components whose behaviour can be verified against known physics.
APA
Gopakumar, V., Gray, A., Giles, D., Zanisi, L., Kusner, M.J., Betcke, T., Pamela, S. & Deisenroth, M.P.. (2026). Learning Physical Operators using Neural Operators . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:3223-3231 Available from https://proceedings.mlr.press/v300/gopakumar26a.html.

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