Energy-consistent Neural Operators for Hamiltonian and Dissipative Partial Differential Equations

Yusuke Tanaka, Takaharu Yaguchi, Tomoharu Iwata, Naonori Ueda
Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, PMLR 258:1882-1890, 2025.

Abstract

The operator learning has received significant attention in recent years, with the aim of learning a mapping between function spaces. Prior works have proposed deep neural networks (DNNs) for learning such a mapping, enabling the learning of solution operators of partial differential equations (PDEs). However, these works still struggle to learn dynamics that obeys the laws of physics. This paper proposes Energy-consistent Neural Operators (ENOs), a general framework for learning solution operators of PDEs that follows the energy conservation or dissipation law from observed solution trajectories. We introduce a novel penalty function inspired by the energy-based theory of physics for training, in which the functional derivative is calculated making full use of automatic differentiation, allowing one to bias the outputs of the DNN-based solution operators to obey appropriate energetic behavior without explicit PDEs. Experiments on multiple systems show that ENO outperforms existing DNN models in predicting solutions from data, especially in super-resolution settings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v258-tanaka25a, title = {Energy-consistent Neural Operators for Hamiltonian and Dissipative Partial Differential Equations}, author = {Tanaka, Yusuke and Yaguchi, Takaharu and Iwata, Tomoharu and Ueda, Naonori}, booktitle = {Proceedings of The 28th International Conference on Artificial Intelligence and Statistics}, pages = {1882--1890}, year = {2025}, editor = {Li, Yingzhen and Mandt, Stephan and Agrawal, Shipra and Khan, Emtiyaz}, volume = {258}, series = {Proceedings of Machine Learning Research}, month = {03--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v258/main/assets/tanaka25a/tanaka25a.pdf}, url = {https://proceedings.mlr.press/v258/tanaka25a.html}, abstract = {The operator learning has received significant attention in recent years, with the aim of learning a mapping between function spaces. Prior works have proposed deep neural networks (DNNs) for learning such a mapping, enabling the learning of solution operators of partial differential equations (PDEs). However, these works still struggle to learn dynamics that obeys the laws of physics. This paper proposes Energy-consistent Neural Operators (ENOs), a general framework for learning solution operators of PDEs that follows the energy conservation or dissipation law from observed solution trajectories. We introduce a novel penalty function inspired by the energy-based theory of physics for training, in which the functional derivative is calculated making full use of automatic differentiation, allowing one to bias the outputs of the DNN-based solution operators to obey appropriate energetic behavior without explicit PDEs. Experiments on multiple systems show that ENO outperforms existing DNN models in predicting solutions from data, especially in super-resolution settings.} }
Endnote
%0 Conference Paper %T Energy-consistent Neural Operators for Hamiltonian and Dissipative Partial Differential Equations %A Yusuke Tanaka %A Takaharu Yaguchi %A Tomoharu Iwata %A Naonori Ueda %B Proceedings of The 28th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2025 %E Yingzhen Li %E Stephan Mandt %E Shipra Agrawal %E Emtiyaz Khan %F pmlr-v258-tanaka25a %I PMLR %P 1882--1890 %U https://proceedings.mlr.press/v258/tanaka25a.html %V 258 %X The operator learning has received significant attention in recent years, with the aim of learning a mapping between function spaces. Prior works have proposed deep neural networks (DNNs) for learning such a mapping, enabling the learning of solution operators of partial differential equations (PDEs). However, these works still struggle to learn dynamics that obeys the laws of physics. This paper proposes Energy-consistent Neural Operators (ENOs), a general framework for learning solution operators of PDEs that follows the energy conservation or dissipation law from observed solution trajectories. We introduce a novel penalty function inspired by the energy-based theory of physics for training, in which the functional derivative is calculated making full use of automatic differentiation, allowing one to bias the outputs of the DNN-based solution operators to obey appropriate energetic behavior without explicit PDEs. Experiments on multiple systems show that ENO outperforms existing DNN models in predicting solutions from data, especially in super-resolution settings.
APA
Tanaka, Y., Yaguchi, T., Iwata, T. & Ueda, N.. (2025). Energy-consistent Neural Operators for Hamiltonian and Dissipative Partial Differential Equations. Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 258:1882-1890 Available from https://proceedings.mlr.press/v258/tanaka25a.html.

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