Neural Operators Meet Energy-based Theory: Operator Learning for Hamiltonian and Dissipative PDEs
This work addresses the challenge of incorporating physical constraints into operator learning for PDEs, which is incremental as it builds on prior neural operator methods.
The paper tackles the problem of learning solution operators for PDEs that obey physical energy laws, proposing Energy-consistent Neural Operators (ENOs) which outperform existing models in predicting solutions, particularly in super-resolution tasks.
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 energy functional is modeled by another DNN, allowing one to bias the outputs of the DNN-based solution operators to ensure energetic consistency without explicit PDEs. Experiments on multiple physical systems show that ENO outperforms existing DNN models in predicting solutions from data, especially in super-resolution settings.