Neural Operators Meet Energy-based Theory: Operator Learning for Hamiltonian and Dissipative PDEs
<|reference_start|>Neural Operators Meet Energy-based Theory: Operator Learning for Hamiltonian and Dissipative PDEs: 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.<|reference_end|>
@article{tanaka2024neural,
    title={Neural Operators Meet Energy-based Theory: Operator Learning for
  Hamiltonian and Dissipative PDEs},
    author={Yusuke Tanaka, Takaharu Yaguchi, Tomoharu Iwata, Naonori Ueda},
    journal={arXiv preprint arXiv:2402.09018},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.09018},
    primaryClass={stat.ML cs.LG}
}