Neural Operators Meet Energy-based Theory: Operator Learning for Hamiltonian and Dissipative PDEs

• URL: http://arxiv.org/abs/2402.09018v1
• Date: Wed, 14 Feb 2024 08:50:14 GMT
• Title: Neural Operators Meet Energy-based Theory: Operator Learning for Hamiltonian and Dissipative PDEs
• Authors: Yusuke Tanaka, Takaharu Yaguchi, Tomoharu Iwata, Naonori Ueda
• Abstract summary: This paper proposes Energy-consistent Neural Operators (ENOs) for learning solution operators of partial differential equations. ENOs 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.
• Score: 35.70739067374375
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 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.

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