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.

翻译：算子学习近年来受到广泛关注，旨在学习函数空间之间的映射。已有研究提出利用深度神经网络（DNN）学习此类映射，从而实现对偏微分方程（PDE）解算子的学习。然而，这些方法在学习遵循物理定律的动力学行为时仍存在困难。本文提出能量一致神经算子（ENO），这是一个从观测的轨迹数据中学习遵循能量守恒或耗散定律的PDE解算子通用框架。我们引入一种受物理学能量基理论启发的新型惩罚函数进行训练，其中能量泛函由另一个DNN建模，从而能够在不显式给定PDE的情况下，使基于DNN的解算子输出偏向于保持能量一致性。在多个物理系统上的实验表明，ENO在从数据预测解方面优于现有DNN模型，尤其在超分辨率设置中表现突出。