Neural Operators 与基于能量的理论相遇：面向 Hamiltonian 和耗散 PDEs 的算子学习
Neural Operators Meet Energy-based Theory: Operator Learning for Hamiltonian and Dissipative PDEs
AI总结：
本文提出能量一致神经算子（ENOs）框架，通过引入基于能量理论的惩罚函数，利用神经网络建模能量泛函，在无显式PDE的情况下确保解算子遵循能量守恒或耗散定律，在多物理系统及超分辨率设置下优于现有模型。
摘要：
近年来，算子学习受到了广泛关注，旨在学习函数空间之间的映射。先前的工作提出了深度神经网络（DNNs）来学习这种映射，从而能够学习偏微分方程（PDEs）的解算子。然而，这些工作在学习遵循物理定律的动力学时仍然面临困难。本文提出了 Energy-consistent Neural Operators（ENOs），这是一个用于学习 PDEs 解算子的通用框架，该框架从观测到的解轨迹中遵循能量守恒或耗散定律。我们引入了一种受物理学基于能量理论启发的新颖惩罚函数用于训练，其中能量泛函由另一个 DNN 建模，允许人们偏置基于 DNN 的解算子的输出，以确保在没有显式 PDEs 的情况下保持能量一致性。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的 DNN 模型，特别是在超分辨率设置中。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
