用深度算子网络学习波的哈密顿密度，无需离散化和微分算子
Learning Hamiltonian Density Using DeepONet
- 通过自动微分计算变分导数，直接从数据中学习哈密顿密度算子
- 在无特定离散化条件下成功恢复波方程的哈密顿结构
- 适合物理建模与偏微分方程学习的研究者使用
近年来，基于深度学习建模由偏微分方程（PDE）描述的物理现象受到广泛关注。例如，针对哈密顿力学的学习，基于深度神经网络的方法如哈密顿神经网络（HNNs）及其变体已取得进展。然而，现有方法通常依赖数据的离散化，且常需手动确定所需的微分算子。本文提出一种用于建模波动方程的算子学习方法。特别地，我们利用自动微分算法计算构造方程所需的变分导数。实验表明，该方法能够在不指定离散化方式、无需显式确定微分算子的前提下，从数据中学习定义波的哈密顿密度的算子。
原文摘要 · Abstract (English)
In recent years, deep learning for modeling physical phenomena which can be described by partial differential equations (PDEs) have received significant attention. For example, for learning Hamiltonian mechanics, methods based on deep neural networks such as Hamiltonian Neural Networks (HNNs) and their variants have achieved progress. However, existing methods typically depend on the discretization of data, and the determination of required differential operators is often necessary. Instead, in this work, we propose an operator learning approach for modeling wave equations. In particular, we present a method to compute the variational derivatives that are needed to formulate the equations using the automatic differentiation algorithm. The experiments demonstrated that the proposed method is able to learn the operator that defines the Hamiltonian density of waves from data with unspecific discretization without determination of the differential operators.
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