Learning Hamiltonian Density Using DeepONet

TL;DR

This paper introduces a novel operator learning method using DeepONet to model wave equations and Hamiltonian densities directly from data, avoiding the need for data discretization and explicit differential operator determination.

Contribution

It proposes a new approach to learn Hamiltonian densities of wave equations using DeepONet, bypassing traditional discretization and differential operator calculation methods.

Findings

Successfully learned Hamiltonian density from data without specific discretization

Demonstrated ability to compute variational derivatives via automatic differentiation

Effective modeling of wave equations using the proposed operator learning approach

Abstract

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.