Learning Hamiltonian Density Using DeepONet
This work addresses the challenge of learning Hamiltonian mechanics for wave equations in physics, offering a method that avoids discretization dependencies, but it appears incremental as it builds on existing deep learning approaches like Hamiltonian Neural Networks.
The authors tackled the problem of modeling wave equations described by partial differential equations (PDEs) by proposing an operator learning approach to compute Hamiltonian density without requiring discretization or determination of differential operators, and the experiments showed the method successfully learned the operator from data with unspecific discretization.
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.