Learning Hamiltonian Density Using DeepONet

2025-02-27

AI Summary

Conventional modeling of wave systems requires pre-specifying differential operators and structured spatial discretization, limiting flexibility and generalizability. Method: This paper proposes a Hamiltonian density functional learning framework that operates directly on unstructured, arbitrarily sampled data—without explicit discretization, predefined differential operators, or assumed differential order. Leveraging the DeepONet architecture, it integrates operator learning with automatic differentiation of variational derivatives for end-to-end learning of the Hamiltonian density operator, guided purely by physical constraints. Contribution/Results: The method eliminates dependence on grids, finite-difference schemes, or prior knowledge of differential structure. Experimental validation on wave equation tasks demonstrates superior accuracy and strong generalization capability compared to conventional Hamiltonian neural networks (HNNs) and related approaches.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Modeling wave equations using operator learning
• Computing variational derivatives with automatic differentiation
• Learning Hamiltonian density without specific discretization

Innovation

• DeepONet for Hamiltonian density
• Automatic differentiation algorithm
• Operator learning without discretization