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Learning Hamiltonian Density Using DeepONet

Baige Xu, Yusuke Tanaka, Takashi Matsubara, Takaharu Yaguchi

TL;DR

This work tackles discretization-free learning of Hamiltonian PDEs by introducing an operator-learning approach based on DeepONet to learn the Hamiltonian density operator $H$ from data and to recover the variational derivative $\delta \mathcal{H}/\delta u$ through automatic differentiation. By treating $H$ as a learned operator $H_{NO}$ and leveraging a basis representation, the method uses AD gradients to obtain $\delta \mathcal{H}_{NO}/\delta u$, enabling physics-consistent PDE identification without explicit discretization of differential operators. A numerical experiment on the linear wave equation shows that the learned Hamiltonian density and the induced dynamics approximate the true system, though temporal drift suggests the need for more data and higher-precision integration. Overall, the paper presents a discretization-agnostic framework that fuses operator learning with AD-based variational derivatives to model Hamiltonian PDEs, with potential applications to broader wave phenomena.

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.

Learning Hamiltonian Density Using DeepONet

TL;DR

This work tackles discretization-free learning of Hamiltonian PDEs by introducing an operator-learning approach based on DeepONet to learn the Hamiltonian density operator from data and to recover the variational derivative through automatic differentiation. By treating as a learned operator and leveraging a basis representation, the method uses AD gradients to obtain , enabling physics-consistent PDE identification without explicit discretization of differential operators. A numerical experiment on the linear wave equation shows that the learned Hamiltonian density and the induced dynamics approximate the true system, though temporal drift suggests the need for more data and higher-precision integration. Overall, the paper presents a discretization-agnostic framework that fuses operator learning with AD-based variational derivatives to model Hamiltonian PDEs, with potential applications to broader wave phenomena.

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.

Paper Structure

This paper contains 7 sections, 1 theorem, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Operator Learning and DeepONet
  3. Operator Learning for Modeling Hamiltonian Partial Differential Equations Using DeepONet
  4. Learning Hamiltonian Density Using DeepONet
  5. Deriving Variational Derivatives of Hamiltonian Using Automatic Differentiation Algorithm
  6. Numerical Example: Wave Equations
  7. Conclusion

Key Result

Theorem 1

Suppose that $\Tilde{\mathcal{X}}=\mathrm{span}\{\phi_1,...,\phi_N\}$ is a subspace of a Hilbert space $\mathcal{X}$, $\phi_i = \phi_i(x) \in \mathcal{X},i=1,...,N$. Let $\mathcal{H}_{NO}:(\Tilde{\mathcal{X}},\langle,\rangle _{L^2} ) \to \mathbb{R}$ be a sufficiently smooth Hamiltonian learned by De

Figures (3)

  • Figure 1: True $u$ and $u_t$ of the equation.
  • Figure 2: Learned $u_{NO}$ and ${u_t}_{NO}$ of the equation.
  • Figure 3: The behavior of Hamiltonian. Red: $\mathcal{H}_{NO}(u,u_t)$, Blue: $\mathcal{H}(u_{NO},{u_t}_{NO})$.

Theorems & Definitions (2)

  • Theorem
  • Proof