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Learning Hamiltonian Systems with Pseudo-symplectic Neural Network

Xupeng Cheng, Lijin Wang, Yanzhao Cao, Chen Chen

TL;DR

This paper addresses learning Hamiltonian dynamics from data, including non-separable Hamiltonians, while preserving the intrinsic symplectic structure. It introduces PSNN, which embeds an explicit pseudo-symplectic integrator as the dynamical core and uses learnable Padé-type activations to model the gradient of the full Hamiltonian $H_y$, enabling near-symplectic behavior with efficient training. The method achieves higher accuracy, long-term stability, and energy preservation across challenging systems (bead on a wire, modified pendulum, 4D galactic dynamics) with fewer samples and parameters than existing structure-preserving models. This work bridges computational efficiency and geometric structure preservation in Hamiltonian learning and suggests that Padé-type activations can enhance explicit structure-preserving neural architectures.

Abstract

In this paper, we introduces a Pseudo-Symplectic Neural Network (PSNN) for learning general Hamiltonian systems (both separable and non-separable) from data. To address the limitations of existing structure-preserving methods (e.g., implicit symplectic integrators restricted to separable systems or explicit approximations requiring high computational costs), PSNN integrates an explicit pseudo-symplectic integrator as its dynamical core, achieving nearly exact symplecticity with minimal structural error. Additionally, the authors propose learnable Padé-type activation functions based on Padé approximation theory, which empirically outperform classical ReLU, Taylor-based activations, and PAU. By combining these innovations, PSNN demonstrates superior performance in learning and forecasting diverse Hamiltonian systems (e.g., modified pendulum, galactic dynamics), surpassing state-of-the-art models in accuracy, long-term stability, and energy preservation, while requiring shorter training time, fewer samples, and reduced parameters. This framework bridges the gap between computational efficiency and geometric structure preservation in Hamiltonian system modeling.

Learning Hamiltonian Systems with Pseudo-symplectic Neural Network

TL;DR

This paper addresses learning Hamiltonian dynamics from data, including non-separable Hamiltonians, while preserving the intrinsic symplectic structure. It introduces PSNN, which embeds an explicit pseudo-symplectic integrator as the dynamical core and uses learnable Padé-type activations to model the gradient of the full Hamiltonian , enabling near-symplectic behavior with efficient training. The method achieves higher accuracy, long-term stability, and energy preservation across challenging systems (bead on a wire, modified pendulum, 4D galactic dynamics) with fewer samples and parameters than existing structure-preserving models. This work bridges computational efficiency and geometric structure preservation in Hamiltonian learning and suggests that Padé-type activations can enhance explicit structure-preserving neural architectures.

Abstract

In this paper, we introduces a Pseudo-Symplectic Neural Network (PSNN) for learning general Hamiltonian systems (both separable and non-separable) from data. To address the limitations of existing structure-preserving methods (e.g., implicit symplectic integrators restricted to separable systems or explicit approximations requiring high computational costs), PSNN integrates an explicit pseudo-symplectic integrator as its dynamical core, achieving nearly exact symplecticity with minimal structural error. Additionally, the authors propose learnable Padé-type activation functions based on Padé approximation theory, which empirically outperform classical ReLU, Taylor-based activations, and PAU. By combining these innovations, PSNN demonstrates superior performance in learning and forecasting diverse Hamiltonian systems (e.g., modified pendulum, galactic dynamics), surpassing state-of-the-art models in accuracy, long-term stability, and energy preservation, while requiring shorter training time, fewer samples, and reduced parameters. This framework bridges the gap between computational efficiency and geometric structure preservation in Hamiltonian system modeling.

Paper Structure

This paper contains 16 sections, 4 theorems, 46 equations, 12 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hamiltonian systems and pseudo-symplectic methods
  4. Related work
  5. The Taylor-nets
  6. The Padé activation unit (PAU)
  7. Our method
  8. The learning framework
  9. Construction of $H_{y,net}$ with Padé-type activation functions
  10. The pseudo-symplectic network structure
  11. Numerical experiments
  12. Example 1: Movement of a bead on a wire
  13. Example 2: A modified pendulum problem
  14. Example 3: A 4-dimensional galactic dynamic problem
  15. Example 4: A separable Hamiltonian system
  16. ...and 1 more sections

Key Result

Proposition 3.1

(Kidger) Let $\rho: \mathbb{R} \rightarrow \mathbb{R}$ be any continuous nonpolynomial function which is continuously differentiable at at least one point, with nonzero derivative at that point. Let $U \subset \mathbb{R}^n$ be compact. Then $\mathcal{N N}_{n, m, n+m+1}^\rho$ is dense in $C\left(U ;

Figures (12)

  • Figure 1: Schematic diagram of $H_{\boldsymbol{y},net}$
  • Figure 2: Schematic diagram of PSNN, where $I$ represents the identity mapping.
  • Figure 3: Evolution of energy learned by PSNN with different activation functions (left); Prediction error by PSNN with different activation functions (right).
  • Figure 4: (a) Predicted phase trajectory on $t\in[0,60]$ by PSNN with different activation functions; (b) Predicted phase trajectory on $t\in[0,60]$ by different neural networks.
  • Figure 5: Evolution of energy learned by PSNN with different activation functions (left); Prediction error by PSNN with different activation functions (right).
  • ...and 7 more figures

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.1
  • Theorem 3.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof