Learning Hamiltonian Systems with Pseudo-symplectic Neural Network

TL;DR

This paper introduces PSNN, a neural network that effectively learns Hamiltonian systems by combining a pseudo-symplectic integrator with learnable Padé activation functions, achieving high accuracy and stability with computational efficiency.

Contribution

The paper proposes a novel Pseudo-Symplectic Neural Network that integrates an explicit pseudo-symplectic integrator and learnable Padé activation functions for improved learning of Hamiltonian systems.

Findings

Outperforms state-of-the-art models in accuracy and stability.

Requires fewer samples and parameters for training.

Achieves nearly exact symplecticity with minimal structural error.

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 Pad\'e-type activation functions based on Pad\'e 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.