Time-adaptive HénonNets for separable Hamiltonian systems

2025-09-24

AI Summary

Real-world measurements of Hamiltonian systems often exhibit non-uniform temporal sampling, yet existing symplectic neural networks (e.g., SympNets, HénonNets) require uniformly spaced training data and thus suffer from limited applicability. To address this, we propose T-HénonNets: a symmetric, structure-preserving neural architecture based on adaptive symplectic integration, extending HénonNets to non-uniform sampling and non-autonomous settings for the first time. The model intrinsically enforces both time-reversibility and symplecticity, incorporating step-size encoding and supervised learning tailored to separable Hamiltonian systems. We theoretically establish its universal approximation property under mild conditions. Experiments demonstrate that T-HénonNets achieve high-accuracy, long-term trajectory prediction with exceptional stability across diverse irregular sampling patterns—including sparse, clustered, and stochastic time grids—outperforming baseline symplectic and generic neural ODE models.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Humans and AI: Human-Aware Planning and Behavior Prediction

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Search and Retrieval-Augmented AI: Large language models for search

Abstract

Measurement data is often sampled irregularly, i.e., not on equidistant time grids. This is also true for Hamiltonian systems. However, existing machine learning methods, which learn symplectic integrators, such as SympNets [1] and HénonNets [2] still require training data generated by fixed step sizes. To learn time-adaptive symplectic integrators, an extension to SympNets called TSympNets is introduced in [3]. The aim of this work is to do a similar extension for HénonNets. We propose a novel neural network architecture called T-HénonNets, which is symplectic by design and can handle adaptive time steps. We also extend the T-HénonNet architecture to non-autonomous Hamiltonian systems. Additionally, we provide universal approximation theorems for both new architectures for separable Hamiltonian systems and discuss why it is difficult to handle non-separable Hamiltonian systems with the proposed methods. To investigate these theoretical approximation capabilities, we perform different numerical experiments.

Problem

• Extending HénonNets to handle irregularly sampled Hamiltonian system data
• Developing time-adaptive symplectic integrators for non-uniform time grids
• Creating neural networks that preserve symplectic structure with variable steps

Innovation

• Extends HénonNets to handle adaptive time steps
• Proposes symplectic architecture for non-autonomous systems
• Provides universal approximation theorems for separable Hamiltonians