Frequency-Separable Hamiltonian Neural Network for Multi-Timescale Dynamics

TL;DR

This paper introduces FS-HNN, a neural network model that captures multi-timescale Hamiltonian dynamics by decomposing Hamiltonians into fast and slow components, improving long-term predictions for complex systems.

Contribution

The paper proposes a novel frequency-separable Hamiltonian neural network that models multi-timescale dynamics by learning separate Hamiltonian components at different sampling rates.

Findings

Enhanced long-horizon extrapolation on complex dynamical systems

Improved generalization across various ODE and PDE problems

Effective modeling of multi-timescale Hamiltonian systems

Abstract

While Hamiltonian mechanics provides a powerful inductive bias for neural networks modeling dynamical systems, Hamiltonian Neural Networks and their variants often fail to capture complex temporal dynamics spanning multiple timescales. This limitation is commonly linked to the spectral bias of deep neural networks, which favors learning low-frequency, slow-varying dynamics. Prior approaches have sought to address this issue through symplectic integration schemes that enforce energy conservation or by incorporating geometric constraints to impose structure on the configuration-space. However, such methods either remain limited in their ability to fully capture multiscale dynamics or require substantial domain specific assumptions. In this work, we exploit the observation that Hamiltonian functions admit decompositions into explicit fast and slow modes and can be reconstructed from these components. We introduce the Frequency-Separable Hamiltonian Neural Network (FS-HNN), which parameterizes the system Hamiltonian using multiple networks, each governed by Hamiltonian dynamics and trained on data sampled at distinct timescales. We further extend this framework to partial differential equations by learning a state- and boundary-conditioned symplectic operators. Empirically, we show that FS-HNN improves long-horizon extrapolation performance on challenging dynamical systems and generalizes across a broad range of ODE and PDE problems.