Machine Learning Hamiltonian Dynamical Systems with Sparse and Noisy Data

TL;DR

This paper introduces ASRNNs, a structure-preserving neural network architecture that accurately learns and predicts Hamiltonian dynamical systems from extremely sparse, noisy, and irregular data, enabling symbolic discovery of governing equations.

Contribution

The paper proposes Adaptable Symplectic Recurrent Neural Networks (ASRNNs), a novel architecture that incorporates physical structure to improve learning stability and accuracy under challenging data conditions.

Findings

ASRNNs accurately predict long-term dynamics from minimal data points.

ASRNNs enable symbolic recovery of governing equations for polynomial systems.

The approach is robust to noise and irregular sampling in data.

Abstract

Machine learning has become a powerful tool for discovering governing laws of dynamical systems from data. However, most existing approaches degrade severely when observations are sparse, noisy, or irregularly sampled. In this work, we address the problem of learning symbolic representations of nonlinear Hamiltonian dynamical systems under extreme data scarcity by explicitly incorporating physical structure into the learning architecture. We introduce Adaptable Symplectic Recurrent Neural Networks (ASRNNs), a parameter-cognizant, structure-preserving model that combines Hamiltonian learning with symplectic recurrent integration, avoiding time derivative estimation, and enabling stable learning under noise. We demonstrate that ASRNNs can accurately predict long-term dynamics even when each training trajectory consists of only two irregularly spaced time points, possibly corrupted by correlated noise. Leveraging ASRNNs as structure-preserving data generators, we further enable symbolic discovery using independent regression methods (SINDy and PySR), recovering exact symbolic equations for polynomial systems and consistent polynomial approximations for non-polynomial Hamiltonians. Our results show that such architectures can provide a robust pathway to interpretable discovery of Hamiltonian dynamics from sparse and noisy data.