Learning Hamiltonian Dynamics with Bayesian Data Assimilation

TL;DR

This work tackles predicting unknown Hamiltonian dynamics from trajectory data while preserving invariants. It introduces an Autoregressive Hamiltonian Neural Network (AHNN) and couples it with Bayesian data assimilation via the Unscented Kalman Filter (UKF) to enable real-time refinement and uncertainty quantification. Through experiments on a frictionless mass-spring system and highly elliptic Molniya-like orbits with gravitational perturbations, the approach achieves energy conservation and superior long-term prediction accuracy, outperforming baseline neural models. The combination of structure-preserving learning, autoregressive training, and UKF-based state estimation has significant implications for robust, real-time trajectory prediction in celestial mechanics and related Hamiltonian systems.

Abstract

In this paper, we develop a neural network-based approach for time-series prediction in unknown Hamiltonian dynamical systems. Our approach leverages a surrogate model and learns the system dynamics using generalized coordinates (positions) and their conjugate momenta while preserving a constant Hamiltonian. To further enhance long-term prediction accuracy, we introduce an Autoregressive Hamiltonian Neural Network, which incorporates autoregressive prediction errors into the training objective. Additionally, we employ Bayesian data assimilation to refine predictions in real-time using online measurement data. Numerical experiments on a spring-mass system and highly elliptic orbits under gravitational perturbations demonstrate the effectiveness of the proposed method, highlighting its potential for accurate and robust long-term predictions.

Figures (12)

• Figure 1: Learning unknown dynamics from spacecraft position and velocity data. The model performance is compared for a highly elliptical orbit, which poses significant challenges when the spacecraft is near the primary body due to rapid velocity changes, with speed variations up to an order of magnitude.
• Figure 2: Illustration of the Hamiltonian Neural Network in \ref{['eq:hnn_pred']}.
• Figure 3: Illustration of the unscented transform for estimated mean and covariance propagation (blue) and true mean and covariance (black).
• Figure 4: Accelerations from gravitational perturbations along the sample highly elliptic orbit.
• Figure 5: (a) The state prediction of the simple mass-spring system. (b) The energy evolution over time.