Title: Learning Generalized Hamiltonian Dynamics with Stability from Noisy Trajectory Data
Primary area: learning on time series and dynamical systems
Keywords: Hamiltonian dynamics, machine learning, Gaussian processes, stability

Abstract: We propose a unified framework for learning generalized Hamiltonian dynamics from noisy, sparse phase-space observations via variational Bayesian inference. Modeling conservative, dissipative, and port-Hamiltonian regimes with a single architecture is challenging because each induces distinct phase-space behavior from similar initial energies. To address this, we extend sparse symplectic Gaussian processes with random Fourier features to build a probabilistic surrogate of the Hamiltonian landscape. Our approach supports all three regimes through a generalized formulation of Hamiltonian dynamics. To ensure physical correctness and stability, we softly enforce conservation and Lyapunov-style constraints. With this relaxation, we recast the original constrained optimization problem as a hyperparameter-balanced multi-term loss trained end-to-end. Experiments on standard Hamiltonian benchmarks show improved long-horizon forecasting, stronger short-term prediction accuracy, and higher physics conformity compared with state-of-the-art methods.

TL;DR: We present a method of probabilistic learning of different classes of Hamiltonian dynamics using a balanced multi-term loss function for conservation and stability.