Neuro-Symbolic Approach for Discovering Completely Separable Hamiltonian Equations

Abstract

A central challenge in uncovering Hamiltonian dynamics from scientific data is the presence of strong nonlinear behavior, which often leads to complex dynamics. These nonlinearities arise due to the coupling between generalized momentum p and generalized coordinates q. Recent advances in data-driven modeling have attracted research efforts to combine this field with discovering the analytic expressions for the dynamics of complex Hamiltonian systems. These efforts have achieved outstanding results in modeling the dynamics of both separable and nonseparable Hamiltonian systems, but there is little to no research when it comes to learning a canonical transformation that converts a nonseparable system into a completely separable one. In this work, we propose SympNeuroSR, a novel architecture capable of transforming nonseparable Hamiltonian systems into totally separable ones, while simultaneously discovering their analytical equations of motion. We demonstrate the effectiveness and robustness of the proposed approach on several nonseparable Hamiltonian systems, showing accurate Hamiltonian recovery, energy conservation, and reliable trajectory reconstruction even under substantial additive noise.