Discovery of Symbolic Hamiltonian Expressions with Buckingham-Symplectic Networks

TL;DR

BuSyNet is a novel deep learning architecture that discovers symbolic, energy-unit consistent Hamiltonian expressions by leveraging symplectic transformations and action-angle coordinates, improving interpretability and prediction stability.

Contribution

It introduces BuSyNet, which combines dimensional consistency and symplectic geometry to accurately recover symbolic Hamiltonians from trajectory data.

Findings

BuSyNet accurately recovers Hamiltonians for harmonic oscillator and Kepler problems.

It outperforms existing neural architectures in long-term prediction accuracy.

BuSyNet maintains interpretability of the learned Hamiltonian expressions.

Abstract

Hamiltonian systems lie at the heart of modeling the physical world. Their defining scalar, the Hamiltonian, encodes both energy conservation and symplectic geometry in its phase-space trajectories. Recent deep learning approaches model Hamiltonian systems by embedding their properties either in the architecture or in the loss function. However, they typically ignore that: i) a Hamiltonian carries units of energy and/or ii) that every integrable Hamiltonian admits a canonical transformation to action-angle coordinates in which the dynamics reduce to a simple rotation on an invariant torus. We propose BuSyNet, a deep learning architecture that combines these two constraints via a dimensionally-consistent, symplectic transformation. A symplectic layer maps input trajectories to lower-dimensional latent action-angle variables, which are then combined with system parameters to discover a symbolic Hamiltonian expression in units of energy. Evaluated on the harmonic oscillator and the Kepler two-body problem (in 2D and 3D), BuSyNet recovers concise, closed-form Hamiltonians that outperform state-of-the-art neural architectures in long-term prediction accuracy and stability, while maintaining interpretability.