Symplectic Learning for Hamiltonian Neural Networks

TL;DR

This paper introduces an improved symplectic training method for Hamiltonian Neural Networks that leverages the mathematical structure of Hamiltonian systems, enhancing their accuracy, explainability, and ability to recover true Hamiltonians from data.

Contribution

It proposes a novel symplectic loss function for HNNs, guarantees the existence of an exact Hamiltonian, and introduces a post-training correction method for better Hamiltonian recovery.

Findings

Enhanced training stability and accuracy for HNNs.

Mathematical guarantees of learning an exact Hamiltonian.

Effective post-training correction for Hamiltonian recovery.

Abstract

Machine learning methods are widely used in the natural sciences to model and predict physical systems from observation data. Yet, they are often used as poorly understood "black boxes," disregarding existing mathematical structure and invariants of the problem. Recently, the proposal of Hamiltonian Neural Networks (HNNs) took a first step towards a unified "gray box" approach, using physical insight to improve performance for Hamiltonian systems. In this paper, we explore a significantly improved training method for HNNs, exploiting the symplectic structure of Hamiltonian systems with a different loss function. This frees the loss from an artificial lower bound. We mathematically guarantee the existence of an exact Hamiltonian function which the HNN can learn. This allows us to prove and numerically analyze the errors made by HNNs which, in turn, renders them fully explainable. Finally, we present a novel post-training correction to obtain the true Hamiltonian only from discretized observation data, up to an arbitrary order.