Symplectic Neural Networks Based on Dynamical Systems

TL;DR

This paper introduces Symplectic Neural Networks (SympNets) based on geometric integrators, which are highly expressive, accurate, and interpretable models for Hamiltonian systems, with applications in Hamiltonian regression.

Contribution

The paper develops SympNets as universal approximators for Hamiltonian diffeomorphisms, providing a representation theory for linear systems and demonstrating superior performance over existing methods.

Findings

SympNets can exactly parameterize quadratic Hamiltonian maps.

Numerical tests show SympNets outperform existing architectures in accuracy.

SympNets enable symbolic Hamiltonian regression for polynomial systems.

Abstract

We present and analyze a framework for designing symplectic neural networks (SympNets) based on geometric integrators for Hamiltonian differential equations. The SympNets are universal approximators in the space of Hamiltonian diffeomorphisms, interpretable and have a non-vanishing gradient property. We also give a representation theory for linear systems, meaning the proposed P-SympNets can exactly parameterize any symplectic map corresponding to quadratic Hamiltonians. Extensive numerical tests demonstrate increased expressiveness and accuracy -- often several orders of magnitude better -- for lower training cost over existing architectures. Lastly, we show how to perform symbolic Hamiltonian regression with SympNets for polynomial systems using backward error analysis.