Approximation of nearly-periodic symplectic maps via structure-preserving neural networks

TL;DR

This paper introduces a novel neural network architecture called symplectic gyroceptron that accurately models nearly-periodic symplectic maps, preserving key geometric properties and invariants for long-term stability in non-dissipative dynamical systems.

Contribution

The paper presents the symplectic gyroceptron, a structure-preserving neural network that approximates nearly-periodic symplectic maps while maintaining symplectic structure and adiabatic invariants.

Findings

Ensures the surrogate map is nearly-periodic and symplectic.

Provides long-time stability and preserves adiabatic invariants.

Automatically steps over short timescales without spurious instabilities.

Abstract

A continuous-time dynamical system with parameter $\varepsilon$ is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as $\varepsilon$ approaches 0. Nearly-periodic maps are discrete-time analogues of nearly-periodic systems, defined as parameter-dependent diffeomorphisms that limit to rotations along a circle action, and they admit formal $U(1)$ symmetries to all orders when the limiting rotation is non-resonant. For Hamiltonian nearly-periodic maps on exact presymplectic manifolds, the formal $U(1)$ symmetry gives rise to a discrete-time adiabatic invariant. In this paper, we construct a novel structure-preserving neural network to approximate nearly-periodic symplectic maps. This neural network architecture, which we call symplectic gyroceptron, ensures that the resulting surrogate map is nearly-periodic and symplectic, and that it gives rise to a discrete-time adiabatic invariant and a long-time stability. This new structure-preserving neural network provides a promising architecture for surrogate modeling of non-dissipative dynamical systems that automatically steps over short timescales without introducing spurious instabilities.