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

• URL: http://arxiv.org/abs/2210.05087v2
• Date: Wed, 10 May 2023 13:55:10 GMT
• Title: Approximation of nearly-periodic symplectic maps via structure-preserving neural networks
• Authors: Valentin Duruisseaux, Joshua W. Burby, Qi Tang
• Abstract summary: 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. We construct a novel structure-preserving neural network to approximate nearly-periodic symplectic maps.
• Score: 0.3913403111891026
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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