Approximation of nearly-periodic symplectic maps via structure-preserving neural networks
This work addresses surrogate modeling for non-dissipative dynamical systems, offering a method to automatically step over short timescales, which is incremental as it builds on existing structure-preserving neural network techniques for specific Hamiltonian systems.
The authors tackled the problem of approximating nearly-periodic symplectic maps, which are discrete-time dynamical systems with Hamiltonian properties, by developing a novel structure-preserving neural network called symplectic gyroceptron, resulting in a surrogate map that maintains symplecticity, preserves a discrete-time adiabatic invariant, and ensures long-time stability without spurious instabilities.
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.