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.

翻译：一个具有参数$\varepsilon$的连续时间动力系统被称为近周期系统，若当$\varepsilon$趋近于0时，其所有轨迹均为周期性的且角频率处处非零。近周期映射是近周期系统的离散时间类比，定义为依赖于参数的微分同胚，其极限为沿圆作用的旋转，且在非共振极限旋转情形下，系统在任意阶形式上都具备$U(1)$对称性。对于精确预辛流形上的哈密顿近周期映射，形式$U(1)$对称性导出一个离散时间绝热不变量。本文构造了一种新型结构保持神经网络来逼近近周期辛映射。该神经网络架构——称为辛陀螺感知器——确保代理映射具有近周期性与辛结构，并由此产生离散时间绝热不变量及长期稳定性。这种新型结构保持神经网络为非耗散动力系统的代理建模提供了有前景的架构，能够自动跨越短时间尺度而不引入虚假不稳定性。