Multiscale stochastic dynamical systems have been widely adopted to scientific and engineering problems due to their capability of depicting complex phenomena in many real world applications. This work is devoted to investigating the effective reduced dynamics for a slow-fast stochastic dynamical system. Given observation data on a short-term period satisfying some unknown slow-fast stochastic system, we propose a novel algorithm including a neural network called Auto-SDE to learn invariant slow manifold. Our approach captures the evolutionary nature of a series of time-dependent autoencoder neural networks with the loss constructed from a discretized stochastic differential equation. Our algorithm is also proved to be accurate, stable and effective through numerical experiments under various evaluation metrics.

翻译：多尺度随机动力系统因其能够刻画许多实际应用中的复杂现象，已被广泛运用于科学与工程问题。本文致力于研究慢-快随机动力系统的有效降阶动力学。给定满足某未知慢-快随机系统的短期观测数据，我们提出一种包含名为Auto-SDE神经网络的新算法，用于学习不变的慢流形。我们的方法通过一系列时间依赖的自编码器神经网络，结合离散化随机微分方程构造的损失函数，捕捉了系统的演化本质。通过多种评价指标下的数值实验，该算法被证明具有准确性、稳定性和有效性。