We introduce a machine-learning framework named statistics-informed neural network (SINN) for learning stochastic dynamics from data. This new architecture was theoretically inspired by a universal approximation theorem for stochastic systems, which we introduce in this paper, and the projection-operator formalism for stochastic modeling. We devise mechanisms for training the neural network model to reproduce the correct \emph{statistical} behavior of a target stochastic process. Numerical simulation results demonstrate that a well-trained SINN can reliably approximate both Markovian and non-Markovian stochastic dynamics. We demonstrate the applicability of SINN to coarse-graining problems and the modeling of transition dynamics. Furthermore, we show that the obtained reduced-order model can be trained on temporally coarse-grained data and hence is well suited for rare-event simulations.

翻译：我们引入了一个名为统计信息神经网络(SINN)的机学学习框架,用于从数据中学习随机动态。这个新架构的理论灵感来自我们在本文件中介绍的随机系统通用近似理论,以及随机模型的投影操作正规主义。我们设计了神经网络模型培训机制,以复制目标随机进程的正确行为。数字模拟结果表明,训练有素的SINN可以可靠地近似Markovian 和非Markovian 的随机动态。我们展示了SINN对粗重问题和转型动态模型的适用性。此外,我们展示了所获得减序模型可以接受时间粗略数据的培训,因此非常适合稀有活动的模拟。