We present a numerical method for learning unknown nonautonomous stochastic
dynamical system, i.e., stochastic system subject to time dependent excitation
or control signals. Our basic assumption is that the governing equations for
the stochastic system are unavailable. However, short bursts of input/output
(I/O) data consisting of certain known excitation signals and their
corresponding system responses are available. When a sufficient amount of such
I/O data are available, our method is capable of learning the unknown dynamics
and producing an accurate predictive model for the stochastic responses of the
system subject to arbitrary excitation signals not in the training data. Our
method has two key components: (1) a local approximation of the training I/O
data to transfer the learning into a parameterized form; and (2) a generative
model to approximate the underlying unknown stochastic flow map in
distribution. After presenting the method in detail, we present a comprehensive
set of numerical examples to demonstrate the performance of the proposed
method, especially for long-term system predictions.

我们提出了一种用于学习未知的非自治随机动力系统的数值方法，该系统受到时间依赖的激励或控制信号的影响。我们的方法的基本假设是随机系统的控制方程不可用，但是可以获得一些由已知激励信号和相应系统响应组成的输入 / 输出（I/O）数据。当具有足够数量的 I/O 数据时，我们的方法能够学习未知的动力学，并为系统对于不在训练数据中的任意激励信号的随机响应生成准确的预测模型。我们的方法包含两个关键组成部分：（1）对训练 I/O 数据的局部逼近以将学习转化为参数化形式；（2）用于近似分布的生成模型来近似未知随机流图。在详细介绍该方法之后，我们提供了一套全面的数值示例来展示所提出方法的性能，尤其是用于长期系统预测。