We present a numerical method to learn an accurate predictive model for an unknown stochastic dynamical system from its trajectory data. The method seeks to approximate the unknown flow map of the underlying system. It employs the idea of autoencoder to identify the unobserved latent random variables. In our approach, we design an encoding function to discover the latent variables, which are modeled as unit Gaussian, and a decoding function to reconstruct the future states of the system. Both the encoder and decoder are expressed as deep neural networks (DNNs). Once the DNNs are trained by the trajectory data, the decoder serves as a predictive model for the unknown stochastic system. Through an extensive set of numerical examples, we demonstrate that the method is able to produce long-term system predictions by using short bursts of trajectory data. It is also applicable to systems driven by non-Gaussian noises.

翻译：本文提出了一种数值方法，用于从轨迹数据中学习未知随机动态系统的精确预测模型。该方法旨在逼近底层系统的未知流映射，并利用自编码器的思想识别未观测的潜在随机变量。在我们的方案中，我们设计了一个编码函数来发现潜在变量（将其建模为单位高斯分布），以及一个解码函数来重构系统的未来状态。编码器和解码器均以深度神经网络（DNNs）实现。当深度神经网络通过轨迹数据训练完成后，解码器便作为未知随机系统的预测模型。通过大量数值算例，我们证明了该方法能够利用短时轨迹数据生成长期系统预测，且同样适用于非高斯噪声驱动的系统。