In this paper we consider adaptive deep neural network approximation for stochastic dynamical systems. Based on the Liouville equation associated with the stochastic dynamical systems, a new temporal KRnet (tKRnet) is proposed to approximate the probability density functions (PDFs) of the state variables. The tKRnet gives an explicit density model for the solution of the Liouville equation, which alleviates the curse of dimensionality issue that limits the application of traditional grid based numerical methods. To efficiently train the tKRnet, an adaptive procedure is developed to generate collocation points for the corresponding residual loss function, where samples are generated iteratively using the approximate density function at each iteration. A temporal decomposition technique is also employed to improve the long-time integration. Theoretical analysis of our proposed method is provided, and numerical examples are presented to demonstrate its performance.

翻译：本文研究随机动力系统的自适应深度神经网络近似方法。基于随机动力系统相关的Liouville方程，提出了一种新型时间域KRnet（tKRnet）来逼近状态变量的概率密度函数。tKRnet为Liouville方程的解提供了显式密度模型，有效缓解了传统网格数值方法因维数灾难而受限的问题。为高效训练tKRnet，我们开发了一种自适应程序来生成对应残差损失函数的配置点，该程序在每次迭代中利用近似密度函数迭代生成样本。同时采用时间分解技术改善长时间积分效果。本文对所提方法进行了理论分析，并通过数值算例验证了其性能。