  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.
