Adaptive deep density approximation for stochastic dynamical systems

TL;DR

This paper introduces an adaptive deep neural network method, tKRnet, for efficiently approximating probability densities in stochastic dynamical systems, overcoming high-dimensional challenges and enabling long-time integration.

Contribution

The paper proposes a novel adaptive deep neural network approach, tKRnet, that explicitly models densities for stochastic systems and incorporates an adaptive sampling strategy to improve accuracy and efficiency.

Findings

tKRnet effectively approximates PDFs in high-dimensional stochastic systems.

The adaptive sampling procedure enhances training efficiency and accuracy.

Numerical examples demonstrate superior performance over traditional methods.

Abstract

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.