Learning Stochastic Dynamical System via Flow Map Operator

TL;DR

This paper introduces sFML, a novel framework that extends flow map learning to stochastic dynamical systems by combining deterministic residual networks with generative adversarial networks to model stochastic evolution.

Contribution

The paper develops a new stochastic flow map learning framework that integrates ResNet and GANs, enabling data-driven modeling of unknown stochastic systems.

Findings

sFML effectively models various stochastic systems.

The method provides a weak distributional approximation.

Numerical examples demonstrate high flexibility and accuracy.

Abstract

We present a numerical framework for learning unknown stochastic dynamical systems using measurement data. Termed stochastic flow map learning (sFML), the new framework is an extension of flow map learning (FML) that was developed for learning deterministic dynamical systems. For learning stochastic systems, we define a stochastic flow map that is a superposition of two sub-flow maps: a deterministic sub-map and a stochastic sub-map. The stochastic training data are used to construct the deterministic sub-map first, followed by the stochastic sub-map. The deterministic sub-map takes the form of residual network (ResNet), similar to the work of FML for deterministic systems. For the stochastic sub-map, we employ a generative model, particularly generative adversarial networks (GANs) in this paper. The final constructed stochastic flow map then defines a stochastic evolution model that is a weak approximation, in term of distribution, of the unknown stochastic system. A comprehensive set of numerical examples are presented to demonstrate the flexibility and effectiveness of the proposed sFML method for various types of stochastic systems.