Detecting Stochastic Governing Laws with Observation on Stationary Distributions

TL;DR

This paper presents a neural network approach to infer stochastic differential equations from stationary distribution data, enabling modeling of complex systems with uncertainties.

Contribution

It introduces a novel loss function based on Hellinger distance for learning drift and diffusion terms from stationary distributions.

Findings

Accurately learns stochastic differential equations from stationary data.

Demonstrates effectiveness through numerical experiments.

Provides a new tool for modeling uncertain complex systems.

Abstract

Mathematical models for complex systems are often accompanied with uncertainties. The goal of this paper is to extract a stochastic differential equation governing model with observation on stationary probability distributions. We develop a neural network method to learn the drift and diffusion terms of the stochastic differential equation. We introduce a new loss function containing the Hellinger distance between the observation data and the learned stationary probability density function. We discover that the learnt stochastic differential equation provides a fair approximation of the data-driven dynamical system after minimizing this loss function during the training method. The effectiveness of our method is demonstrated in numerical experiments.