Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama scheme

TL;DR

This paper introduces a neural network-based method for learning stochastic differential equations by integrating an SDE scheme directly into the model, enabling effective parameter estimation from data.

Contribution

It presents a novel data-driven approach that incorporates an SDE integration scheme within neural networks for improved learning of stochastic dynamical systems.

Findings

Successfully applied to geometric Brownian motion and stochastic Lorenz-63 models.

Outperforms gradient matching and non-stochastic neural network methods.

Demonstrates robustness in handling state-dependent stochastic components.

Abstract

Stochastic differential equations (SDEs) are one of the most important representations of dynamical systems. They are notable for the ability to include a deterministic component of the system and a stochastic one to represent random unknown factors. However, this makes learning SDEs much more challenging than ordinary differential equations (ODEs). In this paper, we propose a data driven approach where parameters of the SDE are represented by a neural network with a built-in SDE integration scheme. The loss function is based on a maximum likelihood criterion, under order one Markov Gaussian assumptions. The algorithm is applied to the geometric brownian motion and a stochastic version of the Lorenz-63 model. The latter is particularly hard to handle due to the presence of a stochastic component that depends on the state. The algorithm performance is attested using different simulations results. Besides, comparisons are performed with the reference gradient matching method used for non linear drift estimation, and a neural networks-based method, that does not consider the stochastic term.