Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama scheme
This work addresses the problem of modeling complex dynamical systems with stochastic components for researchers in computational science, but it appears incremental as it builds on existing neural network methods by incorporating SDE integration.
The authors tackled the challenge of learning stochastic differential equations (SDEs) by proposing a neural network approach with a built-in SDE integration scheme, achieving performance demonstrated through simulations on geometric Brownian motion and a stochastic Lorenz-63 model.
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.