Learning Drift and Diffusion Terms in Stochastic Differential Equations Using Neural Networks

Abstract

This paper presents a comprehensive investigation into learning drift and diffusion terms in stochastic differential equations (SDEs) using neural network-based approaches. We explore three primary methodologies: Physics-Informed Neural Networks (PINNs), Neural Stochastic Differential Equations (Neural SDEs), and deep learning-based parameter estimation techniques. Our study encompasses both theoretical foundations and practical implementations, with extensive numerical experiments on benchmark problems including the Ornstein-Uhlenbeck process, geometric Brownian motion, and multi-dimensional diffusion processes. We develop a unified framework that combines automatic differentiation with stochastic numerical integration schemes to enable end-to-end learning of SDE parameters from observational data. Our results demonstrate that neural network approaches achieve comparable or superior accuracy to traditional statistical methods such as maximum likelihood estimation while offering greater flexibility for complex, high-dimensional problems. The proposed methods show particular promise for applications in financial modeling, physics simulations, and biological systems where the underlying dynamics are governed by stochastic processes. We provide open-source Python implementations and detailed experimental protocols to facilitate reproducibility and future research in this emerging field at the intersection of deep learning and stochastic calculus.