Physics-Informed Neural Networks for Solving Stochastic Differential Equations - Rishav Jha; Kameshwar Sahani; Suresh Kumar Sahani; Ravi Kumar Raj; Dilip Kumar Sah (Journal of Multidisciplinary Science: MIKAILALSYS, 2026)
Stochastic differential equations (SDEs) are fundamental tools for modeling systems with inherent randomness across finance, physics, biology, and engineering; however, traditional numerical methods, including Monte Carlo simulations and Euler–Maruyama schemes, often face substantial computational challenges, particularly in high-dimensional settings. This study aims to develop and evaluate a comprehensive Physics-Informed Neural Networks (PINNs) framework as an efficient approach for solving SDEs. The proposed methodology embeds physical laws and stochastic dynamics directly into the neural network architecture through a carefully designed loss function that incorporates PDE residuals, initial conditions, and boundary constraints. The framework was evaluated through extensive numerical experiments on benchmark problems, including geometric Brownian motion, Ornstein–Uhlenbeck processes, and multi-dimensional stochastic systems. The findings indicate that PINNs achieve accuracy comparable to traditional numerical methods while providing substantial computational advantages, especially for parametric studies and real-time applications. Across all test cases, relative L2 errors consistently remained below 2%, and computational speedups reached up to 1000 times compared with Monte Carlo methods after network training. The proposed framework also demonstrated strong scalability for higher-dimensional problems, addressing the curse of dimensionality commonly associated with conventional numerical approaches. This study concludes that PINNs offer a promising paradigm for the efficient and accurate solution of stochastic differential equations. Its contribution lies in advancing scientific machine learning for stochastic modeling and opening further opportunities for uncertainty quantification and real-time computational applications.