PINNs for Stochastic Dynamics: Modeling Brownian Motion via Verlet Integration

This study presents a Physics-Informed Neural Network (PINN) framework for modeling stochastic systems like Brownian motion, designed to overcome critical challenges in physical consistency and numerical stability that affect classical solvers and standard data-driven models. Traditional numerical methods often struggle with high-dimensional spaces or sparse data, while many machine learning approaches fail to enforce fundamental physical laws. To address this, our proposed PINN architecture integrates a multi-component loss function that explicitly enforces the Fokker-Planck equation, which describes the system’s governing physics, alongside boundary conditions and a global probability conservation law. This physics-informed approach is anchored by high-fidelity training data generated from Verlet-integrated trajectories of the underlying Langevin dynamics. We validate our model against the analytical solution for one-dimensional Brownian motion, demonstrating its ability to accurately recover the true probability density function (PDF). Rigorous comparisons using statistical metrics show superior accuracy over a canonical data-driven operator learning model, DeepONet. Specifically, our PINN achieves a relative L2 error of 5.66% and maintains probability normalization within a 0.03% tolerance, significantly outperforming DeepONet’s 32.46% error and 3.2% probability deviation. Furthermore, a recursive error-bounding technique provides quantifiable confidence in the model’s predictions. While validated in a low-dimensional system, our framework demonstrates a promising and robust methodology for problems in fields like soft matter physics and financial modeling, where both physical consistency and data-driven flexibility are crucial. We also provide a transparent analysis of the model’s computational trade-offs, positioning this physics-informed approach as a reliable tool for complex scientific applications.