International Journal of Information Technology and Computer Science (IJITCS)
IJITCS Vol. 18, No. 2, 8 Apr. 2026
Cover page and Table of Contents: PDF (size: 1575KB)
PINNs for Stochastic Dynamics: Modeling Brownian Motion via Verlet Integration
PDF (1575KB), PP.126-145
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Author(s)
Yulison Herry
1,*
Julian Evan
2
Jeremia Oktavian
3
Ferry Faizal
2
Index Terms
Physics-informed Neural Networks, Brownian Motion, Verlet Integration, Fokker-planck Equation, Numerical Stability, Stochastic Dynamics
Abstract
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.
Cite This Paper
Yulison Herry, Julian Evan, Jeremia Oktavian, Ferry Faizal, "PINNs for Stochastic Dynamics: Modeling Brownian Motion via Verlet Integration", International Journal of Information Technology and Computer Science(IJITCS), Vol.18, No.2, pp.126-145, 2026. DOI:10.5815/ijitcs.2026.02.08
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