Solving the Fokker-Planck Equation with Neural Networks: A Performance Improvement Approach

Document Type : Research Article

Authors

1 Department of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

2 Department of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran; Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, Tehran, Iran

Abstract

The Fokker-Planck equation models the evolution of probability densities in fields such as physics, biology, and finance. Traditional numerical methods for solving this equation can be computationally expensive and struggle with complex, high-dimensional problems. In this work, we propose a Physics-Informed Neural Networks (PINNs) approach to efficiently approximate solutions of the Fokker-Planck equation. Our method employs a fully connected feedforward neural network using two activation functions--Tanh and SiLU--with fixed learning rates (0.001 for Tanh and 0.01 for SiLU) and varied spatial discretization. The loss function is designed to enforce the governing differential equation as well as the initial and boundary conditions. Experimental results, evaluated using standard error metrics (RMS, Relative $L_2$-Norm Error, and MAE), demonstrate that our PINN approach achieves competitive accuracy with improved convergence and lower computational costs compared to traditional methods. This study underscores the potential of neural network-based solvers for complex differential equations and sets the stage for future optimization.

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Analytical and Numerical Solutions for Nonlinear Equations
Volume 9, Issue 1
May 2024
Pages 1-11

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History

  • Receive Date: 27 February 2025
  • Revise Date: 09 April 2025
  • Accept Date: 16 April 2025
  • Publish Date: 20 April 2025

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