Solving Fokker-Planck equation using deep learning.

Area of Science:

• Computational Physics
• Applied Mathematics
• Machine Learning

Background:

• Stochastic differential equations (SDEs) are fundamental in modeling complex systems.
• The Fokker-Planck (FP) equation governs the evolution of probability density functions for SDEs.
• Traditional numerical methods for FP equations can be computationally intensive and require complex transformations.

Purpose of the Study:

• To develop a novel deep learning-based method for solving general Fokker-Planck equations.
• To address limitations of traditional numerical techniques, such as interpolation and coordinate transformations.
• To enhance the accuracy and feasibility of machine learning for FP equation solutions.

Main Methods:

• A deep neural network framework is employed to solve the FP equation.
• Novelty includes the introduction of penalty factors to mitigate local optimization issues.
• A normalization condition is incorporated as a supervision technique to prevent trivial solutions.

Main Results:

• The proposed deep learning algorithm demonstrates feasibility and effectiveness across one-, two-, and three-dimensional systems.
• Numerical examples validate the performance of the machine learning approach.
• The study analyzes the impact of network architecture (hidden layers), penalty factors, and optimization algorithms on performance.

Conclusions:

• Deep learning offers a powerful and effective alternative for solving Fokker-Planck equations.
• Careful construction of neural networks and appropriate use of penalty factors and normalization are crucial for optimal performance.
• The findings suggest significant potential for machine learning in computational physics and applied mathematics.