Solving the non-local Fokker-Planck equations by deep learning.

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Machine Learning

Background:

• Physics-informed neural networks (PiNNs) are effective for solving partial differential equations (PDEs).
• Fractional calculus extends differential equations to non-integer orders, enabling modeling of complex phenomena.
• Accurate numerical evaluation of fractional operators is crucial for solving fractional PDEs.

Purpose of the Study:

• Introduce trapz-PiNNs, a novel approach combining PiNNs with a modified trapezoidal rule.
• Solve space-fractional Fokker-Planck equations in 2D and 3D using the proposed method.
• Analyze and improve the performance of trapz-PiNNs using local error metrics.

Main Methods:

• Developed a modified trapezoidal rule for accurate fractional Laplacian evaluation.
• Integrated the trapezoidal rule into physics-informed neural networks (PiNNs) to create trapz-PiNNs.
• Verified the second-order accuracy of the modified trapezoidal rule.
• Applied trapz-PiNNs to solve 2D and 3D space-fractional Fokker-Planck equations.

Main Results:

• Demonstrated high expressive power of trapz-PiNNs with low L2 relative errors across various numerical examples.
• Verified the second-order accuracy of the modified trapezoidal rule.
• Analyzed local metrics (point-wise absolute and relative errors) for performance assessment.
• Presented a method to enhance trapz-PiNN performance using physical observations or high-fidelity simulations.

Conclusions:

• trapz-PiNNs offer an accurate and effective method for solving space-fractional Fokker-Planck equations.
• The modified trapezoidal rule provides a robust way to handle fractional Laplacians in neural networks.
• The method shows potential for generalization to higher dimensions and other domain types.