Quantitative Analysis of the Fractional Fokker–Planck–Levy Equation via a Modified Physics-Informed Neural Network Architecture

Abstract

An innovative approach is utilized in this paper to solve the fractional Fokker–Planck–Levy (FFPL) equation. A hybrid technique is designed by combining the finite difference method (FDM), Adams numerical technique, and physics-informed neural network (PINN) architecture, namely, the FDM-APINN, to solve the fractional Fokker–Planck–Levy (FFPL) equation numerically. Two scenarios of the FFPL equation are considered by varying the value of (i.e., 1.75, 1.85). Moreover, three cases of each scenario are numerically studied for different discretized domains with 100, 200, and 500 points in x∈[−1, 1] and t∈[0, 1]. For the FFPL equation, solutions are obtained via the FDM-APINN technique via 1000, 2000, and 5000 iterations. The errors, loss function graphs, and statistical tables are presented to validate our claim that the FDM-APINN is a better alternative intelligent technique for handling fractional-order partial differential equations with complex terms. The FDM-APINN can be extended by using nongradient-based bioinspired computing for higher-order fractional partial differential equations.

Keywords

• fractional Fokker–Planck–Levy equation
• physics-informed neural networks (PINNs)
• finite difference method (FDM)
• Adams numerical technique
• hybrid numerical method
• fractional partial differential equations