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Fractal Fract
Volume 9
Issue 12
10.3390/fractalfract9120773
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Solving the Fractional Allen–Cahn Equation and the Fractional Cahn–Hilliard Equation with the Fractional Physics-Informed Neural Networks

by
Xiaorong Kang
1,
Yang Li
1,
Yongzheng Li
2,
Jinsong Hu
3 and
Kelong Zheng
2,*
1
School of Mathematics and Physics, Southwest University of Science and Technology, Mianyang 621010, China
2
Faculty of Science, Civil Aviation Flight University of China, Guanghan 618307, China
3
College of Big Data and Artificial Intelligence, Chengdu Technological University, Chengdu 611730, China
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(12), 773; https://doi.org/10.3390/fractalfract9120773
Submission received: 23 October 2025 / Revised: 20 November 2025 / Accepted: 24 November 2025 / Published: 26 November 2025
(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs, Second Edition)
Versions Notes

Abstract

In this paper, we present a computational framework based on fractional Physics-Informed Neural Networks combined with the L1 approximation of the Caputo fractional derivative to solve the fractional Allen–Cahn and fractional Cahn–Hilliard equations. Considering the limitation of the original fractional Physics-Informed Neural Networks in achieving high accuracy when applied to these highly stiff fractional equations, we propose three improved optimization strategies: adaptive non-uniform sampling, adaptive exponential moving average ratio loss weighting, and two–stage adaptive quasi optimization. By combining these strategies, three improved fPINNs algorithms are developed: f–A–PINNs, f–A–A–PINNs, and f–A–T–PINNs. Numerical experiments demonstrate that the f–A–T–PINNs algorithm achieves superior computational accuracy and improved parameter stability compared to the other algorithms.
Keywords: fractional Physics-Informed Neural Networks; Caputo fractional derivative; fractional Allen–Cahn equation; fractional Cahn–Hilliard equation; optimization strategies fractional Physics-Informed Neural Networks; Caputo fractional derivative; fractional Allen–Cahn equation; fractional Cahn–Hilliard equation; optimization strategies

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MDPI and ACS Style

Kang, X.; Li, Y.; Li, Y.; Hu, J.; Zheng, K. Solving the Fractional Allen–Cahn Equation and the Fractional Cahn–Hilliard Equation with the Fractional Physics-Informed Neural Networks. Fractal Fract. 2025, 9, 773. https://doi.org/10.3390/fractalfract9120773

AMA Style

Kang X, Li Y, Li Y, Hu J, Zheng K. Solving the Fractional Allen–Cahn Equation and the Fractional Cahn–Hilliard Equation with the Fractional Physics-Informed Neural Networks. Fractal and Fractional. 2025; 9(12):773. https://doi.org/10.3390/fractalfract9120773

Chicago/Turabian Style

Kang, Xiaorong, Yang Li, Yongzheng Li, Jinsong Hu, and Kelong Zheng. 2025. "Solving the Fractional Allen–Cahn Equation and the Fractional Cahn–Hilliard Equation with the Fractional Physics-Informed Neural Networks" Fractal and Fractional 9, no. 12: 773. https://doi.org/10.3390/fractalfract9120773

APA Style

Kang, X., Li, Y., Li, Y., Hu, J., & Zheng, K. (2025). Solving the Fractional Allen–Cahn Equation and the Fractional Cahn–Hilliard Equation with the Fractional Physics-Informed Neural Networks. Fractal and Fractional, 9(12), 773. https://doi.org/10.3390/fractalfract9120773

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