Numerical study of fractional-order Ginzburg-Landau equations using fPINNs

Hong Lu; Yujie Zhai; Mingji Zhang; Hong Lu; Yujie Zhai; Mingji Zhang

doi:10.3934/math.20261178

AIMS Mathematics

2026, Volume 11, Issue 9: 29696-29726. doi: 10.3934/math.20261178

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Numerical study of fractional-order Ginzburg-Landau equations using fPINNs

1.
School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, China; ljwenling@163.com, 202317831@mail.sdu.edu.cn
2.
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

Received: 11 March 2026 Revised: 14 August 2026 Accepted: 28 August 2026 Published: 14 September 2026
MSC : 35R11, 65M99

In this work, we propose an efficient fractional-order physics-informed neural network (fPINN) approach to solve the complex fractional Ginzburg-Landau equations. To handle the temporal fractional derivatives, the $ L2-1_{\sigma} $ difference scheme is adopted, outperforming $L1$ and $ L1-2 $ schemes in accuracy. Spatial fractional derivatives are handled via the shifted Grünwald-Letnikov (G-L) formula, enabling high-precision nonlinear simulation. Supported by detailed numerical error analysis, our method provides a robust and reliable tool for modeling dissipative dynamical systems.
- fractional Ginzburg-Landau equations,
- fPINNs,
- $ L2-1_\sigma $ formula,
- shifted G-L formula,
- reverse-value functions,
- multi-output network
Citation: Hong Lu, Yujie Zhai, Mingji Zhang. Numerical study of fractional-order Ginzburg-Landau equations using fPINNs[J]. AIMS Mathematics, 2026, 11(9): 29696-29726. doi: 10.3934/math.20261178

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Abstract

In this work, we propose an efficient fractional-order physics-informed neural network (fPINN) approach to solve the complex fractional Ginzburg-Landau equations. To handle the temporal fractional derivatives, the $ L2-1_{\sigma} $ difference scheme is adopted, outperforming $L1$ and $ L1-2 $ schemes in accuracy. Spatial fractional derivatives are handled via the shifted Grünwald-Letnikov (G-L) formula, enabling high-precision nonlinear simulation. Supported by detailed numerical error analysis, our method provides a robust and reliable tool for modeling dissipative dynamical systems.

References

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