A Caputo Fractional PINN Framework for Fractional Gardner and Wound-Healing Models

Hafsa Khan, Muhammad Sulaiman, Izaz Ur Rahman, Mohammed Abdullah Salman, Ghaylen Laouini and Naveed Ahmad Khan

Journal of Mathematics, 2026, vol. 2026, 1-29

Abstract: Systems with memory, hereditary effects, and nonlocal temporal dynamics are naturally described by the fractional partial differential equations (FPDEs). However, the history-dependent operators in FPDEs pose significant challenges for traditional numerical and physics-informed methods. This study introduces a Caputo fractional physics-informed neural network (PINN-CFA) framework to solve nonlinear time-fractional Gardner and wound-healing models. The main mathematical novelty of this work is the explicit use of a full-memory, Caputo-consistent Grünwald–Letnikov history operator in the physics-informed residual. The fractional residual is dependent on all the earlier neural network states; hence, there is no short-memory truncation in the Caputo operator. This nonlocal temporal convolution is coupled with automatic differentiation of the local spatial derivatives, and the governing equation, initial conditions, and boundary conditions are embedded into a single optimization problem. The applicability is explored for two classes of FPDEs: one- and two-dimensional wound-healing equations with diffusion and convection terms and the nonlinear dispersive Gardner equation, which features quadratic and cubic nonlinearities and a third-order spatial derivative. For the integer-order case, a comparison to the available exact solutions shows good agreement, and a residual evaluation for the fractional orders shows stability of the approximations. Under the same network architectures and training conditions, the proposed GL-based loss yields lower relative L2 errors for the problems studied, smaller, more spatially uniform PDE residuals, and smoother loss convergence than the implemented L1 loss. The GL residuals are less than 10−2 for the fractional Gardner equation over the range of reported final times. Refinement of the temporal history grid from N=50 to N=100 decreases the mean squared error by about 40%. The results illustrate that the discretization of the fractional operator is a fundamental part of the PINN residual and can significantly affect the accuracy, optimization performance, and compliance with the governing physical laws. PINN-CFA then offers a common computational framework for problems in biological transport and nonlinear waves with Caputo fractional dynamics.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jjmath:8302968

DOI: 10.1155/jom/8302968

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