Learning PINN-Based Coupled Fractional Systems With Memory and Nonlocal Dynamics Using Neural Computing

Muhammad Saad, Muhammad Sulaiman, Izaz Ur Rahman, Fahad Sameer Alshammari, Ghaylen Laouini and Mohammed Abdullah Salman

Journal of Mathematics, 2026, vol. 2026, 1-20

Abstract: This manuscript examines a reformulation-based benchmarking system to assess the performance of Physics-Informed Neural Networks (PINNs) to address the coupled system of nonlinear fractional partial differential equations (fPDEs). The proposed method bypasses the incompatibility between automatic differentiation and nonlocal fractional operators by calculating analytically Caputo fractional derivative terms by using known exact solutions and embedding them as modified source terms. The resulting transformation creates the original fPDE system into an equivalent integer-order formulation (valid specifically for benchmarking problems with known analytical solutions) and permits the standard PINN architectures to be used, without depending on unstable or computationally costly numerical approximations of fractional derivatives. This reformulation purposely takes off the fractional operator of the training loop, serving as a high-precision benchmarking tool that isolates neural-network approximation error from fractional-operator discretization error. The paradigm is tested across three benchmark problems using power-law, polynomial, and hyperbolic exact solutions with fractional orders ranging from 1.3 to 1.5. Numerical performance is characterized by high predictive accuracy, with mean absolute errors ranging between 10−3 and 10−4, and convergent behavior is stable at a training less than 10−4 after 15,000 epochs. The main contribution of this work is not a general-purpose fPDE solver but a high-precision validation tool that disconnects the approximation properties of the neural network from the effects of the complexities of the fractional calculus, which determines a predictable performance standard in PINNs on the coupled fractional equations. The paper clearly indicates the limitations of the method, such as its reliance on known analytical solutions, and provides future directions on how to perform generalization through numerical fractional operators or neural approximations. This research provides a fundamental base for the development and verification of the mesh-free solvers of memory-dependent problems in areas like viscoelasticity, biophysics, and finance.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jjmath:7301135

DOI: 10.1155/jom/7301135

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