Hybrid ARFIMA–LSTM Framework for Improving Numerical Solutions of Time-Fractional Partial Differential Equations

Source abstract

Fractional partial differential equations (FPDEs) have become strong mathematical tools for the modeling of complex physical phenomena with memory effects and/or with anomalous diffusion. The traditional numerical solutions to FPDEs, however, have some inbuilt discretization errors that restrict the accuracy of the solutions. In this paper, a new hybrid model is proposed, based on the Autoregressive Fractionally Integrated Moving Average (ARFIMA) model and Long Short-Term Memory (LSTM) neural network, to correct the numerical errors in the solution of a time-fractional diffusion equation. A finite difference scheme (FDS), namely L1, is applied to the uniform grid to discretize the Caputo fractional derivative of order α = 0.5. The exact solution in terms of the Mittag-Leffler function is used to validate the numerical solution, obtained by using the stable identity E₁/₂(z) = e^(z²) erfc(-z). Residual errors between numerical and exact solutions are modeled by ARFIMA to embrace long memory structures, and LSTM networks learn the remaining nonlinearities. At the center x = 0.5 point, the hybrid correction reduces the root mean square error (RMSE) from 7.976875 × 10⁻³ to 7.528688 × 10⁻³, a significant improvement of about 5.6% compared to the original numerical method. The proposed framework shows the effectiveness of the statistical time-series model and deep learning model to solve the fractional PDE problem, providing a new idea for high-accuracy computational fractional calculus.