A Stable Cubic B-Spline Scheme and Data-Driven Error Surrogate for the Time-Fractional Gray–Scott Model

This paper presents a numerical and data-driven study of the one-dimensional (1D) time-fractional Gray–Scott reaction–diffusion model (TFGSM), which is widely used to describe pattern formation in chemical and biological systems. A fully discrete scheme is developed by combining the Caputo fractional derivative (CFD) in time with cubic B-spline (CBS) collocation in space. The nonlinear reaction terms are linearized to obtain an efficient algebraic system at each time step, and a discrete stability analysis shows that the proposed method is stable in the associated energy norm. Numerical investigations demonstrate that the approach is accurate and reliable, with errors decreasing as fractional orders under temporal and spatial refinement. A machine learning-assisted surrogate model is presented to predict the reported numerical error behavior resulting from fractional order and discretization parameters, thereby supplementing the deterministic method. Furthermore, CBS feature mapping in machine learning is highlighted as a natural link between the numerical technique and the data-driven surrogate. The findings demonstrate that the presented method offers an effective tool for investigating the numerical performance of the TFGSM.