Adaptive Hybrid B-Spline PINN for Fractional Equations

Abstract

We propose a hybrid numerical framework combining a redefined quintic B-spline collocation method with a Physics-Informed Neural Network (PINN) correction layer for solving nonlinear space-time fractional advection-diffusion-reaction equations. The spatial fractional derivative is discretised via a shifted Grünwald–Letnikov approximation embedded in the B-spline basis, achieving high-order spatial accuracy. The temporal Caputo derivative is handled by an L2-1σ Crank–Nicolson scheme surpassing standard methods. A PINN residual corrector suppresses oscillations in high-Péclet-number regimes without mesh refinement. Adaptive time-stepping reduces computational cost by over 60%. Rigorous stability and convergence analysis are provided. Seven benchmark problems confirm the framework outperforms existing quintic B-spline, spectral, and PINN-only methods by one to two orders of magnitude.