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Peer-reviewedCC BY-NC-NDAnnals of Mathematics and Computer Science

Adaptive Hybrid B-Spline PINN for Fractional Equations

  • Balaji Padhy1
  • Prabhu Sankardarshi Mahapatra1
  • Tumbanath Samantara1
  1. 1Department of Mathematics, Centurion University of Technology and Management, Odisha, India
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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.

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Cite

BibTeXRIS
BibTeX
@article{padhy2026adaptive,
  title = {{Adaptive Hybrid B-Spline PINN for Fractional Equations}},
  author = {Balaji Padhy and Prabhu Sankardarshi Mahapatra and Tumbanath Samantara},
  journal = {Annals of Mathematics and Computer Science},
  year = {2026},
  month = jul,
  doi = {10.56947/amcs.v35.864},
  url = {https://doi.org/10.56947/amcs.v35.864},
  note = {CC BY-NC-ND},
}
RIS
TY  - JOUR
TI  - Adaptive Hybrid B-Spline PINN for Fractional Equations
AU  - Balaji Padhy
AU  - Prabhu Sankardarshi Mahapatra
AU  - Tumbanath Samantara
PY  - 2026
DA  - 2026/07/21
JO  - Annals of Mathematics and Computer Science
DO  - 10.56947/amcs.v35.864
UR  - https://doi.org/10.56947/amcs.v35.864
AB  - 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.
SN  - 2789-7206
ER  -