Document Type : Research article
Authors
1
Department of Mathematics, Kongu Engineering College, Erode, India
2
Canadian Quantum Research Center, 106-460 Doyle Ave, Kelowna, British Columbia V1Y 0C2, Canada
3
Department of Statistics, University of Mazandaran, Babolsar, Iran
4
Payame Noor University (PNU), Tehran, Iran
5
Department of Physics, Sar.C., Islamic Azad University, Sari, Iran
6
Department of Mathemstics, University of Mazandaran, Babolsar, Iran
Abstract
Nonlinear weakly singular integral equations are challenging to solve numerically because the singular behaviour of the kernel can reduce the effectiveness of conventional uniform discretization and physics-informed learning strategies. This study develops a Self-Refining Kernel-Aware Physics-Informed Neural Network (SKA-PINN) for nonlinear weakly singular Fredholm and Volterra integral equations. The proposed framework combines a kernel-aware difficulty indicator based on residual magnitude, residual variation, and kernel complexity with adaptive residual weighting, self-refining collocation, and adaptive quadrature. Instead of distributing computational points uniformly, the method identifies numerically difficult regions and progressively increases the local resolution during training. The framework is evaluated using nonlinear Fredholm and Volterra benchmark problems with prescribed analytical solutions. For the Fredholm problem, the maximum reported absolute and relative errors are $6.3\times10^{-6}$ and $3.4\times10^{-6}$, respectively. For the Volterra problem, the maximum absolute error is $7.5\times10^{-6}$, with a maximum relative error of $1.2\times10^{-4}$. The refinement experiments show a consistent reduction in the maximum residual and solution error as the collocation and quadrature resolutions increase. In comparison with the classical and adaptive PINN configurations considered, SKA-PINN achieves the smallest reported $L^2$ error of $1.09\times10^{-5}$ while requiring fewer training epochs and lower CPU time. The ablation results further show that kernel-aware weighting and self-refinement both contribute to the overall improvement. These results indicate that incorporating kernel information directly into the learning and refinement process provides an effective computational strategy for nonlinear weakly singular integral equations.
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