A Singularity-İnferred Operator-Consistent Fractional PINN for One- and Two-Dimensional Third-Kind Caputo–Hadamard Auto-Convolution Volterra Equations with Weakly Singular Kernels

Abstract

We propose a singularity-inferred operator-consistent fractional physics-informed neural network for one- and two-dimensional third-kind Caputo–Hadamard auto-convolution Volterra equations with weakly singular kernels. The logarithmic change of variable transforms the Caputo–Hadamard derivative into a Caputo derivative, and a derivative-free Gauss–Jacobi rule is used for its numerical evaluation. The trial function contains a trainable endpoint coordinate, a polynomial term, cubic polyharmonic-spline terms, and a smooth correction. The same endpoint coordinate is used in the fitted quadrature rules. The numerical examples use an independent integral residual and higher-order evaluations of the Caputo and auto-convolution operators. In the controlled one-dimensional comparison, the proposed approximation gives the smallest mean validation error, and the two-dimensional example recovers the predicted endpoint scale.