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<p>We present the first comprehensive benchmark revealing that discretization strategy selection in Physics-Informed Neural Networks (PINNs) profoundly impacts performance for Fredholm integral equations of the second kind. Our kernel-adaptive framework systematically compares discrete coordinate, endpoint, and midpoint methods across smooth, singular, and regularized kernels. Key findings: discrete coordinate methods achieve 35% error reduction for smooth kernels through integral structure preservation, endpoint methods deliver 40% stability improvement for singular cases via explicit singularity handling, and midpoint methods provide 25% accuracy enhancement for regularized problems through optimal quadrature properties. Comparison with classical solvers (Nyström, Galerkin, Collocation) confirms that the PINN midpoint method achieves 2.5–3.5<InlineEquation ID="IEq1"><EquationSource Format="TEX">\(\times\)</EquationSource></InlineEquation> lower error for weakly singular kernels, and convergence rate experiments validate the theoretical <InlineEquation ID="IEq2"><EquationSource Format="TEX">\(O(h^2)\)</EquationSource></InlineEquation> and <InlineEquation ID="IEq3"><EquationSource Format="TEX">\(O(h^{1-\alpha })\)</EquationSource></InlineEquation> predictions. Medical imaging validation achieves 29.1 dB PSNR, confirming practical impact. This work transforms PINN discretization from trial-and-error to principled, kernel-adaptive selection strategies.</p>

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Kernel-Adaptive Discretization Strategies for Physics-Informed Neural Networks: A Comprehensive Framework for Optimal Solution of Fredholm Integral Equations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月22日

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  • Yan Ma,
  • Wei Li

摘要

We present the first comprehensive benchmark revealing that discretization strategy selection in Physics-Informed Neural Networks (PINNs) profoundly impacts performance for Fredholm integral equations of the second kind. Our kernel-adaptive framework systematically compares discrete coordinate, endpoint, and midpoint methods across smooth, singular, and regularized kernels. Key findings: discrete coordinate methods achieve 35% error reduction for smooth kernels through integral structure preservation, endpoint methods deliver 40% stability improvement for singular cases via explicit singularity handling, and midpoint methods provide 25% accuracy enhancement for regularized problems through optimal quadrature properties. Comparison with classical solvers (Nyström, Galerkin, Collocation) confirms that the PINN midpoint method achieves 2.5–3.5\(\times\) lower error for weakly singular kernels, and convergence rate experiments validate the theoretical \(O(h^2)\) and \(O(h^{1-\alpha })\) predictions. Medical imaging validation achieves 29.1 dB PSNR, confirming practical impact. This work transforms PINN discretization from trial-and-error to principled, kernel-adaptive selection strategies.