SinCoTrap: A High-Order Locally Corrected Trapezoidal Rule for Periodic Singular Integrals in Arbitrary Dimensions

What the math gives to ML

The paper develops a constructive way to retain high-order accuracy when uniformly sampled periodic integrals contain an integrable power-law singularity. Its transferable asset is the combination of a fixed local stencil modification, analytically continued lattice-sum coefficients, and an explicit error rate that improves with correction order without abandoning tensor-grid structure. This maps directly to singular convolution layers in neural operators, particle-free graph networks, and periodic PDE surrogates: replace the few kernel samples near zero rather than refining the entire grid. The most useful first experiment is a periodic neural integral layer with a known |x|^{-s} singular component, comparing the corrected layer against naive sampled convolution at equal resolution and FLOPs.