# Cubic-Hermite Product Integration

A neural collocation scheme for semilinear time-fractional reaction–diffusion equations rewrites the Caputo memory term as cubic-Hermite product integration, using the network state and its autodifferentiated time slope as endpoint data on each time panel while the weakly singular weights are precomputed once before training. On the scalar benchmark v(t)=t⁴.2 the Hermite memory approximation is three to six orders of magnitude more accurate than the classical L1 formula on the tested meshes, and for a manufactured problem with α=0.7 a small CPU-trained network reaches a relative L2 error of 1.98×10⁻⁵ and a maximum error of 3.80×10⁻⁵.

A neural collocation scheme for semilinear time-fractional reaction–diffusion equations rewrites the Caputo memory term as cubic-Hermite product integration, using the network state and its autodifferentiated time slope as endpoint data on each time panel while the weakly singular weights are precomputed once before training. On the scalar benchmark v(t)=t⁴.2 the Hermite memory approximation is three to six orders of magnitude more accurate than the classical L1 formula on the tested meshes, and for a manufactured problem with α=0.7 a small CPU-trained network reaches a relative L2 error of 1.98×10⁻⁵ and a maximum error of 3.80×10⁻⁵.