Hermite spectral approximation for functions with endpoint singularities using exponential transforms

What the math gives to ML

The paper provides a principled way to represent functions on a bounded interval whose behavior is difficult at endpoints: map the interval to the real line with a monotone SE, DE, or error-function transform, then expand in rapidly decaying Hermite functions. The transferable asset is not merely using a spectral basis, but matching the coordinate transform and a scale parameter to endpoint regularity, which can reduce the number of coefficients needed. This suggests a boundary-singularity-aware coordinate layer for MLPs, neural operators, or implicit fields, with Hermite features replacing generic Fourier or polynomial features and the transform scale selected or learned. The extracted material also gives a sparse Hermite differentiation rule, making derivative losses cheap.