Shallow ReLU$^s$ Networks in $L^p$-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization

Deep learning has shown remarkable effectiveness in high-dimensional approximation problems, particularly in scientific computing, inverse problems, and operator learning (Han et al., 2018; Adcock et al., 2022; Beck et al., 2023). In many such settings, the ReLUs activation σs(t) = max{0,t}s (s N0) is especially relevant because it yields piecewisepolynomial representations that are well suited to smooth targets and derivative-sensitive tasks (Yang and Zhou, 2025; He et al., 2024).