[GS_M_APP] Fractional Parabolic PDEs in Anisotropic Spectral Barron Spaces: Maximal Regularity and Neural Network Approximation
ABSTRACT
We study fractional parabolic equations with lower-order terms in anisotropic spectral Barron spaces, which reflect the parabolic scaling between temporal and spatial frequencies. We establish dimension-independent maximal regularity estimates and resolve the mismatch between forward evolution and global space–time Fourier analysis by constructing a high-order extension of the fractional heat semigroup across the initial time. We also show, through a frequency-localized counterexample, that the corresponding uniform-in-time spatial Barron estimate generally fails. Finally, the regularity results yield dimension-efficient approximation rates for two-layer neural networks in mixed temporal and spatial Sobolev norms. This presentation is based on joint work with Hyojae Lim, Jinsol Seo, Young-Jin Sim, and Changhoon Song.
