NEURAL NETWORK APPROXIMATION OF SOLUTIONS TO FRACTIONAL PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

This paper addresses the challenge of approximating solutions to high-dimensional fractional parabolic partial differential equations (PDEs) using neural networks. While neural networks are known to avoid the curse of dimensionality for certain problems, a systematic regularity theory that explains this efficiency for fractional parabolic operators—which involve complex space-time coupling—has been lacking.

The authors introduce "anisotropic spectral Barron spaces," a new framework that measures temporal and spatial regularity independently in the frequency domain. This allows them to handle the intrinsic coupling where one temporal derivative corresponds to $\gamma$ spatial derivatives. To overcome the difficulty of analyzing forward-in-time evolution within a global Fourier structure, the authors employ a Vandermonde-based reflection procedure to extend the fractional heat semigroup across the initial time. They then use the method of continuity to incorporate lower-order drift and potential terms, proving a maximal regularity theory that is independent of the spatial dimension.

The study provides a rigorous foundation for neural network approximation of these PDEs. The researchers prove that for a wide class of fractional parabolic equations, the solution operator preserves or improves the regularity within the anisotropic Barron framework. Consequently, they derive $n^{-1/2}$ approximation bounds for two-layer neural networks, where $n$ is the number of neurons. Crucially, the constant in these bounds is independent of the spatial dimension $d$, demonstrating that neural networks can efficiently represent solutions to these complex evolution equations even in high dimensions. The authors also provide a counterexample showing that uniform-in-time estimates for these Barron norms generally fail, highlighting the necessity of their space-time approach.