# Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation

arXiv: https://arxiv.org/abs/2607.27781 · HTML: https://mlsanity.com/paper/2607.27781/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 57 references that are in the map.
- First posted on arXiv 30 Jul 2026.

## Builds on (most cited first, 20 of 20)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory](https://mlsanity.com/paper/2006.15733/) (2020)
- [On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics](https://mlsanity.com/paper/2007.15623/) (2020)
- [Some observations on high-dimensional partial differential equations with Barron data](https://mlsanity.com/paper/2012.01484/) (2020)
- [Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective](https://mlsanity.com/paper/2210.12101/) (2022)
- [Approximations with deep neural networks in Sobolev time-space](https://mlsanity.com/paper/2101.06115/) (2020)
- [A general approximation lower bound in Lp norm, with applications to feed-forward neural networks](https://mlsanity.com/paper/2206.04360/) (2022)
- [Space-time deep neural network approximations for high-dimensional partial differential equations](https://mlsanity.com/paper/2006.02199/) (2020)
- [Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces](https://mlsanity.com/paper/2312.08461/) (2023)
- [Embedding Inequalities for Barron-type Spaces](https://mlsanity.com/paper/2305.19082/) (2023)
- [Time-Frequency Analysis for Neural Networks](https://mlsanity.com/paper/2512.15992/) (2025)
- [Approximation Rates of Shallow Neural Networks: Barron Spaces, Activation Functions and Optimality Analysis](https://mlsanity.com/paper/2510.18388/) (2025)
- [Sobolev Approximation of Deep ReLU Networks in Log-Barron Space](https://mlsanity.com/paper/2601.01295/) (2026)
- [Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces](https://mlsanity.com/paper/2602.19381/) (2026)
- [Elliptic Regularity Theory in Barron Spaces and Applications to the Deep Ritz Method](https://mlsanity.com/paper/2607.25100/) (2026)

## Abstract (arXiv)

We study fractional parabolic initial-value problems with lower-order drift and potential terms in anisotropic spectral Barron spaces, defined by weighted space--time Fourier L1 norms adapted to parabolic scaling. We prove existence, uniqueness, and maximal regularity with a gain of one derivative in time and γ derivatives in space, where γ>0 is the order of the fractional Laplacian. The evolution is defined only for t≥0, whereas the finite-time norm requires a global extension with sufficient temporal Fourier decay. We construct a finite reflected semigroup extension using a Vandermonde system to match derivatives at t=0, obtaining temporal Fourier estimates uniform in the semigroup parameter. Combined with Fourier multiplier estimates for the damped principal operator, it yields maximal regularity. Dimension-independent multiplication estimates support a finite regularity bootstrap, while interpolation and sufficient damping absorb the lower-order terms in the base estimate. The a priori estimate and the method of continuity yield maximal regularity without smallness assumptions on the lower-order coefficients. A frequency-localized counterexample shows that a uniform-in-time spatial Barron bound on the forcing does not imply the corresponding two-derivative solution bound, even for the one-dimensional heat equation. Using this regularity, Fourier sampling yields n-1/2 approximation rates for the solution in mixed space--time Sobolev norms using shallow networks with suitable activations. Sampling in a product Hilbert space yields a population-level PINN consistency estimate for shallow cosine networks on a bounded cylinder. There exists a single width-n network for which the sum of the squared mixed-Sobolev solution error, the squared L2-norm of the residual for the whole-space fractional equation, and the squared initial-data error is O(n-1).