Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces

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 $L^1$ 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 $t\geq0$, 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 $L^2$-norm of the residual for the whole-space fractional equation, and the squared initial-data error is $O(n^{-1})$.

We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial differential equations. In particular, we consider problems set in $d$-dimensional periodic domains, $d=1, 2, \dots$, and with analytic right-hand sides and coefficients. Our analysis covers linear, elliptic second order divergence-form PDEs as, e.g., diffusion-reaction problems, parametric diffusion equations, and elliptic systems such as linear isotropic elastostatics in heterogeneous materials. We leverage the exponential convergence of spectral collocation methods for boundary value problems whose solutions are analytic. In the present periodic and analytic setting, this follows from classical elliptic regularity. Within the ONet branch and trunk construction of [Chen and Chen, 1993] and of [Lu et al., 2021], we show the existence of deep ONets which emulate the coefficient-to-solution map to a desired accuracy in the $H^1$ norm, uniformly over the coefficient set. We prove that the neural networks in the ONet have size $\mathcal{O}(\left|\log(\varepsilon)\right|^κ)$, where $\varepsilon>0$ is the approximation accuracy, for some $κ>0$ depending on the physical space dimension.

We study how well deep neural networks approximate and learn spectral Barron functions. Recent studies have shown that these function classes can be efficiently approximated by shallow neural networks without suffering from the curse of dimensionality. We complement these results by providing new approximation bounds for deep networks with ReLU activation and establishing the minimax rates for learning these function classes. Specifically, we show that $d$-dimensional spectral Barron functions with smoothness index $s>0$ can be approximated by deep ReLU neural networks with approximation rate $\widetilde{\mathcal{O}} (S^{-\frac{1}{2}-\frac{s}{d}})$, where $S$ denotes the number of nonzero parameters in the network. Using this approximation result, we further show that deep ReLU neural networks can learn spectral Barron functions in a fast rate $n^{-\frac{d+2s}{2d+2s}}$ with $n$ training samples. Finally, we prove that this convergence rate is minimax optimal up to logarithmic factors.