arXiv subjects

Ahmed Abdeljawad

Publications and source records attributed to Ahmed Abdeljawad.

15 recordsLinked to original sources

arXiv2026

Approximation Rates for Metaplectic Neural Networks

In this paper we develop quantitative approximation results for shallow neural networks constructed using a dictionary based on metaplectic operators. First, we extend the concept of Barron spaces by considering a symplectically motivated extension of the Fourier transform, known as the metaplectic transform. Then, after establishing embedding between metaplectic Barron spaces and Sobolev spaces we consider a neural metaplectic dictionary and we prove Monte-Carlo approximation bounds for metaplectic Barron functions using finite linear combinations of atoms of the dictionary. Finally, we validate the introduction of the neural metaplectic dictionary by devising a deep neural network architecture that uses as building blocks the atoms of the dictionary. We test it to approximate solutions of time-dependent Schrödinger equations, demonstrating better performance compared to classical phyisics informed neural networks architectures.

cs.LG↗

arXiv2026

Time-Frequency Analysis for Neural Networks

We develop a quantitative approximation theory for shallow neural networks using tools from time-frequency analysis. Working in weighted modulation spaces $M^{p,q}_m(\mathbf{R}^{d})$, we prove dimension-independent approximation rates in Sobolev norms $W^{n,r}(Ω)$ for networks whose units combine standard activations with localized time-frequency windows. Our main result shows that for $f \in M^{p,q}_m(\mathbf{R}^{d})$ one can achieve \[ \|f - f_N\|_{W^{n,r}(Ω)} \lesssim N^{-1/2}\,\|f\|_{M^{p,q}_m(\mathbf{R}^{d})}, \] on bounded domains, with explicit control of all constants. We further obtain global approximation theorems on $\mathbf{R}^{d}$ using weighted modulation dictionaries, and derive consequences for Feichtinger's algebra, Fourier-Lebesgue spaces, and Barron spaces. Numerical experiments in one and two dimensions confirm that modulation-based networks achieve substantially better Sobolev approximation than standard ReLU networks, consistent with the theoretical estimates.

math.NA↗

arXiv2025

Approximation Rates in Fréchet Metrics: Barron Spaces, Paley-Wiener Spaces, and Fourier Multipliers

Operator learning is a recent development in the simulation of Partial Differential Equations (PDEs) by means of neural networks. The idea behind this approach is to learn the behavior of an operator, such that the resulting neural network is an (approximate) mapping in infinite-dimensional spaces that is capable of (approximately) simulating the solution operator governed by the PDE. In our work, we study some general approximation capabilities for linear differential operators by approximating the corresponding symbol in the Fourier domain. Analogous to the structure of the class of Hörmander-Symbols, we consider the approximation with respect to a topology that is induced by a sequence of semi-norms. In that sense, we measure the approximation error in terms of a Fréchet metric, and our main result identifies sufficient conditions for achieving a predefined approximation error. Secondly, we then focus on a natural extension of our main theorem, in which we manage to reduce the assumptions on the sequence of semi-norms. Based on existing approximation results for the exponential spectral Barron space, we then present a concrete example of symbols that can be approximated well.

math.NA↗

arXiv2024

Uniform Approximation with Quadratic Neural Networks

In this work, we examine the approximation capabilities of deep neural networks utilizing the Rectified Quadratic Unit (ReQU) activation function, defined as $\max(0,x)^2$, for approximating Hölder-regular functions with respect to the uniform norm. We constructively prove that deep neural networks with ReQU activation can approximate any function within the $R$-ball of $r$-Hölder-regular functions ($\mathcal{H}^{r, R}([-1,1]^d)$) up to any accuracy $ε$ with at most $\mathcal{O}\left(ε^{-d /2r}\right)$ neurons and fixed number of layers. This result highlights that the effectiveness of the approximation depends significantly on the smoothness of the target function and the characteristics of the ReQU activation function. Our proof is based on approximating local Taylor expansions with deep ReQU neural networks, demonstrating their ability to capture the behavior of Hölder-regular functions effectively. Furthermore, the results can be straightforwardly generalized to any Rectified Power Unit (RePU) activation function of the form $\max(0,x)^p$ for $p \geq 2$, indicating the broader applicability of our findings within this family of activations.

cs.LG↗

arXiv2024

Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains

In this work, we consider the approximation capabilities of shallow neural networks in weighted Sobolev spaces for functions in the spectral Barron space. The existing literature already covers several cases, in which the spectral Barron space can be approximated well, i.e., without curse of dimensionality, by shallow networks and several different classes of activation function. The limitations of the existing results are mostly on the error measures that were considered, in which the results are restricted to Sobolev spaces over a bounded domain. We will here treat two cases that extend upon the existing results. Namely, we treat the case with bounded domain and Muckenhoupt weights and the case, where the domain is allowed to be unbounded and the weights are required to decay. We first present embedding results for the more general weighted Fourier-Lebesgue spaces in the weighted Sobolev spaces and then we establish asymptotic approximation rates for shallow neural networks that come without curse of dimensionality.

cs.LG↗

arXiv2023

Sampling Complexity of Deep Approximation Spaces

While it is well-known that neural networks enjoy excellent approximation capabilities, it remains a big challenge to compute such approximations from point samples. Based on tools from Information-based complexity, recent work by Grohs and Voigtlaender [Journal of the FoCM (2023)] developed a rigorous framework for assessing this so-called "theory-to-practice gap". More precisely, in that work it is shown that there exist functions that can be approximated by neural networks with ReLU activation function at an arbitrary rate while requiring an exponentially growing (in the input dimension) number of samples for their numerical computation. The present study extends these findings by showing analogous results for the ReQU activation function.

cs.LG↗

arXiv2023

Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces

Approximation capabilities of shallow neural networks (SNNs) form an integral part in understanding the properties of deep neural networks (DNNs). In the study of these approximation capabilities some very popular classes of target functions are the so-called spectral Barron spaces. This spaces are of special interest when it comes to the approximation of partial differential equation (PDE) solutions. It has been shown that the solution of certain static PDEs will lie in some spectral Barron space. In order to alleviate the limitation to static PDEs and include a time-domain that might have a different regularity than the space domain, we extend the notion of spectral Barron spaces to anisotropic weighted Fourier-Lebesgue spaces. In doing so, we consider target functions that have two blocks of variables, among which each block is allowed to have different decay and integrability properties. For these target functions we first study the inclusion of anisotropic weighted Fourier-Lebesgue spaces in the Bochner-Sobolev spaces. With that we can now also measure the approximation error in terms of an anisotropic Sobolev norm, namely the Bochner-Sobolev norm. We use this observation in a second step where we establish a bound on the approximation rate for functions from the anisotropic weighted Fourier-Lebesgue spaces and approximation via SNNs in the Bochner-Sobolev norm.

cs.LG↗

arXiv2021

Integral representations of shallow neural network with Rectified Power Unit activation function

In this effort, we derive a formula for the integral representation of a shallow neural network with the Rectified Power Unit activation function. Mainly, our first result deals with the univariate case of representation capability of RePU shallow networks. The multidimensional result in this paper characterizes the set of functions that can be represented with bounded norm and possibly unbounded width.

cs.NE↗

arXiv2020

Approximations with deep neural networks in Sobolev time-space

Solutions of evolution equation generally lies in certain Bochner-Sobolev spaces, in which the solution may has regularity and integrability properties for the time variable that can be different for the space variables. Therefore, in this paper, we develop a framework shows that deep neural networks can approximate Sobolev-regular functions with respect to Bochner-Sobolev spaces. In our work we use the so-called Rectified Cubic Unit (ReCU) as an activation function in our networks, which allows us to deduce approximation results of the neural networks while avoiding issues caused by the non regularity of the most commonly used Rectivied Linear Unit (ReLU) activation function.

cs.LG↗

arXiv2020

Characterizations of a class of Pilipovi{ć} spaces by powers of harmonic oscillator

We show that a smooth function $f$ on $\mathbf R^d$ belongs to the Pilipovi{ć} space $\mathcal H_{\flat _σ}(\mathbf R^d)$ or the Pilipovi{ć} space $\mathcal H_{0,\flat _σ}(\mathbf R^d)$, if and only if the $L^p$ norm of $H_d^Nf$ for $N\ge 0$, satisfy certain types of estimates. Here $H_d=|x|^2-Δ_x$ is the harmonic oscillator.

math.FA↗

arXiv2019

Bilinear pseudo-differential operators with Gevrey-Hörmander symbols

We consider bilinear pseudo-differential operators whose symbols posses Gevrey type regularity and may have a sub-exponential growth at infinity, together with all their derivatives. It is proved that those symbol classes can be described by the means of the short-time Fourier transform and modulation spaces. Our first main result is the invariance property of the corresponding bilinear operators. Furthermore we prove the continuity of such operators when acting on modulation spaces. As a consequence, we derive their continuity on anisotropic Gelfand-Shilov type spaces. We consider both Beurling and Roumieu type symbol classes and Gelfand-Shilov spaces.

math.FA↗

arXiv2018

Deterministic and Stochastic Cauchy problems for a class of weakly hyperbolic operators on R^n

We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on R^n. We prove well-posedness in Sobolev-Kato spaces, with loss of smoothness and decay at infinity. We also obtain results about propagation of singularities, in terms of wave-front sets describing the evolution of both smoothness and decay singularities of temperate distributions. Moreover, we can prove the existence of random-field solutions for the associated stochastic Cauchy problems. To this aim, we first discuss algebraic properties for iterated integrals of suitable parameter-dependent families of Fourier integral operators, associated with the characteristic roots, which are involved in the construction of the fundamental solution. In particular, we show that, also for this operator class, the involutiveness of the characteristics implies commutative properties for such expressions.

math.AP↗

arXiv2018

Anisotropic Gevrey-Hörmander pseudo-differential operators on modulation spaces

We show continuity properties for the pseudo-differential operator $\operatorname{Op} (a)$ from $M(ω_0ω,\mathscr B )$ to $M(ω,\mathscr B )$, for fixed $s,σ\ge 1$, $ω,ω_0\in \mathscr P _{s,σ}^0$ ($ω,ω_0\in \mathscr P _{s,σ}$), $a\in Γ^{σ,s}_{(ω_0)}$ ($a\in Γ^{σ,s;0}_{(ω_0)}$) , and $\mathscr B$ is an invariant Banach function space.

math.FA↗

arXiv2018

Pseudo-differential calculus in anisotropic Gelfand-Shilov setting

We study some classes of pseudo-differential operators with symbols $a$ admitting anisotropic exponential growth at infinity and we prove mapping properties for these operators on Gelfand-Shilov spaces of type S. Moreover, we deduce algebraic and certain invariance properties of these classes.

math.FA↗

arXiv2017

Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey type pseudo-differential operators

We deduce one-parameter group properties for pseudo-differential operators $\operatorname{Op} (a)$, where $a$ belongs to the class $Γ^{(ω_0)}_*$ of certain Gevrey symbols. We use this to show that there are pseudo-differential operators $\operatorname{Op} (a)$ and $\operatorname{Op} (b)$ which are inverses to each others, where $a\in Γ^{(ω_0)}_*$ and $b\in Γ^{(1/ω_0)}_*$. We apply these results to deduce lifting property for modulation spaces and construct explicit isomorpisms between them. For each weight functions $ω,ω_0$ moderated by GRS submultiplicative weights, we prove that the Toeplitz operator (or localization operator) $\operatorname{Tp} (ω_0)$ is an isomorphism from $M^{p,q}_{(ω)}$ onto $M^{p,q}_{(ω/ω_0)}$ for every $p,q \in (0,\infty ]$.

math.FA↗