Related papers: Approximation by Neural Network operators in $L^p$…

We establish $L^p$ error estimates for monotone numerical schemes approximating Hamilton-Jacobi equations on the $d$-dimensional torus. Using the adjoint method, we first prove a $L^1$ error bound of order one for finite-difference and…

We present a unified framework to construct well-posed formulations for large classes of linear operator equations including elliptic, parabolic and hyperbolic partial differential equations. This general approach incorporates known weak…

We study the power of deep neural networks (DNNs) with sigmoid activation function. Recently, it was shown that DNNs approximate any $d$-dimensional, smooth function on a compact set with a rate of order $W^{-p/d}$, where $W$ is the number…

Recently, versions of neural networks with infinite-dimensional affine operators inside the computational units (``neural operator'' networks) have been applied to learn solutions to differential equations. To enable practical computations,…

Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and $(p_-(L),\, p_+(L))$ be the maximal interval of exponents $q\in[1,\,\infty]$ such that the semigroup…

As demonstrated in many areas of real-life applications, neural networks have the capability of dealing with high dimensional data. In the fields of optimal control and dynamical systems, the same capability was studied and verified in many…

We study uniform $\epsilon-$BPB approximations of bounded linear operators between Banach spaces from a geometric perspective. We show that for sufficiently small positive values of $\epsilon,$ many geometric properties like smoothness,…

We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function $\sigma : \mathbb{C} \to…

Neural operator architectures approximate operators between infinite-dimensional Banach spaces of functions. They are gaining increased attention in computational science and engineering, due to their potential both to accelerate…

We investigate the approximation capabilities of dense neural networks. While universal approximation theorems establish that sufficiently large architectures can approximate arbitrary continuous functions if there are no restrictions on…

In recent articles it was proved that when $\mu$ is a finite, radial measure in $\real^n$ with a bounded, radially decreasing density, the $L^p(\mu)$ norm of the associated maximal operator $M_\mu$ grows to infinity with the dimension for a…

We introduce a multiplicative neural network architecture in which multiplicative interactions constitute the fundamental representation, rather than appearing as auxiliary components within an additive model. We establish a universal…

We prove $L^p_{comp}\to L^p_{s}$ boundedness for averaging operators associated to a class of curves in the Heisenberg group $\mathbb{H}^1$ via $L^2$ estimates for related oscillatory integrals and Bourgain-Demeter decoupling inequalities…

Motivated by the abundance of functional data such as time series and images, there has been a growing interest in integrating such data into neural networks and learning maps from function spaces to R (i.e., functionals). In this paper, we…

This survey paper contains a detailed self-contained introduction to Korovkin-type theorems and to some of their applications concerning the approximation of continuous functions as well as of L^p-functions, by means of positive linear…

This paper introduces the Kernel Neural Operator (KNO), a provably convergent operator-learning architecture that utilizes compositions of deep kernel-based integral operators for function-space approximation of operators (maps from…

Neural operators have emerged as transformative tools for learning mappings between infinite-dimensional function spaces, offering useful applications in solving complex partial differential equations (PDEs). This paper presents a rigorous…

Operator learning refers to the application of ideas from machine learning to approximate (typically nonlinear) operators mapping between Banach spaces of functions. Such operators often arise from physical models expressed in terms of…

Three density theorems for three suitable subspaces of $SBD$ functions, in the strong $BD$ topology, are proven. The spaces are $SBD$, $SBD^p_\infty$, where the absolutely continuous part of the symmetric gradient is in $L^p$, with $p>1$,…

This paper extends the proof of density of neural networks in the space of continuous (or even measurable) functions on Euclidean spaces to functions on compact sets of probability measures. By doing so the work parallels a more then a…