Related papers: Neural Network Operator-Based Fractal Approximatio…
In this study the general formula for differential and integral operations of fractional calculus via fractal operators by the method of cumulative diminution and cumulative growth is obtained. The under lying mechanism in the success of…
Fractals are self-similar and scale-invariant patterns found ubiquitously in nature. A lot of evidences implying fractal properties such as 1/f power spectrums have been also observed in resting state fMRI time series. To explain the…
Neural operators extend data-driven models to map between infinite-dimensional functional spaces. While these operators perform effectively in either the time or frequency domain, their performance may be limited when applied to…
In Convolutional Neural Network (CNN) based image processing, most of the studies propose networks that are optimized for a single-level (or a single-objective); thus, they underperform on other levels and must be retrained for delivery of…
A novel physics-informed operator learning technique based on spectral methods is introduced to model the complex behavior of heterogeneous materials. The Lippmann-Schwinger operator in Fourier space is employed to construct physical…
Classical Laguerre spectral approximations are highly effective on the half-line when the target function is smooth in the usual polynomial scale. However, their accuracy deteriorates for nonsmooth functions. Such behavior appears naturally…
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
Structured convex optimization problems typically involve a mix of smooth and nonsmooth functions. The common practice is to activate the smooth functions via their gradient and the nonsmooth ones via their proximity operator. We show that,…
We present a novel and unifying framework for constructing spectral approximations to fractional integral operators. These spectral approximations are based on transplanted Chebyshev polynomials, which are obtained by composing Chebyshev…
A novel approach to zipper fractal interpolation theory for functions of several variables is proposed. We develop multivariate zipper fractal functions in a constructive manner. We then perturb a multivariate function to construct its…
In this paper we discuss a new method to blend fractal attractors using the code map for the IFS formed by the Hutchinson--Barnsley operators of a finite family of hyperbolic IFSs. We introduce a parameter called blending coefficient to…
In this article we present a modification of classical Radial Basis Function (RBF) interpolation techniques aimed at reducing oscillations near discontinuities in one and two dimensions. Our approach introduces an adaptive mechanism by…
Neural operators have shown remarkable performance in approximating solutions of partial differential equations. However, their convergence behavior under grid refinement is still not well understood from the viewpoint of numerical…
We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with $\alpha$-H\"older derivatives (for some $0<\alpha\leq 1$). The smooth approximation is given by means of an…
We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel…
Graphons, as limits of graph sequences, provide an operator-theoretic framework for analyzing the asymptotic behavior of graph neural operators. Spectral convergence of sampled graphs to graphons induces convergence of the corresponding…
The two-parametric Mittag-Leffler function (MLF), $E_{\alpha,\beta}$, is fundamental to the study and simulation of fractional differential and integral equations. However, these functions are computationally expensive and their numerical…
In this paper, we introduce the fractional Fourier series on the fractional torus and study some basic facts of fractional Fourier series, such as fractional convolution and fractional approximation. Meanwhile, fractional Fourier inversion…
In this paper, we propose a new trigonometric interpolation algorithm and establish relevant convergent properties. The method adjusts an existing trigonometric interpolation algorithm such that it can better leverage Fast Fourier Transform…
Function fitting/approximation plays a fundamental role in computer graphics and other engineering applications. While recent advances have explored neural networks to address this task, these methods often rely on architectures with many…