Neural Network Operator-Based Fractal Approximation: Smoothness Preservation and Convergence Analysis

• URL: http://arxiv.org/abs/2505.06229v1
• Date: Sat, 22 Mar 2025 07:12:57 GMT
• Title: Neural Network Operator-Based Fractal Approximation: Smoothness Preservation and Convergence Analysis
• Authors: Aaqib Ayoub Bhat, Asif Khan, M. Mursaleen,
• Abstract summary: This paper presents a new approach of constructing $alpha$-fractal functions (FIFs) using neural network operators.<n>We have developed fractal functions that utilize only the values of the original function at the nodes or partition points.<n>We analyze the convergence of these $alpha$-fractals to the original function under suitable conditions.
• Score: 1.0879986496362417
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper presents a new approach of constructing $\alpha$-fractal interpolation functions (FIFs) using neural network operators, integrating concepts from approximation theory. Initially, we construct $\alpha$-fractals utilizing neural network-based operators, providing an approach to generating fractal functions with interpolation properties. Based on the same foundation, we have developed fractal interpolation functions that utilize only the values of the original function at the nodes or partition points, unlike traditional methods that rely on the entire original function. Further, we have constructed \(\alpha\)-fractals that preserve the smoothness of functions under certain constraints by employing a four-layered neural network operator, ensuring that if \(f \in C^{r}[a,b]\), then the corresponding fractal \(f^{\alpha} \in C^{r}[a,b]\). Furthermore, we analyze the convergence of these $\alpha$-fractals to the original function under suitable conditions. The work uses key approximation theory tools, such as the modulus of continuity and interpolation operators, to develop convergence results and uniform approximation error bounds.

Related papers

• Function Forms of Simple ReLU Networks with Random Hidden Weights [1.2289361708127877]
  We investigate the function space dynamics of a two-layer ReLU neural network in the infinite-width limit.<n>We highlight the Fisher information matrix's role in steering learning.<n>This work offers a robust foundation for understanding wide neural networks.
  arXiv  Detail & Related papers  (2025-05-23T13:53:02Z)
• Extension of Symmetrized Neural Network Operators with Fractional and Mixed Activation Functions [0.0]
  We propose a novel extension to symmetrized neural network operators by incorporating fractional and mixed activation functions.<n>Our framework introduces a fractional exponent in the activation functions, allowing adaptive non-linear approximations with improved accuracy.
  arXiv  Detail & Related papers  (2025-01-17T14:24:25Z)
• Neural Operators with Localized Integral and Differential Kernels [77.76991758980003]
  We present a principled approach to operator learning that can capture local features under two frameworks. We prove that we obtain differential operators under an appropriate scaling of the kernel values of CNNs. To obtain local integral operators, we utilize suitable basis representations for the kernels based on discrete-continuous convolutions.
  arXiv  Detail & Related papers  (2024-02-26T18:59:31Z)
• A Globally Convergent Algorithm for Neural Network Parameter Optimization Based on Difference-of-Convex Functions [29.58728073957055]
  We propose an algorithm for optimizing parameters of hidden layer networks. Specifically, we derive a blockwise (DC-of-the-art) difference function.
  arXiv  Detail & Related papers  (2024-01-15T19:53:35Z)
• Stable Nonconvex-Nonconcave Training via Linear Interpolation [51.668052890249726]
  This paper presents a theoretical analysis of linearahead as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear can help by leveraging the theory of nonexpansive operators.
  arXiv  Detail & Related papers  (2023-10-20T12:45:12Z)
• Approximation and interpolation of deep neural networks [0.0]
  In the overparametrized regime, deep neural network provide universal approximations and can interpolate any data set. In the last section, we provide a practical probabilistic method of finding such a point under general conditions on the activation function.
  arXiv  Detail & Related papers  (2023-04-20T08:45:16Z)
• Factorized Fourier Neural Operators [77.47313102926017]
  The Factorized Fourier Neural Operator (F-FNO) is a learning-based method for simulating partial differential equations. We show that our model maintains an error rate of 2% while still running an order of magnitude faster than a numerical solver.
  arXiv  Detail & Related papers  (2021-11-27T03:34:13Z)
• Deep neural network approximation of analytic functions [91.3755431537592]
  entropy bound for the spaces of neural networks with piecewise linear activation functions. We derive an oracle inequality for the expected error of the considered penalized deep neural network estimators.
  arXiv  Detail & Related papers  (2021-04-05T18:02:04Z)
• A semigroup method for high dimensional committor functions based on neural network [1.7205106391379026]
  Instead of working with partial differential equations, the new method works with an integral formulation based on the semigroup of the differential operator. gradient descent type algorithms can be applied in the training of the committor function without the need of computing any mixed second-order derivatives. Unlike the previous methods that enforce the boundary conditions through penalty terms, the new method takes into account the boundary conditions automatically.
  arXiv  Detail & Related papers  (2020-12-12T05:00:47Z)
• Fourier Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  We formulate a new neural operator by parameterizing the integral kernel directly in Fourier space. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. It is up to three orders of magnitude faster compared to traditional PDE solvers.
  arXiv  Detail & Related papers  (2020-10-18T00:34:21Z)
• Complexity of Finding Stationary Points of Nonsmooth Nonconvex Functions [84.49087114959872]
  We provide the first non-asymptotic analysis for finding stationary points of nonsmooth, nonsmooth functions. In particular, we study Hadamard semi-differentiable functions, perhaps the largest class of nonsmooth functions.
  arXiv  Detail & Related papers  (2020-02-10T23:23:04Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.