Taylor-Accelerated Neural Network Interpolation Operators on Irregular Grids with Higher Order Approximation

TL;DR

This paper introduces Taylor-accelerated neural network interpolation operators on irregular grids that leverage Taylor polynomials to achieve higher-order approximation and improved convergence rates for smooth functions.

Contribution

It presents a novel interpolation method combining neural networks with Taylor polynomials, enhancing accuracy and convergence on irregular grids.

Findings

Achieves higher-order convergence on irregular grids.

Outperforms existing neural network interpolation schemes.

Provides theoretical error estimates and numerical validation.

Abstract

In this paper, a new class of \emph{Taylor-accelerated neural network interpolation operators} is introduced on quasi-uniform irregular grids. These operators improve existing neural network interpolation operators by incorporating Taylor polynomials at the sampling nodes, thereby exploiting higher smoothness of the target function. The proposed operators are shown to be well defined, uniformly bounded, and to satisfy an exact interpolation property at the grid points. In addition, polynomial reproduction up to a prescribed degree is established. Jackson-type approximation estimates are derived in terms of higher-order moduli of smoothness, yielding enhanced convergence rates for sufficiently smooth functions. Numerical experiments are presented to support the theoretical analysis and to demonstrate the significant accuracy improvement achieved through the Taylor-accelerated construction. In particular, higher-order convergence on irregular grids is obtained, and the proposed approach outperforms existing neural network interpolation operators on irregular grids, including Lagrange-based schemes.