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

Sachin Saini

TL;DR

This work tackles high-order interpolation on irregular grids by introducing a Taylor-accelerated neural network interpolation operator $\widetilde{T}_{n,r}(g,y)$ that combines local Taylor polynomials $P_r(g;y,z_k)$ with a compactly supported sigmoidal kernel. The operator is shown to be well-defined and uniformly bounded, while exactly interpolating at grid nodes and reproducing all polynomials of degree $\le r$. It yields Jackson-type error bounds $\|\widetilde{T}_{n,r}(g)-g\|_\infty \le C_r\,\omega_{r+1}(g,d_2)$ and, under mesh refinement with $h_n\le C/n$, achieves the optimal rate $O(n^{-(r+1)})$ for $g\in C^{r+1}[c,d]$. Numerical experiments on quasi-uniform irregular grids validate the theory and show that the Taylor-accelerated operator provides significantly higher accuracy than existing Lagrange-based irregular-grid interpolants, highlighting the practical impact of incorporating local Taylor information. The framework paves the way for higher-order, stable interpolation on non-uniform grids and invites extensions to multivariate and adaptive settings.

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 the 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. Direct 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 partitions, including Lagrange-based schemes.

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

TL;DR

This work tackles high-order interpolation on irregular grids by introducing a Taylor-accelerated neural network interpolation operator that combines local Taylor polynomials with a compactly supported sigmoidal kernel. The operator is shown to be well-defined and uniformly bounded, while exactly interpolating at grid nodes and reproducing all polynomials of degree . It yields Jackson-type error bounds and, under mesh refinement with , achieves the optimal rate for . Numerical experiments on quasi-uniform irregular grids validate the theory and show that the Taylor-accelerated operator provides significantly higher accuracy than existing Lagrange-based irregular-grid interpolants, highlighting the practical impact of incorporating local Taylor information. The framework paves the way for higher-order, stable interpolation on non-uniform grids and invites extensions to multivariate and adaptive settings.

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 the 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. Direct 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 partitions, including Lagrange-based schemes.
Paper Structure (8 sections, 7 theorems, 72 equations, 4 figures)

This paper contains 8 sections, 7 theorems, 72 equations, 4 figures.

Table of Contents

  1. Introduction and Preliminaries
  2. Taylor-Accelerated Neural Network Interpolation Operator on Irregular Grids
  3. Approximation Estimate
  4. Numerical Validation
  5. Activation Function and Grid Construction
  6. Error Convergence and Comparison
  7. Discussion
  8. Conclusion

Key Result

Proposition 2.2

For every $g\in C^{r}[c,d]$ and every $y\in[c,d]$, the operator $\widetilde{T}_{n,r}(g,y)$ is well defined.

Figures (4)

  • Figure 1: Uniform approximation error $E^{\mathrm{Tay}}_{n,r}=\|\widetilde{T}_{n,r}(f)-f\|_\infty$ as a function of $n$ for Taylor degrees $r=1,2,3$ on nested quasi-uniform irregular grids. Higher values of $r$ yield significantly improved convergence rates.
  • Figure 2: Error comparison between Taylor and Lagrange neural network operators for $r=1$.
  • Figure 3: Error comparison between Taylor and Lagrange neural network operators for $r=2$.
  • Figure 4: Error comparison between Taylor and Lagrange neural network operators for $r=3$.

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5: Polynomial Reproduction
  • proof
  • ...and 9 more