Boundary Interpolation on Triangles via Neural Network Operators

TL;DR

This paper introduces a new class of neural network operators for boundary interpolation on triangles that do not require training, using adjustable hidden neurons to achieve accurate approximation, validated through numerical and graphical analysis.

Contribution

It presents a novel boundary interpolation operator based on neural networks that avoids training and adjusts weights via hidden neurons, with proven approximation properties.

Findings

Operators effectively interpolate boundary values on triangles.

Numerical examples demonstrate high accuracy and efficiency.

Comparative analysis confirms the method's validity.

Abstract

The primary objective of this study is to develop novel interpolation operators that interpolate the boundary values of a function defined on a triangle. This is accomplished by constructing New Generalized Boolean sum neural network operator $\mathcal{B}_{n_1, n_2, \xi }$ using a class of activation functions. Its interpolation properties are established and the estimates for the error of approximation corresponding to operator $\mathcal{B}_{n_1, n_2, \xi }$ is computed in terms of mixed modulus of continuity. The advantage of our method is that it does not require training the network. Instead, the number of hidden neurons adjusts the weights and bias. Numerical examples are illustrated to show the efficacy of these newly constructed operators. Further, with the help of MATLAB, comparative and graphical analysis is given to show the validity and efficiency of the results obtained for these operators.