Neural network interpolation operators optimized by Lagrange polynomial.

Area of Science:

• Numerical Analysis
• Approximation Theory
• Machine Learning

Background:

• Interpolation operators are crucial for approximating functions.
• Feedforward neural networks offer powerful function approximation capabilities.
• Lagrange polynomials provide a basis for constructing interpolation operators.

Purpose of the Study:

• Introduce a new class of interpolation operators based on Lagrange polynomials.
• Analyze the approximation properties and theoretical guarantees of these operators.
• Extend the operators to the multivariate case and evaluate their performance.

Main Methods:

• Utilizing Lagrange polynomials of degree r to define interpolation operators.
• Estimating approximation rates using the (r+1)-th modulus of smoothness.
• Establishing inequalities for operator derivatives under smooth activation function assumptions.
• Applying K-functional and Berens-Lorentz lemma for converse approximation theorems.
• Deriving Voronovskaja-type asymptotic estimations for smooth functions.

Main Results:

• The new operators are shown to be equivalent to four-layer feedforward neural networks.
• Theoretical bounds on approximation rates are established.
• Converse theorems and asymptotic estimations are derived.
• The operators are successfully extended to multivariate function approximation.
• Numerical examples validate the theoretical findings and demonstrate operator superiority.

Conclusions:

• The introduced interpolation operators possess strong theoretical foundations in approximation theory.
• These operators demonstrate potential as efficient feedforward neural network models.
• The study provides a rigorous mathematical framework for understanding their approximation capabilities.