Saturation theorems for neural network operators by solving elliptic and hyperbolic PDEs with analytical and semi-analytical inverse problems

TL;DR

This paper investigates inverse problems for neural network operators using PDEs, establishing saturation theorems, inverse theorems, and a semi-analytical reconstruction method with potential real-world applications.

Contribution

It introduces new saturation and inverse theorems for neural network operators based on solving elliptic and hyperbolic PDEs, and develops a semi-analytical data reconstruction approach.

Findings

Derived saturation classes related to harmonic and transport equations

Established inverse theorems involving sub-harmonic and Sobolev functions

Proposed a semi-analytical reconstruction method for noisy data

Abstract

This paper addresses inverse problems (in a broad sense) for two classes of multivariate neural network (NN) operators, with particular emphasis on saturation results, and both analytical and semi-analytical inverse theorems. One of the key aspects in addressing these issues is solving of certain elliptic and hyperbolic partial differential equations (PDEs), as well as suitable asymptotic formulas for the NN operators based on sufficiently smooth functions; the connection between these two topics lies in the application of the so-called generalized parabola technique by Ditzian. From the saturation theorems characterizations of the saturation classes are derived; these are respectively related to harmonic functions and to the solution of a certain transport equation. Analytical inverse theorems, on the other hand, are related to sub-harmonic functions as well as to functions in the Sobolev space $W^2_\infty$. Finally, the problem of reconstructing data affected by noise is addressed, along with a semi-analytical inverse problem. The latter serves as the starting point for deriving a retrieval procedure that may be useful in real world applications.