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

Danilo Costarelli

TL;DR

This work studies inverse problems for two families of multivariate neural-network operators, $F_n$ and $K_n$, focusing on saturation and both analytical and semi-analytical inverse theorems. A central theme is relating saturation behavior to solutions of elliptic and hyperbolic PDEs via the generalized parabola technique of $Ditzian$, yielding local and global saturation results tied to harmonic and transport equations. The authors derive inverse characterization results in Lipschitz spaces, establish local Voronovskaja-type asymptotics, address reconstruction under noise, and develop a semi-analytical retrieval framework using ramp-based sigmoids, with concrete 1D and 2D examples. The findings provide theoretical guarantees and practical retrieval tools for real-world data, while outlining important open questions and extensions, particularly for Kantorovich operators and higher-dimensional settings.

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.

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

TL;DR

This work studies inverse problems for two families of multivariate neural-network operators, and , focusing on saturation and both analytical and semi-analytical inverse theorems. A central theme is relating saturation behavior to solutions of elliptic and hyperbolic PDEs via the generalized parabola technique of , yielding local and global saturation results tied to harmonic and transport equations. The authors derive inverse characterization results in Lipschitz spaces, establish local Voronovskaja-type asymptotics, address reconstruction under noise, and develop a semi-analytical retrieval framework using ramp-based sigmoids, with concrete 1D and 2D examples. The findings provide theoretical guarantees and practical retrieval tools for real-world data, while outlining important open questions and extensions, particularly for Kantorovich operators and higher-dimensional settings.

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 . 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.
Paper Structure (9 sections, 17 theorems, 261 equations, 2 tables)

This paper contains 9 sections, 17 theorems, 261 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Some recall on the multivariate neural network operators
  3. Analytical inverse results
  4. Local Voronovskaja-type theorems
  5. Saturation theorems by the solution of the Laplace equation
  6. Saturation theorems for the Kantorovich NN by the solution of an hyperbolic PDE
  7. Reconstruction and contrast to noise perturbations
  8. Semi-analytical inversion of the NN operators
  9. Final remarks and open problems

Key Result

Theorem 3.1

Let $\alpha>1$ and assume in addition that $M_1(\phi_{\sigma}')$ is finite. Let now $f \in C(I)$ such that: with $0<\nu<1$. Then $f \in Lip(\nu)$.

Theorems & Definitions (43)

  • Example 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • Example 3.4
  • Remark 3.5
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 33 more