Quadratic Neural Networks for Solving Inverse Problems

Abstract

In this paper we investigate the solution of inverse problems with neural network ansatz functions with generalized decision functions. The relevant observation for this work is that such functions can approximate typical test cases, such as the Shepp-Logan phantom, better, than standard neural networks. Moreover, we show that the convergence analysis of numerical methods for solving inverse problems with shallow generalized neural network functions leads to more intuitive convergence conditions, than for deep affine linear neural networks.

KEYWORDS:

• Generalized neural network functions
• inverse problems

MATHEMATICS SUBJECT CLASSIFICATION:

• 65J22
• 45Q05
• 42C40
• 65F20

Acknowledgments

The authors would like to thank some referees for their valuable suggestions and their patience.

Notes

1 For a detailed exposition on generalized inverses see [29].

2 In the following $\nabla$ and ${\nabla }^2$ (without subscripts) always denote derivatives with respect to an n-dimensional variable such as $\overrightarrow{x}$. $\prime$ and $″$ denotes derivatives of a one-dimensional function.

Additional information

Funding

This research was funded in whole, or in part, by the Austrian Science Fund (FWF) 10.55776/P34981 (OS & LF) – New Inverse Problems of Super-Resolved Microscopy (NIPSUM), SFB 10.55776/F68 (OS) “Tomography Across the Scales,” project F6807-N36 (Tomography with Uncertainties), and 10.55776/T1160 (CS) “Photoacoustic Tomography: Analysis and Numerics.” For open access purposes, the author has applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission. The financial support by the Austrian Federal Ministry for Digital and Economic Affairs, the National Foundation for Research, Technology and Development and the Christian Doppler Research Association is gratefully acknowledged.