Shallow neural network yields regularization for ill-posed inverse problems

Abstract

In this paper, we develop regularization theory for neural network approximations of general ill-posed operator equations with noisy data. Within the framework of iterative regularization, we propose two expanding neural network (ENN) based schemes under different a priori assumptions. Instead of prescribing a fixed architecture, ENNs adaptively select the number of neurons through an a posteriori stopping rule, so that the network size serves as a regularization parameter balancing approximation accuracy and stability with respect to data noise. We prove their regularizing properties and quantitative relationships between the network size and the noise level. Within the framework of variational regularization, we propose a neural network-based Tikhonov scheme and derive both convergence and convergence-rate results under mild assumptions. The estimates depend on the noise level, the network size, and the underlying smoothness expressed through general variational source conditions. Numerical experiments demonstrate the effectiveness and robustness of the algorithms, and show that, for highly noisy data, relatively small network architectures can already produce stable reconstructions.

arXiv comment24 pages, 27 figures