NETT swMATH ID: 41773 Software Authors: Housen Li, Johannes Schwab, Stephan Antholzer, Markus Haltmeier Description: NETT: Solving Inverse Problems with Deep Neural Networks. Recovering a function or high-dimensional parameter vector from indirect measurements is a central task in various scientific areas. Several methods for solving such inverse problems are well developed and well understood. Recently, novel algorithms using deep learning and neural networks for inverse problems appeared. While still in their infancy, these techniques show astonishing performance for applications like low-dose CT or various sparse data problems. However, there are few theoretical results for deep learning in inverse problems. In this paper, we establish a complete convergence analysis for the proposed NETT (Network Tikhonov) approach to inverse problems. NETT considers data consistent solutions having small value of a regularizer defined by a trained neural network. We derive well-posedness results and quantitative error estimates, and propose a possible strategy for training the regularizer. Our theoretical results and framework are different from any previous work using neural networks for solving inverse problems. A possible data driven regularizer is proposed. Numerical results are presented for a tomographic sparse data problem, which demonstrate good performance of NETT even for unknowns of different type from the training data. To derive the convergence and convergence rates results we introduce a new framework based on the absolute Bregman distance generalizing the standard Bregman distance from the convex to the non-convex case. Homepage: https://arxiv.org/abs/1803.00092 Related Software: DeepAdverserialRegulariser; U-Net; Adam; PDE-Net; Wasserstein GAN; TensorFlow; DnCNN; PULSE; LoDoPaB-CT; convex_learning; GitHub; BSDS; DGM; UNLocBoX; StyleGAN; BigGAN; RRR; MNIST; OPAL; BADMM Cited in: 19 Publications Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year NETT: solving inverse problems with deep neural networks. Zbl 1456.65038Li, Housen; Schwab, Johannes; Antholzer, Stephan; Haltmeier, Markus 2020 all top 5 Cited by 49 Authors 3 Arridge, Simon R. 3 Haltmeier, Markus 3 Schönlieb, Carola-Bibiane 3 Schwab, Johannes 2 Antholzer, Stephan 2 Maass, Peter 2 Xu, Hao 2 Ye, Xiaojing 2 Zhang, Dongxiao 1 Aspri, Andrea 1 Bao, Feng 1 Bao, Gang 1 Bar, Leah 1 Chang, Haibin 1 Chen, Yunmei 1 Cho, Taewon 1 Chung, Julianne M. 1 de Hoop, Maarten V. 1 Effland, Alexander 1 Habring, Andreas 1 Hauptmann, Andreas 1 Holler, Martin 1 Jiang, Jiahua 1 Kobler, Erich 1 Korolev, Yury M. 1 Lassas, Matti J. 1 Leuschner, Johannes 1 Li, Housen 1 Liu, Hongcheng 1 Lunz, Sebastian 1 Maier, Thomas 1 Obmann, Daniel 1 Öktem, Ozan 1 Otero Baguer, Daniel 1 Pereyra, Marcelo 1 Pinetz, Thomas 1 Pock, Thomas 1 Scherzer, Otmar 1 Schmidt, Maximilian 1 Sochen, Nir 1 Tarvainen, Tanja 1 Wang, Nanzhe 1 Webster, Clayton G. 1 Wong, Christopher A. 1 Xie, Xuping 1 Zang, Yaohua 1 Zhang, Qingchao 1 Zhou, Hao-Min 1 Zygalakis, Konstantinos C. all top 5 Cited in 9 Serials 6 Inverse Problems 5 SIAM Journal on Imaging Sciences 2 Journal of Computational Physics 1 Journal of Mathematical Imaging and Vision 1 Acta Numerica 1 Oberwolfach Reports 1 Discrete and Continuous Dynamical Systems. Series S 1 Mathematical Statistics and Learning 1 SIAM Journal on Mathematics of Data Science all top 5 Cited in 14 Fields 15 Numerical analysis (65-XX) 15 Computer science (68-XX) 4 Partial differential equations (35-XX) 4 Statistics (62-XX) 4 Information and communication theory, circuits (94-XX) 3 Biology and other natural sciences (92-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 1 General and overarching topics; collections (00-XX) 1 Functions of a complex variable (30-XX) 1 Integral equations (45-XX) 1 Operator theory (47-XX) 1 Geophysics (86-XX) 1 Operations research, mathematical programming (90-XX) 1 Systems theory; control (93-XX) Citations by Year