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.

References in zbMATH (referenced in 15 articles , 1 standard article )

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  1. Habring, Andreas; Holler, Martin: A generative variational model for inverse problems in imaging (2022)
  2. Bar, Leah; Sochen, Nir: Strong solutions for PDE-based tomography by unsupervised learning (2021)
  3. Chen, Yunmei; Liu, Hongcheng; Ye, Xiaojing; Zhang, Qingchao: Learnable descent algorithm for nonsmooth nonconvex image reconstruction (2021)
  4. Cho, Taewon; Chung, Julianne; Jiang, Jiahua: Hybrid projection methods for large-scale inverse problems with mixed Gaussian priors (2021)
  5. de Hoop, Maarten V.; Lassas, Matti; Wong, Christopher A.: Deep learning architectures for nonlinear operator functions and nonlinear inverse problems (2021)
  6. Lunz, Sebastian; Hauptmann, Andreas; Tarvainen, Tanja; Schönlieb, Carola-Bibiane; Arridge, Simon: On learned operator correction in inverse problems (2021)
  7. Obmann, Daniel; Schwab, Johannes; Haltmeier, Markus: Deep synthesis network for regularizing inverse problems (2021)
  8. Pinetz, Thomas; Kobler, Erich; Pock, Thomas; Effland, Alexander: Shared prior learning of energy-based models for image reconstruction (2021)
  9. Aspri, Andrea; Korolev, Yury; Scherzer, Otmar: Data driven regularization by projection (2020)
  10. Baguer, Daniel Otero; Leuschner, Johannes; Schmidt, Maximilian: Computed tomography reconstruction using deep image prior and learned reconstruction methods (2020)
  11. Bao, Gang; Ye, Xiaojing; Zang, Yaohua; Zhou, Haomin: Numerical solution of inverse problems by weak adversarial networks (2020)
  12. Li, Housen; Schwab, Johannes; Antholzer, Stephan; Haltmeier, Markus: NETT: solving inverse problems with deep neural networks (2020)
  13. Schwab, Johannes; Antholzer, Stephan; Haltmeier, Markus: Big in Japan: regularizing networks for solving inverse problems (2020)
  14. Xu, Hao; Chang, Haibin; Zhang, Dongxiao: DLGA-PDE: discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm (2020)
  15. Arridge, Simon; Maass, Peter; Öktem, Ozan; Schönlieb, Carola-Bibiane: Solving inverse problems using data-driven models (2019)