Solution of Inverse Problems Using Multilayer Quaternion Neural Networks

2014 ◽  
Author(s):  
Takehiko Ogawa
Keyword(s):  
Neural Networks ◽  
Inverse Problems

Jointly learning variational data assimilation models and solvers for geophysical dynamics

2021 ◽  
Author(s):  
Ronan Fablet ◽  
Bertrand Chapron ◽  
Lucas Drumetz ◽  
Etienne Memin ◽  
Olivier Pannekoucke ◽  
...  
Keyword(s):  
Neural Networks ◽  
Data Assimilation ◽  
Inverse Problems ◽  
Automatic Differentiation ◽  
Variational Data Assimilation ◽  
Point Of View ◽  
Variational Model ◽  
Short Term ◽  
End To End ◽  
Short Term Forecasting

<p>This paper addresses representation learning for the resolution of inverse problems  with geophysical dynamics. Among others, examples of inverse problems of interest include space-time interpolation, short-term forecasting, conditional simulation w.r.t. available observations, downscaling problems… From a methodological point of view, we rely on a variational data assimilation framework. Data assimilation (DA) aims to reconstruct the time evolution of some state given a series of  observations, possibly noisy and irregularly-sampled. Here, we investigate DA from a machine learning point of view backed by an underlying variational representation.  Using automatic differentiation tools embedded in deep learning frameworks, we introduce end-to-end neural network architectures for variational data assimilation. It comprises two key components: a variational model and a gradient-based solver both implemented as neural networks. A key feature of the proposed end-to-end learning architecture is that we may train the neural networks models using both supervised and unsupervised strategies. We first illustrate applications to the reconstruction of Lorenz-63 and Lorenz-96 systems from partial and noisy observations. Whereas the gain issued from the supervised learning setting emphasizes the relevance of groundtruthed observation dataset for real-world case-studies, these results also suggest new means to design data assimilation models from data. Especially, they suggest that learning task-oriented representations of the underlying dynamics may be beneficial. We further discuss applications to short-term forecasting and sampling design along with preliminary results for the reconstruction of sea surface currents from satellite altimetry data. </p><p>This abstract is supported by a preprint available online: https://arxiv.org/abs/2007.12941</p>

scholarly journals Neural networks with excitatory and inhibitory components: Direct and inverse problems by a mean-field approach

2016 ◽  
Vol 93 (1) ◽  
Author(s):  
Matteo di Volo ◽  
Raffaella Burioni ◽  
Mario Casartelli ◽  
Roberto Livi ◽  
Alessandro Vezzani
Keyword(s):  
Neural Networks ◽  
Inverse Problems ◽  
Mean Field ◽  
Direct And Inverse Problems ◽  
Field Approach ◽  
Inhibitory Components

scholarly journals Big in Japan: Regularizing Networks for Solving Inverse Problems

2019 ◽  
Vol 62 (3) ◽  
pp. 445-455
Author(s):  
Johannes Schwab ◽  
Stephan Antholzer ◽  
Markus Haltmeier
Keyword(s):  
Neural Networks ◽  
Inverse Problems ◽  
Regularization Method ◽  
Deep Neural Networks ◽  
Convergence Rates ◽  
Null Space ◽  
Space Network ◽  
Special Cases ◽  
Data Problem ◽  
Sparse Data Problem

Abstract Deep learning and (deep) neural networks are emerging tools to address inverse problems and image reconstruction tasks. Despite outstanding performance, the mathematical analysis for solving inverse problems by neural networks is mostly missing. In this paper, we introduce and rigorously analyze families of deep regularizing neural networks (RegNets) of the form $$\mathbf {B}_\alpha + \mathbf {N}_{\theta (\alpha )} \mathbf {B}_\alpha $$Bα+Nθ(α)Bα, where $$\mathbf {B}_\alpha $$Bα is a classical regularization and the network $$\mathbf {N}_{\theta (\alpha )} \mathbf {B}_\alpha $$Nθ(α)Bα is trained to recover the missing part $${\text {Id}}_X - \mathbf {B}_\alpha $$IdX-Bα not found by the classical regularization. We show that these regularizing networks yield a convergent regularization method for solving inverse problems. Additionally, we derive convergence rates (quantitative error estimates) assuming a sufficient decay of the associated distance function. We demonstrate that our results recover existing convergence and convergence rates results for filter-based regularization methods as well as the recently introduced null space network as special cases. Numerical results are presented for a tomographic sparse data problem, which clearly demonstrate that the proposed RegNets improve classical regularization as well as the null space network.

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