DeepTV: A neural network approach for total variation minimization
Langer, Andreas LU
and Behnamian, Sara
LU
(2024)
- Abstract
- Neural network approaches have been demonstrated to work quite well to solve partial differential equations in practice. In this context approaches like physics-informed neural networks and the Deep Ritz method have become popular. In this paper, we propose a similar approach to solve an infinite-dimensional total variation minimization problem using neural networks. We illustrate that the resulting neural network problem does not have a solution in general. To circumvent this theoretic issue, we consider an auxiliary neural network problem, which indeed has a solution, and show that it converges in the sense of -convergence to the original problem. For computing a numerical solution we further propose a discrete version of the auxiliary... (More)
- Neural network approaches have been demonstrated to work quite well to solve partial differential equations in practice. In this context approaches like physics-informed neural networks and the Deep Ritz method have become popular. In this paper, we propose a similar approach to solve an infinite-dimensional total variation minimization problem using neural networks. We illustrate that the resulting neural network problem does not have a solution in general. To circumvent this theoretic issue, we consider an auxiliary neural network problem, which indeed has a solution, and show that it converges in the sense of -convergence to the original problem. For computing a numerical solution we further propose a discrete version of the auxiliary neural network problem and again show its -convergence to the original infinite-dimensional problem. In particular, the -convergence proof suggests a particular discretization of the total variation. Moreover, we connect the discrete neural network problem to a finite difference discretization of the infinite-dimensional total variation minimization problem. Numerical experiments are presented supporting our theoretical findings. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/d1601ccb-2a75-44b3-80fb-9b92607dd4de
- author
- Langer, Andreas
LU
and Behnamian, Sara
LU
- organization
- publishing date
- 2024
- type
- Working paper/Preprint
- publication status
- published
- subject
- DOI
- 10.48550/arXiv.2409.05569
- language
- English
- LU publication?
- yes
- id
- d1601ccb-2a75-44b3-80fb-9b92607dd4de
- date added to LUP
- 2025-04-26 11:20:40
- date last changed
- 2026-06-26 12:34:42
@misc{d1601ccb-2a75-44b3-80fb-9b92607dd4de,
abstract = {{Neural network approaches have been demonstrated to work quite well to solve partial differential equations in practice. In this context approaches like physics-informed neural networks and the Deep Ritz method have become popular. In this paper, we propose a similar approach to solve an infinite-dimensional total variation minimization problem using neural networks. We illustrate that the resulting neural network problem does not have a solution in general. To circumvent this theoretic issue, we consider an auxiliary neural network problem, which indeed has a solution, and show that it converges in the sense of -convergence to the original problem. For computing a numerical solution we further propose a discrete version of the auxiliary neural network problem and again show its -convergence to the original infinite-dimensional problem. In particular, the -convergence proof suggests a particular discretization of the total variation. Moreover, we connect the discrete neural network problem to a finite difference discretization of the infinite-dimensional total variation minimization problem. Numerical experiments are presented supporting our theoretical findings.}},
author = {{Langer, Andreas and Behnamian, Sara}},
language = {{eng}},
note = {{Preprint}},
title = {{DeepTV: A neural network approach for total variation minimization}},
url = {{http://dx.doi.org/10.48550/arXiv.2409.05569}},
doi = {{10.48550/arXiv.2409.05569}},
year = {{2024}},
}