# DeepTV: A neural network approach for total variation minimization

arXiv: https://arxiv.org/abs/2409.05569 · HTML: https://mlsanity.com/paper/2409.05569/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 49 references that are in the map.
- First posted on arXiv 9 Sep 2024.

## Builds on (most cited first, 7 of 7)

- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Image Restoration using Total Variation Regularized Deep Image Prior](https://mlsanity.com/paper/1810.12864/) (2018)
- [Unsupervised Learning of the Total Variation Flow](https://mlsanity.com/paper/2206.04406/) (2022)
- [Approximation of BV functions by neural networks: A regularity theory approach](https://mlsanity.com/paper/2012.08291/) (2020)

## Built on by (most cited first, 2 of 2)

- [Ensemble of Weak Spectral Total Variation Learners: a PET-CT Case Study](https://mlsanity.com/paper/2507.08735/) (2025)
- [FractalPINN-Flow: A Fractal-Inspired Network for Unsupervised Optical Flow Estimation with Total Variation Regularization](https://mlsanity.com/paper/2509.08670/) (2025)

## Abstract (arXiv)

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 Gamma-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 Gamma-convergence to the original infinite-dimensional problem. In particular, the Gamma-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.