DeepTV: A neural network approach for total variation minimization
<|reference_start|>DeepTV: A neural network approach for total variation minimization: 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.<|reference_end|>
@article{langer2024deeptv:,
    title={DeepTV: A neural network approach for total variation minimization},
    author={Andreas Langer and Sara Behnamian},
    journal={arXiv preprint arXiv:2409.05569},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.05569},
    primaryClass={math.NA cs.NA}
}