DeepTV: A neural network approach for total variation minimization

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