{"title":"DeepTV: A neural network approach for total variation minimization","authors":"Andreas Langer, Sara Behnamian","doi":"arxiv-2409.05569","DOIUrl":null,"url":null,"abstract":"Neural network approaches have been demonstrated to work quite well to solve\npartial differential equations in practice. In this context approaches like\nphysics-informed neural networks and the Deep Ritz method have become popular.\nIn this paper, we propose a similar approach to solve an infinite-dimensional\ntotal variation minimization problem using neural networks. We illustrate that\nthe resulting neural network problem does not have a solution in general. To\ncircumvent this theoretic issue, we consider an auxiliary neural network\nproblem, which indeed has a solution, and show that it converges in the sense\nof $\\Gamma$-convergence to the original problem. For computing a numerical\nsolution we further propose a discrete version of the auxiliary neural network\nproblem and again show its $\\Gamma$-convergence to the original\ninfinite-dimensional problem. In particular, the $\\Gamma$-convergence proof\nsuggests a particular discretization of the total variation. Moreover, we\nconnect the discrete neural network problem to a finite difference\ndiscretization of the infinite-dimensional total variation minimization\nproblem. Numerical experiments are presented supporting our theoretical\nfindings.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05569","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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 $\Gamma$-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.
在实践中,神经网络方法在求解偏微分方程时已被证明非常有效。在这种情况下,物理信息神经网络和 Deep Ritz 方法等方法开始流行起来。在本文中,我们提出了一种类似的方法,利用神经网络解决无限维总变异最小化问题。本文提出了利用神经网络求解无限维总变异最小化问题的类似方法,并说明了由此产生的神经网络问题在一般情况下没有解。为了避免这个理论问题,我们考虑了一个辅助神经网络问题,它确实有一个解,并证明它在$\Gamma$-收敛的意义上收敛于原始问题。为了计算数值解,我们进一步提出了离散版的辅助神经网络问题,并再次证明了它与原始无限维问题的$\Gamma$收敛性。特别是,$\Gamma$-收敛证明提出了总变异的特定离散化。此外,我们还将离散神经网络问题与无限维总变化最小化问题的有限差分离散化联系起来。数值实验支持我们的理论发现。