用神经网络求解总变差最小化问题，解决理论无解难题。
DeepTV: A neural network approach for total variation minimization
- 设计辅助神经网络模型克服原问题无解缺陷
- 证明辅助模型在Γ-收敛意义下逼近原问题
- 连接离散神经网络与有限差分法，适合优化领域研究者
神经网络在求解偏微分方程方面表现良好，如物理信息神经网络和Deep Ritz方法。本文提出一种类似方法，用于求解无限维的总变差最小化问题。我们发现该神经网络问题通常无解。为克服这一理论困境，引入一个辅助神经网络问题，其存在解，并证明该辅助问题在Γ-收敛意义下收敛到原问题。进一步提出辅助问题的离散版本，同样证明其Γ-收敛至原问题。Γ-收敛证明揭示了总变差的一种特定离散化方式。此外，将离散神经网络问题与无限维总变差最小化问题的有限差分离散化相联系。数值实验验证了理论结果的有效性。
原文摘要 · Abstract (English)
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.
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