Recently, progress has been made in the application of neural networks to the numerical analysis of partial differential equations (PDEs). In the latter the variational formulation of the Poisson problem is used in order to obtain an objective function - a regularised Dirichlet energy - that was used for the optimisation of some neural networks. In this notes we use the notion of $\Gamma$-convergence to show that ReLU networks of growing architecture that are trained with respect to suitably regularised Dirichlet energies converge to the true solution of the Poisson problem. We discuss how this approach generalises to arbitrary variational problems under certain universality assumptions of neural networks and see that this covers some nonlinear stationary PDEs like the $p$-Laplace.

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重访深丽兹酒店

最近，在将神经网络应用于偏微分方程 (PDE) 的数值分析方面取得了进展。在后者中，泊松问题的变分公式用于获得目标函数 - 正则化狄利克雷能量 - 用于优化某些神经网络。在本笔记中，我们使用 $\Gamma$-convergence 的概念来表明，根据适当正则化 Dirichlet 能量训练的不断增长的架构的 ReLU 网络收敛到泊松问题的真正解。我们讨论了这种方法如何在神经网络的某些普遍性假设下推广到任意变分问题，并看到这涵盖了一些非线性平稳 PDE，如 $p$-Laplace。