We show that optimal mesh refinement algorithms for a large class of PDEs can be learned by a recurrent neural network with a fixed number of trainable parameters independent of the desired accuracy and the input size, i.e., number of elements of the mesh. This includes problems for which no optimal adaptive strategy is known yet. The proposed algorithm is problem independent in the sense that it only requires the current numerical approximation in order to optimally refine the mesh. Thus, the method is a provably optimal black-box mesh refinement tool for a wide variety of PDE problems.

中文翻译：

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循环神经网络作为最佳网格细化策略

我们表明，对于一大类偏微分方程的最佳网格细化算法可以通过具有固定数量的可训练参数的循环神经网络学习，独立于所需的精度和输入大小，即网格元素的数量。这包括尚无最佳自适应策略的问题。所提出的算法与问题无关，因为它只需要当前的数值近似值来优化网格。因此，该方法是一种可证明适用于各种 PDE 问题的最佳黑盒网格细化工具。