Recurrent neural networks as optimal mesh refinement strategies

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.

Automated summary

A recurrent neural network with a fixed number of trainable parameters can learn optimal mesh refinement algorithms for a wide variety of PDE problems, regardless of the desired accuracy and input size. The proposed algorithm is problem-independent and only requires the current numerical approximation to optimally refine the mesh, making it a provably optimal black-box mesh refinement tool.