Recurrent Neural Networks as Optimal Mesh Refinement Strategies

TL;DR

This paper demonstrates that recurrent neural networks can learn optimal mesh refinement strategies for elliptic PDEs and other PDEs with solutions approximable by deep neural networks, independent of input size.

Contribution

It introduces a method to train recurrent neural networks to learn optimal mesh refinement strategies, even for PDEs lacking known adaptive solutions.

Findings

RNNs can learn optimal mesh refinement with fixed parameters

The approach applies to a broad class of PDEs

No prior adaptive strategy needed for some PDEs

Abstract

We show that an optimal finite element mesh refinement algorithm for a prototypical elliptic PDE 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. Moreover, for a general class of PDEs with solutions which are well-approximated by deep neural networks, we show that an optimal mesh refinement strategy can be learned by recurrent neural networks. This includes problems for which no optimal adaptive strategy is known yet.