Recurrent Neural Networks as Optimal Mesh Refinement Strategies

2019/09/10 by Jan Bohn, Michael Feischl, Bohn, Jan +1 · 3 citations
Computer Science · Mathematics · #FOS: Mathematics #Numerical Analysis (math.NA) #cs.NA #math.NA

paper · pdf · doi:10.48550/arxiv.1909.04275

arxiv created 2020/08/07 · arxiv updated 2020/08/10

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.

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• HypeR Adaptivity: Joint hr-Adaptive Meshing via Hypergraph Multi-Agent Deep Reinforcement Learning
• A Quantifier-Reversal Approximation Paradigm for Recurrent Neural Networks
• AMBER: Adaptive Mesh Generation by Iterative Mesh Resolution Prediction

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