# Quasi-optimal hp-finite element refinements towards singularities via deep neural network prediction

arXiv: https://arxiv.org/abs/2209.05844 · HTML: https://mlsanity.com/paper/2209.05844/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 48 references that are in the map.
- First posted on arXiv 13 Sep 2022.

## Builds on (most cited first, 6 of 6)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Graph Element Networks: adaptive, structured computation and memory](https://mlsanity.com/paper/1904.09019/) (2019)
- [MeshingNet: A New Mesh Generation Method based on Deep Learning](https://mlsanity.com/paper/2004.07016/) (2020)
- [Reinforcement learning for automatic quadrilateral mesh generation: a soft actor-critic approach](https://mlsanity.com/paper/2203.11203/) (2022)
- [Refinement of polygonal grids using Convolutional Neural Networks with applications to polygonal Discontinuous Galerkin and Virtual Element methods](https://mlsanity.com/paper/2102.05738/) (2021)

## Built on by (most cited first, 6 of 6)

- [r-Adaptive Deep Learning Method for Solving Partial Differential Equations](https://mlsanity.com/paper/2210.10900/) (2022)
- [A Deep Double Ritz Method (D2RM) for solving Partial Differential Equations using Neural Networks](https://mlsanity.com/paper/2211.03627/) (2022)
- [Learning topological operations on meshes with application to block decomposition of polygons](https://mlsanity.com/paper/2309.06484/) (2023)
- [Adaptive Swarm Mesh Refinement using Deep Reinforcement Learning with Local Rewards](https://mlsanity.com/paper/2406.08440/) (2024)
- [UGM2N: An Unsupervised and Generalizable Mesh Movement Network via M-Uniform Loss](https://mlsanity.com/paper/2508.08615/) (2025)
- [Multilevel CNNs for Parametric PDEs based on Adaptive Finite Elements](https://mlsanity.com/paper/2408.10838/) (2024)

## Abstract (arXiv)

We show how to construct the deep neural network (DNN) expert to predict quasi-optimal hp-refinements for a given computational problem. The main idea is to train the DNN expert during executing the self-adaptive hp-finite element method (hp-FEM) algorithm and use it later to predict further hp refinements. For the training, we use a two-grid paradigm self-adaptive hp-FEM algorithm. It employs the fine mesh to provide the optimal hp refinements for coarse mesh elements. We aim to construct the DNN expert to identify quasi-optimal hp refinements of the coarse mesh elements. During the training phase, we use the direct solver to obtain the solution for the fine mesh to guide the optimal refinements over the coarse mesh element. After training, we turn off the self-adaptive hp-FEM algorithm and continue with quasi-optimal refinements as proposed by the DNN expert trained. We test our method on three-dimensional Fichera and two-dimensional L-shaped domain problems. We verify the convergence of the numerical accuracy with respect to the mesh size. We show that the exponential convergence delivered by the self-adaptive hp-FEM can be preserved if we continue refinements with a properly trained DNN expert. Thus, in this paper, we show that from the self-adaptive hp-FEM it is possible to train the DNN expert the location of the singularities, and continue with the selection of the quasi-optimal hp refinements, preserving the exponential convergence of the method.