# Reinforcement Learning for Adaptive Mesh Refinement

arXiv: https://arxiv.org/abs/2103.01342 · HTML: https://mlsanity.com/paper/2103.01342/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 60 references that are in the map.
- First posted on arXiv 1 Mar 2021.

## Builds on (most cited first, 18 of 18)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Dota 2 with Large Scale Deep Reinforcement Learning](https://mlsanity.com/paper/1912.06680/) (2019)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Graph Convolutional Policy Network for Goal-Directed Molecular Graph Generation](https://mlsanity.com/paper/1806.02473/) (2018)
- [#Exploration: A Study of Count-Based Exploration for Deep Reinforcement Learning](https://mlsanity.com/paper/1611.04717/) (2016)
- [Distral: Robust Multitask Reinforcement Learning](https://mlsanity.com/paper/1707.04175/) (2017)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Behaviour Suite for Reinforcement Learning](https://mlsanity.com/paper/1908.03568/) (2019)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (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)
- [Growing Action Spaces](https://mlsanity.com/paper/1906.12266/) (2019)
- [GraphOpt: Learning Optimization Models of Graph Formation](https://mlsanity.com/paper/2007.03619/) (2020)

## Built on by (most cited first, 30 of 39)

- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Physics Informed Token Transformer for Solving Partial Differential Equations](https://mlsanity.com/paper/2305.08757/) (2023)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Learning Controllable Adaptive Simulation for Multi-resolution Physics](https://mlsanity.com/paper/2305.01122/) (2023)
- [Reinforcement learning for automatic quadrilateral mesh generation: a soft actor-critic approach](https://mlsanity.com/paper/2203.11203/) (2022)
- [Swarm Reinforcement Learning For Adaptive Mesh Refinement](https://mlsanity.com/paper/2304.00818/) (2023)
- [M2N: Mesh Movement Networks for PDE Solvers](https://mlsanity.com/paper/2204.11188/) (2022)
- [Machine learning for modelling unstructured grid data in computational physics: a review](https://mlsanity.com/paper/2502.09346/) (2025)
- [Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flows](https://mlsanity.com/paper/2405.17211/) (2024)
- [Learning robust marking policies for adaptive mesh refinement](https://mlsanity.com/paper/2207.06339/) (2022)
- [Better Neural PDE Solvers Through Data-Free Mesh Movers](https://mlsanity.com/paper/2312.05583/) (2023)
- [E2N: Error Estimation Networks for Goal-Oriented Mesh Adaptation](https://mlsanity.com/paper/2207.11233/) (2022)
- [Growing Q-Networks: Solving Continuous Control Tasks with Adaptive Control Resolution](https://mlsanity.com/paper/2404.04253/) (2024)
- [Multi-Agent Reinforcement Learning for Adaptive Mesh Refinement](https://mlsanity.com/paper/2211.00801/) (2022)
- [A reinforcement learning strategy for p-adaptation in high order solvers](https://mlsanity.com/paper/2306.08292/) (2023)
- [Hyena Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2306.16524/) (2023)
- [Towards Universal Mesh Movement Networks](https://mlsanity.com/paper/2407.00382/) (2024)
- [AMBER: Adaptive Mesh Generation by Iterative Mesh Resolution Prediction](https://mlsanity.com/paper/2505.23663/) (2025)
- [MaskMA: Towards Zero-Shot Multi-Agent Decision Making with Mask-Based Collaborative Learning](https://mlsanity.com/paper/2310.11846/) (2023)
- [A Two-Phase Deep Learning Framework for Adaptive Time-Stepping in High-Speed Flow Modeling](https://mlsanity.com/paper/2506.07969/) (2025)
- [A Survey of AI Methods for Geometry Preparation and Mesh Generation in Engineering Simulation](https://mlsanity.com/paper/2512.23719/) (2025)
- [NUNet: Deep Learning for Non-Uniform Super-Resolution of Turbulent Flows](https://mlsanity.com/paper/2203.14154/) (2022)
- [Reinforcement Learning for Block Decomposition of CAD Models](https://mlsanity.com/paper/2302.11066/) (2023)
- [Learning topological operations on meshes with application to block decomposition of polygons](https://mlsanity.com/paper/2309.06484/) (2023)
- [RL-PINNs: Reinforcement Learning-Driven Adaptive Sampling for Efficient Training of PINNs](https://mlsanity.com/paper/2504.12949/) (2025)
- [UGM2N: An Unsupervised and Generalizable Mesh Movement Network via M-Uniform Loss](https://mlsanity.com/paper/2508.08615/) (2025)
- [Accelerating PDE Surrogates via RL-Guided Mesh Optimization](https://mlsanity.com/paper/2603.02066/) (2026)
- [Iterative Sizing Field Prediction for Adaptive Mesh Generation From Expert Demonstrations](https://mlsanity.com/paper/2406.14161/) (2024)
- [Reinforcement learning for anisotropic p-adaptation and error estimation in high-order solvers](https://mlsanity.com/paper/2407.19000/) (2024)
- [Multilevel CNNs for Parametric PDEs based on Adaptive Finite Elements](https://mlsanity.com/paper/2408.10838/) (2024)

## Abstract (arXiv)

Large-scale finite element simulations of complex physical systems governed by partial differential equations (PDE) crucially depend on adaptive mesh refinement (AMR) to allocate computational budget to regions where higher resolution is required. Existing scalable AMR methods make heuristic refinement decisions based on instantaneous error estimation and thus do not aim for long-term optimality over an entire simulation. We propose a novel formulation of AMR as a Markov decision process and apply deep reinforcement learning (RL) to train refinement policies directly from simulation. AMR poses a new problem for RL as both the state dimension and available action set changes at every step, which we solve by proposing new policy architectures with differing generality and inductive bias. The model sizes of these policy architectures are independent of the mesh size and hence can be deployed on larger simulations than those used at train time. We demonstrate in comprehensive experiments on static function estimation and time-dependent equations that RL policies can be trained on problems without using ground truth solutions, are competitive with a widely-used error estimator, and generalize to larger, more complex, and unseen test problems.