# Deep Reinforcement Learning for Adaptive Mesh Refinement

arXiv: https://arxiv.org/abs/2209.12351 · HTML: https://mlsanity.com/paper/2209.12351/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 71 references that are in the map.
- First posted on arXiv 25 Sep 2022.

## Builds on (most cited first, 15 of 15)

- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
- [Addressing Function Approximation Error in Actor-Critic Methods](https://mlsanity.com/paper/1802.09477/) (2018)
- [Asynchronous Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1602.01783/) (2016)
- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [A Brief Survey of Deep Reinforcement Learning](https://mlsanity.com/paper/1708.05866/) (2017)
- [Deep Reinforcement Learning: An Overview](https://mlsanity.com/paper/1701.07274/) (2017)
- [A Practical Guide to Multi-Objective Reinforcement Learning and Planning](https://mlsanity.com/paper/2103.09568/) (2021)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [Reinforcement learning for automatic quadrilateral mesh generation: a soft actor-critic approach](https://mlsanity.com/paper/2203.11203/) (2022)
- [Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning](https://mlsanity.com/paper/1905.11079/) (2019)
- [Sparse Regression and Adaptive Feature Generation for the Discovery of Dynamical Systems](https://mlsanity.com/paper/1902.02719/) (2019)
- [General solutions for nonlinear differential equations: a rule-based self-learning approach using deep reinforcement learning](https://mlsanity.com/paper/1805.07297/) (2018)

## Built on by (most cited first, 20 of 20)

- [Physics Informed Token Transformer for Solving Partial Differential Equations](https://mlsanity.com/paper/2305.08757/) (2023)
- [Swarm Reinforcement Learning For Adaptive Mesh Refinement](https://mlsanity.com/paper/2304.00818/) (2023)
- [Machine learning for modelling unstructured grid data in computational physics: a review](https://mlsanity.com/paper/2502.09346/) (2025)
- [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)
- [G-Adaptivity: optimised graph-based mesh relocation for finite element methods](https://mlsanity.com/paper/2407.04516/) (2024)
- [Adaptive Swarm Mesh Refinement using Deep Reinforcement Learning with Local Rewards](https://mlsanity.com/paper/2406.08440/) (2024)
- [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)
- [Autonomous Adaptive Solver Selection for Chemistry Integration via Reinforcement Learning](https://mlsanity.com/paper/2604.00264/) (2026)
- [MeshTok: Efficient Multi-Scale Tokenization for Scalable PDE Transformers](https://mlsanity.com/paper/2606.04366/) (2026)
- [Playing to Par: Reinforcement Learning for Provably Optimal Quadrilateral Block Decompositions](https://mlsanity.com/paper/2609.32146/) (2026)
- [Learning When to Refine: Long-Horizon Reinforcement Learning for Budgeted Neural-Operator PDE Solvers](https://mlsanity.com/paper/2610.06883/) (2026)

## Abstract (arXiv)

Finite element discretizations of problems in computational physics often rely on adaptive mesh refinement (AMR) to preferentially resolve regions containing important features during simulation. However, these spatial refinement strategies are often heuristic and rely on domain-specific knowledge or trial-and-error. We treat the process of adaptive mesh refinement as a local, sequential decision-making problem under incomplete information, formulating AMR as a partially observable Markov decision process. Using a deep reinforcement learning approach, we train policy networks for AMR strategy directly from numerical simulation. The training process does not require an exact solution or a high-fidelity ground truth to the partial differential equation at hand, nor does it require a pre-computed training dataset. The local nature of our reinforcement learning formulation allows the policy network to be trained inexpensively on much smaller problems than those on which they are deployed. The methodology is not specific to any particular partial differential equation, problem dimension, or numerical discretization, and can flexibly incorporate diverse problem physics. To that end, we apply the approach to a diverse set of partial differential equations, using a variety of high-order discontinuous Galerkin and hybridizable discontinuous Galerkin finite element discretizations. We show that the resultant deep reinforcement learning policies are competitive with common AMR heuristics, generalize well across problem classes, and strike a favorable balance between accuracy and cost such that they often lead to a higher accuracy per problem degree of freedom.