In this work, we revisit the marking decisions made in the standard adaptive finite element method (AFEM). Experience shows that a na\"{i}ve marking policy leads to inefficient use of computational resources for adaptive mesh refinement (AMR). Consequently, using AFEM in practice often involves ad-hoc or time-consuming offline parameter tuning to set appropriate parameters for the marking subroutine. To address these practical concerns, we recast AMR as a Markov decision process in which refinement parameters can be selected on-the-fly at run time, without the need for pre-tuning by expert users. In this new paradigm, the refinement parameters are also chosen adaptively via a marking policy that can be optimized using methods from reinforcement learning. We use the Poisson equation to demonstrate our techniques on $h$- and $hp$-refinement benchmark problems, and our experiments suggest that superior marking policies remain undiscovered for many classical AFEM applications. Furthermore, an unexpected observation from this work is that marking policies trained on one family of PDEs are sometimes robust enough to perform well on problems far outside the training family. For illustration, we show that a simple $hp$-refinement policy trained on 2D domains with only a single re-entrant corner can be deployed on far more complicated 2D domains, and even 3D domains, without significant performance loss. For reproduction and broader adoption, we accompany this work with an open-source implementation of our methods.

翻译：在本工作中，我们重新审视了标准自适应有限元方法（AFEM）中的标记决策。实践经验表明，简单的标记策略会导致自适应网格细化（AMR）计算资源的低效利用。因此，实际应用AFEM时常常需要针对标记子程序进行临时或耗时的离线参数调优以设定合适参数。为解决这些实际问题，我们将AMR重新建模为马尔可夫决策过程，其中细化参数可在运行时动态选择，无需专家用户预先调优。在此新范式下，细化参数通过可借助强化学习方法优化的标记策略自适应选择。我们以泊松方程为例子，在$h$和$hp$细化基准问题上展示技术方法，实验表明经典AFEM应用中仍存在大量未被发现的更优标记策略。此外，本工作的一项意外发现是：基于某类偏微分方程训练的标记策略，有时具有足够鲁棒性，能在远超出训练问题族的场景中表现优异。通过示例说明，仅在含单一凹角的二维域上训练的简单$hp$细化策略，可直接部署在更复杂的二维域乃至三维域上而无显著性能损失。为便于复现和推广，我们随文附上方法开源实现。