Adaptive mesh refinement (AMR) is necessary for efficient finite element simulations of complex physical phenomenon, as it allocates limited computational budget based on the need for higher or lower resolution, which varies over space and time. We present a novel formulation of AMR as a fully-cooperative Markov game, in which each element is an independent agent who makes refinement and de-refinement choices based on local information. We design a novel deep multi-agent reinforcement learning (MARL) algorithm called Value Decomposition Graph Network (VDGN), which solves the two core challenges that AMR poses for MARL: posthumous credit assignment due to agent creation and deletion, and unstructured observations due to the diversity of mesh geometries. For the first time, we show that MARL enables anticipatory refinement of regions that will encounter complex features at future times, thereby unlocking entirely new regions of the error-cost objective landscape that are inaccessible by traditional methods based on local error estimators. Comprehensive experiments show that VDGN policies significantly outperform error threshold-based policies in global error and cost metrics. We show that learned policies generalize to test problems with physical features, mesh geometries, and longer simulation times that were not seen in training. We also extend VDGN with multi-objective optimization capabilities to find the Pareto front of the tradeoff between cost and error.

翻译：自适应网格细化（AMR）对于复杂物理现象的高效有限元模拟至关重要，因为它根据空间和时间上变化的分辨率需求分配有限的计算预算。我们提出了一种新颖的AMR公式，将其建模为完全协作的马尔可夫博弈，其中每个单元基于局部信息独立做出细化和粗化决策的智能体。我们设计了一种名为价值分解图网络（VDGN）的新型深度多智能体强化学习（MARL）算法，该算法解决了AMR对MARL提出的两个核心挑战：由于智能体创建和删除导致的死后信用分配问题，以及由于网格几何结构多样性导致的非结构化观测问题。我们首次证明，MARL能够实现对未来时刻将遇到复杂特征的区域进行预期性细化，从而解锁了传统基于局部误差估计器方法无法触及的误差-成本目标函数全新区域。综合实验表明，VDGN策略在全局误差和成本指标上显著优于基于误差阈值的策略。我们证明，学习到的策略能够泛化到训练中未涉及的物理特征、网格几何结构及更长仿真时间的测试问题。我们还扩展了VDGN的多目标优化能力，以找到成本与误差权衡的帕累托前沿。