The Finite Element Method, an important technique in engineering, is aided by Adaptive Mesh Refinement (AMR), which dynamically refines mesh regions to allow for a favorable trade-off between computational speed and simulation accuracy. Classical methods for AMR depend on task-specific heuristics or expensive error estimators, hindering their use for complex simulations. Recent learned AMR methods tackle these problems, but so far scale only to simple toy examples. We formulate AMR as a novel Adaptive Swarm Markov Decision Process in which a mesh is modeled as a system of simple collaborating agents that may split into multiple new agents. This framework allows for a spatial reward formulation that simplifies the credit assignment problem, which we combine with Message Passing Networks to propagate information between neighboring mesh elements. We experimentally validate the effectiveness of our approach, Adaptive Swarm Mesh Refinement (ASMR), showing that it learns reliable, scalable, and efficient refinement strategies on a set of challenging problems. Our approach significantly speeds up computation, achieving up to 30-fold improvement compared to uniform refinements in complex simulations. Additionally, we outperform learned baselines and achieve a refinement quality that is on par with a traditional error-based AMR strategy without expensive oracle information about the error signal.

翻译：有限元方法是工程中的重要技术，自适应网格细化（AMR）通过动态细化网格区域，在计算速度与仿真精度之间实现有利权衡。传统的AMR方法依赖于特定任务的启发式规则或昂贵的误差估计器，这阻碍了其在复杂仿真中的应用。近期基于学习的AMR方法解决了这些问题，但迄今仅能扩展至简单的玩具示例。我们将AMR形式化为一种新颖的自适应群体马尔可夫决策过程，其中网格被建模为由简单协作代理构成的系统，这些代理可分裂为多个新代理。该框架允许采用空间奖励表述来简化信用分配问题，我们将其与消息传递网络相结合，在相邻网格单元间传播信息。我们通过实验验证了所提方法——自适应群体网格细化（ASMR）的有效性，表明该方法能在一系列具有挑战性的问题上学习可靠、可扩展且高效的细化策略。我们的方法显著加速了计算，在复杂仿真中相比均匀细化实现了高达30倍的性能提升。此外，我们超越了基于学习的基线方法，在无需关于误差信号的昂贵先验信息情况下，达到与传统基于误差的AMR策略相媲美的细化质量。