Adaptive Mesh Refinement (AMR) enhances the Finite Element Method, an important technique for simulating complex problems in engineering, by dynamically refining mesh regions, enabling a favorable trade-off between computational speed and simulation accuracy. Classical methods for AMR depend on heuristics or expensive error estimators, hindering their use for complex simulations. Recent learning-based AMR methods tackle these issues, 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 our approach, Adaptive Swarm Mesh Refinement (ASMR), on challenging refinement tasks. Our approach learns reliable and efficient refinement strategies that can robustly generalize to different domains during inference. Additionally, it achieves a speedup of up to $2$ orders of magnitude compared to uniform refinements in more demanding simulations. We outperform learned baselines and heuristics, achieving a refinement quality that is on par with costly error-based oracle AMR strategies.

翻译：自适应网格细化（AMR）通过动态细化网格区域，增强了有限元方法这一工程中模拟复杂问题的重要技术，从而在计算速度和模拟精度之间实现有利权衡。经典的AMR方法依赖于启发式策略或昂贵的误差估计器，限制了其在复杂模拟中的应用。近期基于学习的AMR方法解决了这些问题，但迄今为止仅能扩展到简单的玩具示例。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，其中网格被建模为简单的协作代理系统，这些代理可以分裂成多个新的代理。该框架允许一种空间奖励公式，简化了信用分配问题，我们将其与消息传递网络相结合，在相邻网格元素之间传播信息。我们在具有挑战性的细化任务上实验验证了我们的方法——自适应群体网格细化（ASMR）。我们的方法学习到可靠且高效的细化策略，可在推理过程中稳健地泛化到不同领域。此外，与更苛刻模拟中的均匀细化相比，它实现了高达2个数量级的加速。我们超越了学习到的基线和启发式方法，实现了与基于代价高昂误差的oracle AMR策略相当的细化质量。