Simulating physical systems is essential in engineering, but analytical solutions are limited to straightforward problems. Consequently, numerical methods like the Finite Element Method (FEM) are widely used. However, the FEM becomes computationally expensive as problem complexity and accuracy demands increase. Adaptive Mesh Refinement (AMR) improves the FEM by dynamically placing mesh elements on the domain, balancing computational speed and accuracy. Classical AMR depends on heuristics or expensive error estimators, which may lead to suboptimal performance for complex simulations. While AMR methods based on machine learning are promising, they currently only scale to simple problems. In this work, we formulate AMR as a system of collaborating, homogeneous agents that iteratively split into multiple new agents. This agent-wise perspective enables a spatial reward formulation focused on reducing the maximum mesh element error. Our approach, Adaptive Swarm Mesh Refinement++ (ASMR++), offers efficient, stable optimization and generates highly adaptive meshes at user-defined resolution at inference time. Extensive experiments demonstrate that ASMR++ outperforms heuristic approaches and learned baselines, matching the performance of expensive error-based oracle AMR strategies. ASMR additionally generalizes to different domains during inference, and produces meshes that simulate up to 2 orders of magnitude faster than uniform refinements in more demanding settings.

翻译：物理系统仿真在工程领域至关重要，但解析解仅限于处理简单问题。因此，有限元法（FEM）等数值方法得到了广泛应用。然而，随着问题复杂性和精度要求的提升，FEM的计算成本急剧增加。自适应网格细化（AMR）通过动态地在计算域上布置网格单元，在计算速度与精度之间取得平衡，从而改进了FEM。传统的AMR依赖于启发式方法或计算成本高昂的误差估计器，这可能导致复杂仿真中的性能欠佳。尽管基于机器学习的AMR方法前景广阔，但目前仅能扩展到简单问题。在本工作中，我们将AMR建模为一个由协作的、同质智能体组成的系统，这些智能体迭代地分裂为多个新智能体。这种基于智能体的视角使得我们能够构建一种空间奖励机制，其核心在于降低最大网格单元误差。我们提出的方法——自适应群体网格细化++（ASMR++），提供了高效、稳定的优化，并在推理时以用户定义的分辨率生成高度自适应的网格。大量实验表明，ASMR++的性能优于启发式方法和学习基线，与计算昂贵的、基于误差的预言式AMR策略性能相当。此外，ASMR在推理时能够泛化到不同的计算域，并且在要求更高的场景中，其生成的网格仿真速度比均匀细化方法快高达2个数量级。