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
allocating mesh elements on the domain, balancing computational speed and
accuracy. Classical AMR depends on heuristics or expensive error estimators,
limiting its use in complex simulations. While learning-based AMR methods 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
during inference. Extensive experiments, including volumetric meshes and
Neumann boundary conditions, demonstrate that ASMR exceeds 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.

我们将自适应网格细化（AMR）作为一种协作、均质代理系统，通过迭代地分裂成多个新的代理来降低最大网格元素误差，从而提供了高效、稳定的优化，并在用户定义的分辨率下生成高度自适应网格。ASMR 在大量实验中展现了优于启发式方法和学习基准的性能，并在推理过程中在不同领域上推广，而且在更高要求的设置中产生比均匀细化快两个数量级的模拟网格。