Adaptive Mesh Refinement (AMR) is crucial for mesh-based simulations, as it allows for dynamically adjusting the resolution of a mesh to trade off computational cost with the simulation accuracy. Yet, existing methods for AMR either use task-dependent heuristics, expensive error estimators, or do not scale well to larger meshes or more complex problems. In this paper, we formalize AMR as a Swarm Reinforcement Learning problem, viewing each element of a mesh as part of a collaborative system of simple and homogeneous agents. We combine this problem formulation with a novel agent-wise reward function and Graph Neural Networks, allowing us to learn reliable and scalable refinement strategies on arbitrary systems of equations. We experimentally demonstrate the effectiveness of our approach in improving the accuracy and efficiency of complex simulations. Our results show that we outperform learned baselines and achieve a refinement quality that is on par with a traditional error-based AMR refinement strategy without requiring error indicators during inference.

翻译：自适应网格细化（AMR）对于基于网格的模拟至关重要，因为它能动态调整网格分辨率，以平衡计算成本与模拟精度。然而，现有的AMR方法要么依赖任务特定的启发式规则，要么依赖昂贵的误差估计器，要么无法很好地扩展至更大网格或更复杂问题。在本文中，我们将AMR形式化为一个群体强化学习问题，将网格中的每个元素视为由简单同质智能体组成的协作系统的一部分。我们将此问题表述与一种新颖的智能体级奖励函数及图神经网络相结合，从而能够在任意方程组上学习可靠且可扩展的细化策略。我们通过实验证明了该方法在提升复杂模拟精度与效率方面的有效性。结果表明，我们优于基于学习的基线方法，并且在推理过程中无需误差指示器即可达到与传统基于误差的AMR细化策略相当的细化质量。