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(Adaptive Mesh Refinement,AMR)对于基于网格的模拟非常关键,它可以动态调整网格的分辨率,以在计算成本和模拟精度之间进行折衷。然而,现有的AMR方法要么使用任务相关的启发式算法,要么使用昂贵的误差估计器,要么不能很好地扩展到更大的网格或更复杂的问题。在本文中,我们将AMR形式化为Swarm增强学习问题,将网格的每个元素视为简单而同质的智能体协作系统的一部分。我们结合了这个问题的公式化表达、一种新颖的智能体奖励函数和图神经网络,使我们能够学习任意方程系统上可靠和可扩展的精细策略。我们通过实验证明了我们的方法在提高复杂模拟的准确性和效率方面的有效性。我们的结果表明,在推理过程中,我们不需要误差指标就可以优于学习基线,并实现与传统基于误差的AMR精细策略同等的精细质量。