Adaptive mesh refinement is central to the efficient solution of partial differential equations (PDEs) via the finite element method (FEM). Classical $r$-adaptivity optimizes vertex positions but requires solving expensive auxiliary PDEs such as the Monge-Ampère equation, while classical $h$-adaptivity modifies topology through element subdivision but suffers from expensive error indicator computation and is constrained by isotropic refinement patterns that impose accuracy ceilings. Combined $hr$-adaptive techniques naturally outperform single-modality approaches, yet inherit both computational bottlenecks and the restricted cost-accuracy trade-off. Emerging machine learning methods for adaptive mesh refinement seek to overcome these limitations, but existing approaches address $h$-adaptivity or $r$-adaptivity in isolation. We present HypeR, a deep reinforcement learning framework that jointly optimizes mesh relocation and refinement. HypeR casts the joint adaptation problem using tools from hypergraph neural networks and multi-agent reinforcement learning. Refinement is formulated as a heterogeneous multi-agent Markov decision process (MDP) where element agents decide discrete refinement actions, while relocation follows an anisotropic diffusion-based policy on vertex agents with provable prevention of mesh tangling. The reward function combines local and global error reduction to promote general accuracy. Across benchmark PDEs, HypeR reduces approximation error by up to 6--10$\times$ versus state-of-art $h$-adaptive baselines at comparable element counts, breaking through the uniform refinement accuracy ceiling that constrains subdivision-only methods. The framework produces meshes with improved shape metrics and alignment to solution anisotropy, demonstrating that jointly learned $hr$-adaptivity strategies can substantially enhance the capabilities of automated mesh generation.

翻译：自适应网格细化是采用有限元方法高效求解偏微分方程的核心技术。经典 $r$ 自适应方法通过优化顶点位置提升网格质量，但需求解昂贵的辅助偏微分方程（如Monge-Ampère方程）；而经典 $h$ 自适应方法通过单元细分修改网格拓扑，却面临误差指标计算代价高昂、各向同性细化模式导致精度天花板等问题。虽然联合 $hr$ 自适应技术能天然超越单一模态方法，但既继承了上述计算瓶颈，又受限于成本-精度权衡的严格约束。新兴的机器学习自适应网格细化方法试图突破这些局限，但现有研究仅孤立解决 $h$ 自适应或 $r$ 自适应问题。本文提出HypeR——一种深度强化学习框架，可联合优化网格重定位与细化。该框架利用超图神经网络与多智能体强化学习工具，将联合自适应问题形式化为异构多智能体马尔可夫决策过程：单元智能体决策离散细化操作，顶点智能体则基于各向异性扩散策略执行重定位，并具有可证明的网格缠绕防止机制。奖励函数融合局部与全局误差约简，促进整体精度提升。在基准偏微分方程测试中，相较于当前最优$h$自适应基线，HypeR在单元数相当条件下将逼近误差降低达6–10倍，突破了纯细分方法的均匀细化精度天花板。该框架生成的网格在形状度量与解各向异性对齐性上均显著改善，证明联合学习的$hr$自适应策略可大幅增强自动化网格生成能力。