在哪里计算以及如何交互：具有规范感知传输的算子可读自适应
Where to Compute and How to Interact: Operator-Readable Adaptation with Gauge-Aware Transport
- Central South University（中南大学）
- Nanjing University（南京大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对自适应网格神经算子忽视交互的问题，提出规范感知自适应网格神经算子（GA-AMNO），通过物理信息分配和几何条件化低秩规范传输实现可读的自适应，在五个PDE基准上提升精度并验证交互作用。
AI中文摘要：
自适应网格使得用于偏微分方程（PDE）的神经算子能够根据局部物理结构分配空间样本和计算资源。然而，现有方法主要关注在哪里计算，而对节点重定位后信息应如何交互关注较少。网格自适应改变了局部采样尺度、邻域结构和几何上下文，因此在不同节点处形成的表示可能无法直接比较。直接聚合因此可能将物理变化与离散化引起的表示变化纠缠在一起。由于分配和交互通过相同的输出目标进行联合优化，它们的各自作用也难以仅从最终误差中区分。我们引入了算子可读性，要求自适应算子能够解释并测试为何将计算分配到特定位置，以及在由此产生的非均匀离散化下表示如何交互。基于这一原则，我们提出了规范感知自适应网格神经算子（GA-AMNO）。物理信息自适应分配回答了在哪里计算，而几何条件化的低秩规范传输在聚合前将源特征映射到目标表示上下文中，回答了如何交互。这使得网格到求解器的信息交换可检查且可干预。我们建立了表示一致性聚合的充分条件，并分析了在保持拓扑的网格变形下的近似传输误差和连续性。在五个PDE基准上的实验表明，预测精度有所提高，而受控干预和几何失配分析验证了分配和交互的作用，并表明规范传输在强几何失配下提高了跨离散化表示的兼容性。
英文摘要：
Adaptive meshes enable neural operators for partial differential equations (PDEs) to allocate spatial samples and computation according to local physical structures. Existing approaches, however, mainly address where to compute, with less attention to how information should interact after node relocation. Mesh adaptation changes local sampling scales, neighborhood structures, and geometric contexts, so representations formed at different nodes may not be directly comparable. Direct aggregation can therefore entangle physical variation with discretization-induced representation variation. Because allocation and interaction are jointly optimized through the same output objective, their individual roles are also difficult to distinguish from final errors alone. We introduce operator readability, requiring an adaptive operator to account for and test why computation is allocated to particular locations and how representations interact under the resulting nonuniform discretization. Based on this principle, we propose the Gauge-Aware Adaptive Mesh Neural Operator (GA-AMNO). Physics-informed adaptive allocation answers where to compute, while geometry-conditioned low-rank Gauge transport maps source features into target representation contexts before aggregation, answering how to interact. This makes mesh-to-solver information exchange inspectable and intervenable. We establish sufficient conditions for representation-consistent aggregation and analyze approximate transport errors and continuity under topology-preserving mesh deformations. Experiments on five PDE benchmarks demonstrate improved predictive accuracy, while controlled interventions and geometric-mismatch analyses verify the roles of allocation and interaction and show that Gauge transport improves cross-discretization representation compatibility under strong geometric mismatch.
