标题： LRQ-Solver：一种用于快速准确求解大规模3D偏微分方程的基于Transformer的神经算子 显示英文标题

标题： LRQ-Solver: A Transformer-Based Neural Operator for Fast and Accurate Solving of Large-scale 3D PDEs

摘要： 求解复杂三维几何上的大规模偏微分方程（PDE）是科学和工程计算中的核心挑战，通常受到昂贵的预处理阶段和巨大的计算开销的阻碍。我们引入了基于低秩查询的PDE求解器（LRQ-Solver），这是一个集成物理的框架，专为工业级模型的快速、准确和高度可扩展的模拟而设计。该框架建立在两个主要的技术创新之上。首先，我们的参数条件拉格朗日建模（PCLM）方法将局部物理状态与全局设计参数显式耦合，从而在各种模拟配置中实现稳健的预测。通过将物理一致性直接嵌入学习架构中，PCLM确保即使在未见过的设计条件下，预测仍保持物理意义，显著增强了泛化能力和可靠性。其次，低秩查询注意力（LR-QA）模块利用物理场的二阶统计量构建全局一致性核，将注意力的计算复杂度从O(N2)降低到O(NC2 + C3)。通过用协方差分解代替点对聚类，LRQ-Solver实现了卓越的可扩展性，高效处理多达200万个点。在标准基准测试中，LRQ-Solver在DrivAer++数据集上实现了38.9%的误差减少，在3D Beam数据集上实现了28.76%，同时训练速度提高了多达50倍。我们的结果表明，LRQ-Solver为多配置物理模拟提供了一个强大的范式，提供了最先进的准确度、可扩展性和效率组合。用于重现实验的代码可在https://github.com/LilaKen/LRQ-Solver获取。 显示英文摘要

摘要： Solving large-scale Partial Differential Equations (PDEs) on complex three-dimensional geometries represents a central challenge in scientific and engineering computing, often impeded by expensive pre-processing stages and substantial computational overhead. We introduce Low-Rank Query-based PDE Solver (LRQ-Solver), a physics-integrated framework engineered for rapid, accurate, and highly scalable simulations of industrial-grade models. This framework is built upon two primary technical innovations. First, our Parameter Conditioned Lagrangian Modeling (PCLM) approach explicitly couples local physical states with global design parameters, enabling robust predictions across varied simulation configurations. By embedding physical consistency directly into the learning architecture, PCLM ensures that predictions remain physically meaningful even under unseen design conditions, significantly enhancing generalization and reliability. Second, the Low-Rank Query Attention (LR-QA) module leverages the second-order statistics of physical fields to construct a global coherence kernel, reducing the computational complexity of attention from O(N2) to O(NC2 + C3). By replacing point-wise clustering with covariance decomposition, LRQ-Solver achieves exceptional scalability efficiently processing up to 2 million points on a single GPU. Validated on standard benchmarks, LRQ-Solver achieves a 38.9% error reduction on the DrivAer++ dataset and 28.76% on the 3D Beam dataset, alongside a training speedup of up to 50 times. Our results establish that LRQ-Solver offers a powerful paradigm for multi-configuration physics simulations, delivering a SOTA combination of accuracy, scalability, and efficiency. Code to reproduce the experiments is available at https://github.com/LilaKen/LRQ-Solver.