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.

翻译：在复杂三维几何上求解大规模偏微分方程是科学与工程计算的核心挑战，通常受到昂贵的预处理阶段和巨大的计算开销的阻碍。我们提出了基于低秩查询的偏微分方程求解器，这是一个集成了物理原理的框架，专为快速、精确且高度可扩展的工业级模型仿真而设计。该框架建立在两项主要技术创新之上。首先，我们的参数条件化拉格朗日建模方法显式地将局部物理状态与全局设计参数耦合，从而能够在不同的仿真配置下进行鲁棒的预测。通过将物理一致性直接嵌入学习架构，PCLM确保即使在未见过的设计条件下，预测结果仍保持物理意义，显著增强了泛化能力和可靠性。其次，低秩查询注意力模块利用物理场的二阶统计量构建全局相干核，将注意力计算复杂度从O(N²)降低至O(NC² + C³)。通过用协方差分解替代逐点聚类，LRQ-Solver实现了卓越的可扩展性，能够在单个GPU上高效处理高达200万个点。在标准基准测试上的验证表明，LRQ-Solver在DrivAer++数据集上实现了38.9%的误差降低，在3D Beam数据集上实现了28.76%的误差降低，同时训练速度提升高达50倍。我们的结果表明，LRQ-Solver为多配置物理仿真提供了一个强大的范式，在精度、可扩展性和效率方面实现了最先进的结合。用于复现实验的代码可在 https://github.com/LilaKen/LRQ-Solver 获取。