发表机构
School of Mathematics, Nanjing University of Aeronautics and Astronautics; Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT; School of Mathematics and Information Sciences, Yantai University（南京航空航天大学数学学院; 工业和信息化部南京航空航天大学数学建模与高性能计算重点实验室（NUAA）; 烟台大学数学与信息科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
HiLNO通过构建细-粗-细潜空间、多尺度监督和各向异性高斯注意力，在一般几何上高效学习PDE算子，以84.4%参数和69.2%FLOPs的削减实现竞争性精度。
AI 中文摘要
潜变量神经算子通过在紧凑的潜变量表示上执行主要计算，提高了偏微分方程（PDE）算子学习的效率。然而，直接压缩输入表示以获得这种紧凑表示可能会丢弃与解相关的空间信息，特别是对于具有多尺度结构的PDE解。为了解决这个问题，我们提出了HiLNO，一种分层潜变量神经算子，它构建了一个从细到粗再到细的潜空间，并进一步引入了多尺度监督（MSS）和各向异性高斯注意力。这种层次结构减轻了压缩过程中潜在的信息损失，而MSS将中间预测与下采样的目标场对齐，鼓励在多个空间尺度上捕获与解相关的结构。各向异性高斯注意力使得特征能够在层次结构之间传递，从而使HiLNO适用于一般几何。在代表性PDE基准和一个大规模汽车空气动力学任务上的实验表明，与LinearNO相比，HiLNO在实现具有竞争力的预测精度的同时，平均减少了84.4%的参数数量和69.2%的FLOPs。额外的实验证明了对未见过的空间分辨率的有效泛化。代码可在该https URL获取。
英文摘要
Latent neural operators improve the efficiency of operator learning for partial differential equations (PDEs) by performing the main computation on compact latent representations. However, directly compressing the input representation to obtain such compact representations may discard solution-relevant spatial information, especially for PDE solutions with multiscale structures. To address this problem, we propose HiLNO, a hierarchical latent neural operator that constructs a fine-to-coarse-to-fine latent space and further introduces multi-scale supervision (MSS) and anisotropic Gaussian attention. The hierarchy mitigates potential information loss during compression, while MSS aligns intermediate predictions with downsampled target fields, encouraging solution-relevant structures to be captured across multiple spatial scales. Anisotropic Gaussian attention enables feature transfer across the hierarchy, making HiLNO applicable to general geometries. Experiments on representative PDE benchmarks and a large-scale automotive aerodynamics task show that HiLNO achieves competitive predictive accuracy, while reducing the parameter count by an average of 84.4% and FLOPs by an average of 69.2% compared with LinearNO. Additional experiments demonstrate effective generalization to unseen spatial resolutions. Code is available at https://github.com/JcLimath/HiLNO.
