双尺度局部化PCA-Net：用于减少伪影的PDE算子学习的粗全局与局部残差表示
Two-Scale Localized PCA-Net: Coarse-Global and Local-Residual Representations for Artifact-Reduced PDE Operator Learning
- University of Tennessee（田纳西大学）
- Boston University（波士顿大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出双尺度局部化PCA-Net，通过粗全局与局部残差分解，减少块伪影并降低计算成本，在泊松和达西流基准上显著提升重建精度。
中文摘要 AI 辅助
局部化降维提高了高维偏微分方程（PDE）算子学习的可扩展性，但独立解码的局部块可能引入块偏移、界面不匹配和虚假的高波数内容。我们提出了双尺度局部化PCA-Net，将解分解为粗全局分量和局部残差修正。一个紧凑的全局PCA基捕获域尺度结构，而非重叠的局部PCA基表示剩余的细尺度残差。一个块平衡的潜在目标耦合了这两种表示，可选的界面感知微调通过重建和迹损失进一步促进连续性。在泊松基准上，双尺度表示相对于普通和基于重叠的局部化PCA-Net显著降低了重建误差和可见块伪影，同时相对于重叠方法将PCA拟合成本大约减半。在非均质达西流上，它大幅减少了界面和离散残差误差，重建改进较为温和。消融研究表明，主要改进来自双尺度输出表示，而界面感知微调提供了互补的连续性细化。总体而言，将全局相干结构与局部残差细节分离为减少伪影的PDE算子学习提供了一种高效表示。
英文摘要
Localized dimensionality reduction improves the scalability of operator learning for high-dimensional partial differential equations (PDEs), but independently decoded local patches can introduce block offsets, interface mismatches, and spurious high-wavenumber content. We introduce Two-Scale Localized PCA-Net, which decomposes the solution into a coarse-global component and local residual corrections. A compact global PCA basis captures domain-scale structure, while nonoverlapping local PCA bases represent the remaining fine-scale residual. A block-balanced latent objective couples the two representations, and optional interface-aware fine-tuning further promotes continuity through reconstruction and trace losses. On Poisson benchmarks, the two-scale representation substantially reduces reconstruction error and visible block artifacts relative to plain and overlap-based localized PCA-Net while approximately halving PCA fitting cost relative to overlap. On heterogeneous Darcy flow, it strongly reduces interface and discrete-residual errors, with more modest reconstruction gains. Ablations show that the primary improvement arises from the two-scale output representation, while interface-aware fine-tuning provides complementary continuity refinement. Overall, separating globally coherent structure from localized residual detail provides an efficient representation for artifact-reduced PDE operator learning.
