arXiv:2609.08034cs.LG2026-09

通过分层结构减少偏微分方程求解中的块状伪影,提升精度与效率。

Two-Scale Localized PCA-Net: Coarse-Global and Local-Residual Representations for Artifact-Reduced PDE Operator Learning

论文配图:Two-Scale Localized PCA-Net: Coarse-Global and Local-Residual Representations for Artifact-Reduced PDE Operator Learning
图 1 · 摘自论文原文
  • 将解分解为全局粗结构与局部残差,分离不同尺度特征
  • 在泊松问题上重建误差降低,计算成本减半,伪影显著减少
  • 适合需要高精度和连续性的科学计算场景

局部降维可提升高维偏微分方程(PDE)算子学习的可扩展性,但独立解码的局部块易引入块偏移、界面不连续及虚假高频成分。本文提出两尺度局部PCA-Net,将解分解为粗全局成分与局部残差修正。紧凑的全局PCA基捕捉整体结构,非重叠局部PCA基表征精细残差。块平衡潜在目标耦合两类表示,可选的界面感知微调通过重构与迹损失进一步提升连续性。在泊松基准测试中,该方法显著降低重建误差与可见块状伪影,同时相比重叠式方法约节省一半的PCA拟合开销。在异质达西流问题中,有效抑制界面误差与离散残差,重建提升较温和。消融实验表明,主要改进来自两尺度输出表示,界面微调提供互补的连续性优化。总体而言,分离全局相干结构与局部残差细节,为低伪影的PDE算子学习提供了高效表征。

原文摘要 · Abstract (English)

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.

PDE求解降维神经网络误差控制

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