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

2026-09-07

AI Summary

为减少高维PDE学习中的伪影，提出Two-Scale Localized PCA-Net方法，通过分解解为粗全局和局部残差，并结合块平衡潜在目标及接口感知微调来解决。

Abstract

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.

Problem

• Localized dimensionality reduction
• PDE operator learning
• Artifact reduction
• Block offsets
• Interface mismatches

Innovation

• Two-Scale Localized PCA-Net
• Coarse-Global Component
• Local Residual Corrections
• Block-Balanced Latent Objective
• Interface-Aware Fine-Tuning