Localized PCA-Net Neural Operators for Scalable Solution Reconstruction of Elliptic PDEs

Neural operator learning has emerged as a powerful approach for approximating solution operators of partial differential equations (PDEs) in a data-driven manner. However, applying principal component analysis (PCA) to high-dimensional input and solution fields incurs significant computational overhead, particularly because the global singular value decomposition becomes expensive as the spatial resolution increases. To address this limitation, we propose a localized PCA-Net framework that decomposes the computational domain into smaller patch-wise coverings, applies PCA within each local patch, and trains a neural operator in the resulting reduced latent space. We investigate two patch-based formulations that balance computational efficiency, global coupling, and reconstruction accuracy: (1) local-to-global patch PCA, in which input fields are compressed locally while solution fields are represented globally, and (2) local-to-local patch PCA, in which both input and solution fields are compressed locally. To mitigate patch-interface artifacts in the local-to-local setting, we further study two refinement strategies: overlapping patch reconstruction with Hann-type weighted blending and a two-stage CNN-based RefinementNet. Experiments are conducted under a fixed-split, multi-seed protocol with relative error, structural similarity, interface-jump, PDE-residual, and runtime diagnostics. On the 2D Poisson benchmark, localized PCA reduces PCA fitting time by up to approximately 15×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{15 \times }$$\end{document} relative to global PCA, while the fastest local-to-local configuration achieves a 1.7×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{1.7 \times }$$\end{document} end-to-end speedup when PCA fitting, latent transformations, training, and inference are all included. The overlap-enhanced local-to-local model provides the best accuracy and physical consistency, reducing mean relative error from 7.52%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{7.52\%}$$\end{document} for global PCA-Net to 2.31%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{2.31\%}$$\end{document} while still achieving a 1.4×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{1.4 \times }$$\end{document} end-to-end speedup. Additional studies on variable-coefficient Darcy flow, randomized SVD, patch-wise neural-network heads, and Gaussian-random-field roughness demonstrate the robustness, limitations, and extensibility of the proposed localized PCA-Net framework for scalable operator learning.