<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varvec{15 \times }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn mathvariant="bold">15</mn> <mo mathvariant="bold">×</mo> </mrow> </math></EquationSource> </InlineEquation> relative to global PCA, while the fastest local-to-local configuration achieves a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{1.7 \times }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn mathvariant="bold">1.7</mn> <mo mathvariant="bold">×</mo> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varvec{7.52\%}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn mathvariant="bold">7.52</mn> <mo mathvariant="bold">%</mo> </mrow> </math></EquationSource> </InlineEquation> for global PCA-Net to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varvec{2.31\%}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn mathvariant="bold">2.31</mn> <mo mathvariant="bold">%</mo> </mrow> </math></EquationSource> </InlineEquation> while still achieving a <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varvec{1.4 \times }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn mathvariant="bold">1.4</mn> <mo mathvariant="bold">×</mo> </mrow> </math></EquationSource> </InlineEquation> 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.</p>

Localized PCA-Net neural operators for scalable solution reconstruction of elliptic PDEs

期刊论文
发表日期：

2026年5月23日

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Mrigank Dhingra,
Romit Maulik,
Adil Rasheed,
Omer San

摘要

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 $\varvec{15 \times }$ $15 \times$ relative to global PCA, while the fastest local-to-local configuration achieves a $\varvec{1.7 \times }$ $1.7 \times$ 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 $\varvec{7.52\%}$ $7.52 %$ for global PCA-Net to $\varvec{2.31\%}$ $2.31 %$ while still achieving a $\varvec{1.4 \times }$ $1.4 \times$ 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.