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

• URL: http://arxiv.org/abs/2509.18110v1
• Date: Tue, 09 Sep 2025 20:13:51 GMT
• Title: Localized PCA-Net Neural Operators for Scalable Solution Reconstruction of Elliptic PDEs
• Authors: Mrigank Dhingra, Romit Maulik, Adil Rasheed, Omer San,
• Abstract summary: We propose a patch-based PCA-Net framework that decomposes the solution fields into smaller patches, applies PCA within each patch, and trains a neural operator in the reduced PCA space.<n>Our results demonstrate that patch-based PCA significantly reduces computational complexity while maintaining high accuracy, reducing end-to-end pipeline processing time by a factor of 3.7 to 4 times.
• Score: 5.788187988343425
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Neural operator learning has emerged as a powerful approach for solving partial differential equations (PDEs) in a data-driven manner. However, applying principal component analysis (PCA) to high-dimensional solution fields incurs significant computational overhead. To address this, we propose a patch-based PCA-Net framework that decomposes the solution fields into smaller patches, applies PCA within each patch, and trains a neural operator in the reduced PCA space. We investigate two different patch-based approaches that balance computational efficiency and reconstruction accuracy: (1) local-to-global patch PCA, and (2) local-to-local patch PCA. The trade-off between computational cost and accuracy is analyzed, highlighting the advantages and limitations of each approach. Furthermore, within each approach, we explore two refinements for the most computationally efficient method: (i) introducing overlapping patches with a smoothing filter and (ii) employing a two-step process with a convolutional neural network (CNN) for refinement. Our results demonstrate that patch-based PCA significantly reduces computational complexity while maintaining high accuracy, reducing end-to-end pipeline processing time by a factor of 3.7 to 4 times compared to global PCA, thefore making it a promising technique for efficient operator learning in PDE-based systems.

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