{"ID":2873185,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.18110","arxiv_id":"2509.18110","title":"Localized PCA-Net Neural Operators for Scalable Solution Reconstruction of Elliptic PDEs","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.","short_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 fra...","url_abs":"https://arxiv.org/abs/2509.18110","url_pdf":"https://arxiv.org/pdf/2509.18110v1","authors":"[\"Mrigank Dhingra\",\"Romit Maulik\",\"Adil Rasheed\",\"Omer San\"]","published":"2025-09-09T20:13:51Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"cs.CV\"]","methods":"[\"Convolutional Neural Network\"]","has_code":false}