Localized PCA-Net Neural Operators for Scalable Solution Reconstruction of Elliptic PDEs
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918145747320832 |
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| author | Dhingra, Mrigank Maulik, Romit Rasheed, Adil San, Omer |
| author_facet | Dhingra, Mrigank Maulik, Romit Rasheed, Adil San, Omer |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_18110 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Localized PCA-Net Neural Operators for Scalable Solution Reconstruction of Elliptic PDEs Dhingra, Mrigank Maulik, Romit Rasheed, Adil San, Omer Machine Learning Computer Vision and Pattern Recognition 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. |
| title | Localized PCA-Net Neural Operators for Scalable Solution Reconstruction of Elliptic PDEs |
| topic | Machine Learning Computer Vision and Pattern Recognition |
| url | https://arxiv.org/abs/2509.18110 |