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.