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

arXiv: https://arxiv.org/abs/2509.18110 · HTML: https://mlsanity.com/paper/2509.18110/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 59 references that are in the map.
- First posted on arXiv 9 Sep 2025.

## Builds on (most cited first, 30 of 31)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs](https://mlsanity.com/paper/2310.00120/) (2023)
- [Tensor Methods in Computer Vision and Deep Learning](https://mlsanity.com/paper/2107.03436/) (2021)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains](https://mlsanity.com/paper/2104.10873/) (2021)
- [Introducing Hann windows for reducing edge-effects in patch-based image segmentation](https://mlsanity.com/paper/1910.07831/) (2019)
- [Deciphering and integrating invariants for neural operator learning with various physical mechanisms](https://mlsanity.com/paper/2311.14361/) (2023)
- [Machine learning and domain decomposition methods -- a survey](https://mlsanity.com/paper/2312.14050/) (2023)
- [Model-Parallel Fourier Neural Operators as Learned Surrogates for Large-Scale Parametric PDEs](https://mlsanity.com/paper/2204.01205/) (2022)
- [Artificial intelligence for partial differential equations in computational mechanics: A review](https://mlsanity.com/paper/2410.19843/) (2024)
- [Modeling Multivariable High-resolution 3D Urban Microclimate Using Localized Fourier Neural Operator](https://mlsanity.com/paper/2411.11348/) (2024)
- [An Advanced Physics-Informed Neural Operator for Comprehensive Design Optimization of Highly-Nonlinear Systems: An Aerospace Composites Processing Case Study](https://mlsanity.com/paper/2406.14715/) (2024)
- [A Generalized Schwarz-type Non-overlapping Domain Decomposition Method using Physics-constrained Neural Networks](https://mlsanity.com/paper/2307.12435/) (2023)

## Built on by (most cited first, 1 of 1)

- [A Trainable-by-Parts Operator Learning Framework: Bridging DeepONet and Karhunen-Loeve Expansions for Large-Scale Applications](https://mlsanity.com/paper/2606.28519/) (2026)

## Abstract (arXiv)

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.