# Nonlinear model reduction for operator learning

arXiv: https://arxiv.org/abs/2403.18735 · HTML: https://mlsanity.com/paper/2403.18735/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 10 references that are in the map.
- First posted on arXiv 27 Mar 2024.

## Builds on (most cited first, 5 of 5)

- [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)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Machine Learning for Partial Differential Equations](https://mlsanity.com/paper/2303.17078/) (2023)

## Built on by (most cited first, 8 of 8)

- [Spatio-spectral graph neural operator for solving computational mechanics problems on irregular domain and unstructured grid](https://mlsanity.com/paper/2409.00604/) (2024)
- [A general reduced-order neural operator for spatio-temporal predictive learning on complex spatial domains](https://mlsanity.com/paper/2409.05508/) (2024)
- [A Neural Operator based Hybrid Microscale Model for Multiscale Simulation of Rate-Dependent Materials](https://mlsanity.com/paper/2506.16918/) (2025)
- [Variational Rank Reduction Autoencoders for Generative Thermal Design](https://mlsanity.com/paper/2509.08515/) (2025)
- [Enhancing Multiscale Simulations with Constitutive Relations-Aware Deep Operator Networks](https://mlsanity.com/paper/2405.13759/) (2024)
- [The Error of Deep Operator Networks Is the Sum of Its Parts: Branch-Trunk and Mode Error Decompositions](https://mlsanity.com/paper/2602.21910/) (2026)
- [Physics-Informed Reduced-Order Operator Learning for Hyperelasticity in Continuum Micromechanics](https://mlsanity.com/paper/2605.07738/) (2026)
- [NOFE - Neural Operator Function Embedding](https://mlsanity.com/paper/2605.11970/) (2026)

## Abstract (arXiv)

Operator learning provides methods to approximate mappings between infinite-dimensional function spaces. Deep operator networks (DeepONets) are a notable architecture in this field. Recently, an extension of DeepONet based on model reduction and neural networks, proper orthogonal decomposition (POD)-DeepONet, has been able to outperform other architectures in terms of accuracy for several benchmark tests. We extend this idea towards nonlinear model order reduction by proposing an efficient framework that combines neural networks with kernel principal component analysis (KPCA) for operator learning. Our results demonstrate the superior performance of KPCA-DeepONet over POD-DeepONet.