Nonlinear model reduction for operator learning

TL;DR

This paper introduces a novel nonlinear model reduction framework combining neural networks with kernel PCA to enhance operator learning, outperforming existing POD-DeepONet in accuracy.

Contribution

It extends DeepONet by integrating kernel PCA for nonlinear model order reduction, achieving superior performance in operator learning tasks.

Findings

KPCA-DeepONet outperforms POD-DeepONet in accuracy

The proposed method efficiently captures nonlinear dynamics

Experimental results validate the effectiveness of the approach

Abstract

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.