Nonlinear model reduction for operator learning
This work addresses a domain-specific problem in operator learning for researchers and practitioners, offering an incremental improvement over existing methods.
The paper tackles the problem of approximating mappings between infinite-dimensional function spaces in operator learning by proposing a framework that combines neural networks with kernel principal component analysis (KPCA) for nonlinear model reduction, resulting in KPCA-DeepONet outperforming POD-DeepONet in accuracy.
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.