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.

翻译：算子学习提供了逼近无限维函数空间之间映射的方法。深度算子网络（DeepONet）是该领域一种显著的架构。近期，基于模型降阶和神经网络的DeepONet扩展——本征正交分解（POD）-DeepONet，在多个基准测试中已在精度上超越其他架构。我们通过提出一种将神经网络与核主成分分析（KPCA）相结合的高效框架，将这一思想推广至非线性模型降阶领域，用于算子学习。实验结果表明，KPCA-DeepONet的性能优于POD-DeepONet。