Nonlinear model reduction for operator learning
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.
- Model reduction and neural networks for parametric PDEs. arXiv preprint arXiv:2005.03180, 2020. doi: 10.48550/arXiv.2005.03180.
- Machine learning for partial differential equations. arXiv preprint arXiv:2303.17078, 2023. doi: 10.48550/arXiv.2303.17078.
- Nonlinear model reduction via discrete empirical interpolation. SIAM Journal on Scientific Computing, 32(5):2737–2764, 2010. doi: 10.1137/090766498.
- Thomas J. R. Hughes. The finite element method: Linear static and dynamic finite element analysis. Computer-Aided Civil and Infrastructure Engineering, 4(3):245–246, 1989. doi: https://doi.org/10.1111/j.1467-8667.1989.tb00025.x.
- Neural operator: Learning maps between function spaces. arXiv preprint arXiv:2108.08481, 2023. doi: 10.48550/arXiv.2108.08481.
- Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895, 2020. doi: https://doi.org/10.48550/arXiv.2010.08895.
- Decoupled weight decay regularization. In International Conference on Learning Representations, 2019. doi: 10.48550/arXiv.1711.05101.
- DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators. arXiv preprint arXiv:1910.03193, 2019. doi: 10.48550/arXiv.1910.03193.
- Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature Machine Intelligence, 3(3):218–229, 2021. doi: 10.1038/s42256-021-00302-5.
- A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data. Computer Methods in Applied Mechanics and Engineering, 393:114778, 2022. doi: https://doi.org/10.1016/j.cma.2022.114778.
- Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. The MIT Press, 06 2018. ISBN 9780262256933. doi: 10.7551/mitpress/4175.001.0001.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.
