Efficient and dimension independent methods for neural network surrogate construction and training




O'Leary-Roseberry, Thomas Finnian

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In this dissertation I investigate how to efficiently construct neural network surrogates for parametric maps defined by PDEs, and how to use second order information to improve solutions to the related neural network training problem. Many-query problems arising in scientific applications (such as optimization, uncertainty quantification and inference problems) require evaluation of an input output mapping parametrized by a high dimensional nonlinear PDE model. The cost of these evaluations makes solution using the model prohibitive, and efficient accurate surrogates are the key to solving these problems in practice. In this work I investigate neural network surrogates that use model information to detect informed subspaces of the input and output where the parametric map can be represented efficiently. These compact representations require relatively few data to train and outperform conventional data-driven approaches which require large training data sets. Once a neural network is designed, training is a major issue. One seeks to find optimal weights for a neural network that generalize to data not seen during training. In this work I investigate how second order information can be efficiently exploited to design optimizers that have fast convergence and good generalization properties. These optimizers are shown to outperform conventional methods in numerical experiments.


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