Search

We present a novel algorithmic approach and an error analysis leveraging Quasi-Monte Carlo (QMC) points for training deep neural network (DNN) surrogates of holomorphic Data-to-Observable (DtO) maps in engineering design. Our analysis reveals higher-order consistent, deterministic choices of training points in the input parameter space for both deep and shallow neural networks with holomorphic activation functions such as tanh. We prove that higher-order QMC training points facilitate higher-order decay (in terms of the number of training samples) of the underlying generalization error, with consistency error bounds that are free from the curse of dimensionality in terms of the number of input parameters, provided that DNN weights in hidden layers satisfy certain summability conditions. We present numerical experiments for DtO maps from elliptic and parabolic PDEs with uncertain inputs that confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

We propose a supervised deep learning algorithm based on low-discrepancy sequences as the training set. By a combination of theoretical arguments and extensive numerical experiments we demonstrate that the proposed algorithm significantly outperforms standard deep learning algorithms that are based on randomly chosen training data for problems in moderately high dimensions. The proposed algorithm provides an efficient method for building inexpensive surrogates for many underlying maps in the context of scientific computing. [ABSTRACT FROM AUTHOR]