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We propose a multi-level method to increase the accuracy of machine learning algorithms for approximating observables in scientific computing, particularly those that arise in systems modelled by differential equations. The algorithm relies on judiciously combining a large number of computationally cheap training data on coarse resolutions with a few expensive training samples on fine grid resolutions. Theoretical arguments for lowering the generalisation error, based on reducing the variance of the underlying maps, are provided and numerical evidence, indicating significant gains over underlying single-level machine learning algorithms, are presented. Moreover, we also apply the multi-level algorithm in the context of forward uncertainty quantification and observe a considerable speedup over competing algorithms. [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]