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 modeled 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 generalization
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 speed-up over competing algorithms
