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.

中文翻译：

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一种提高机器学习算法准确性的多级程序

我们提出了一种多层次的方法来提高机器学习算法的准确性，以逼近科学计算中的可观察量，特别是那些出现在由微分方程建模的系统中的可观察量。该算法依赖于明智地将大量计算成本低的粗分辨率训练数据与一些昂贵的细网格分辨率训练样本相结合。提供了基于减少底层映射的方差来降低泛化误差的理论论据，并提供了数值证据，表明对底层单级机器学习算法的显着收益。此外，我们还在前向不确定性量化的背景下应用了多级算法，并观察到与竞争算法相比有相当大的加速。