# A Multi-level procedure for enhancing accuracy of machine learning algorithms

arXiv: https://arxiv.org/abs/1909.09448 · HTML: https://mlsanity.com/paper/1909.09448/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Sep 2019.

## Built on by (most cited first, 12 of 12)

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- [Error analysis for deep neural network approximations of parametric hyperbolic conservation laws](https://mlsanity.com/paper/2207.07362/) (2022)
- [Adaptive Multilevel Neural Networks for Parametric PDEs with Error Estimation](https://mlsanity.com/paper/2403.12650/) (2024)
- [AsyReC: A Multimodal Graph-based Framework for Spatio-Temporal Asymmetric Dyadic Relationship Classification](https://mlsanity.com/paper/2504.05030/) (2025)
- [Multilevel CNNs for Parametric PDEs based on Adaptive Finite Elements](https://mlsanity.com/paper/2408.10838/) (2024)
- [Theoretical and Practical Advances in Hierarchical Bayesian Deep Learning Methodology](https://mlsanity.com/paper/2610.19074/) (2026)

## Abstract (arXiv)

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.