We have developed a technique combining the accuracy of quantum Monte Carlo in describing the electron correlation with the efficiency of a machine learning potential (MLP). We use kernel linear regression in combination with SOAP (Smooth Overlap Atomic Position) approach, implemented here in a very efficient way. The key ingredients are: i) a sparsification technique, based on farthest point sampling, ensuring generality and transferability of our MLPs and ii) the so called $\Delta$-learning, allowing a small training data set, a fundamental property for highly accurate but computationally demanding calculations, such as the ones based on quantum Monte Carlo. As a first application we present a benchmark study of the liquid-liquid transition of high-pressure hydrogen and show the quality of our MLP, by emphasizing the importance of high accuracy for this very debated subject, where experiments are difficult in the lab, and theory is still far from being conclusive.

翻译：我们开发了一种技术,将蒙特卡洛量子的准确性与机器学习潜力的效率(MLP)描述电子相关性结合起来。我们使用内核线性回归结合SOAP(烟雾重叠原子位置)方法,并在此非常高效地实施。关键成分是:i)基于最远点抽样的封闭技术,确保我们MLP的普遍性和可转移性,以及ii)所谓的“Delta$学习”,允许一个小型的培训数据集,这是非常准确但计算要求很高的计算基础,例如基于蒙特卡洛量的计算。作为第一个应用,我们介绍了高压氢液体-液体过渡的基准研究,并展示了我们MLP的质量。我们强调高精度精度对于这个在实验室难以进行实验而且理论还远远没有定论的非常有争议的主题的重要性。