The Atomic Cluster Expansion (Drautz, Phys. Rev. B 99, 2019) provides a framework to systematically derive polynomial basis functions for approximating isometry and permutation invariant functions, particularly with an eye to modelling properties of atomistic systems. Our presentation extends the derivation by proposing a precomputation algorithm that yields immediate guarantees that a complete basis is obtained. We provide a fast recursive algorithm for efficient evaluation and illustrate its performance in numerical tests. Finally, we discuss generalisations and open challenges, particularly from a numerical stability perspective, around basis optimisation and parameter estimation, paving the way towards a comprehensive analysis of the convergence to a high-fidelity reference model.

翻译：原子集群扩展(Drautz,Phys.Rev.B 99, 2019)提供了一个框架,以便系统地得出近似异度测量和变异性功能的多元基函数,特别是着眼于原子系统的建模特性。我们的介绍扩展了这一推论,提出了一种预估算法,即能立即保证获得完整基础。我们提供了一种快速的递归算法,用于高效评估,并在数字测试中说明其性能。最后,我们讨论了一般化和公开的挑战,特别是从数字稳定性的角度,围绕基础优化和参数估计,为全面分析与高不贞参考模型的趋同情况铺平了道路。