In this paper, we address the challenge of obtaining a comprehensive and symmetric representation of point particle groups, such as atoms in a molecule, which is crucial in physics and theoretical chemistry. The problem has become even more important with the widespread adoption of machine-learning techniques in science, as it underpins the capacity of models to accurately reproduce physical relationships while being consistent with fundamental symmetries and conservation laws. However, the descriptors that are commonly used to represent point clouds -- most notably those adopted to describe matter at the atomic scale -- are unable to distinguish between special arrangements of particles. This makes it impossible to machine learn their properties. Frameworks that are provably complete exist but are only so in the limit in which they simultaneously describe the mutual relationship between all atoms, which is impractical. We present a novel approach to construct descriptors of finite correlations based on the relative arrangement of particle triplets, which can be employed to create symmetry-adapted models with universal approximation capabilities. Our strategy is demonstrated on a class of atomic arrangements that are specifically built to defy a broad class of conventional symmetric descriptors, showcasing its potential for addressing their limitations.

翻译：在本文中，我们解决了获取点粒子群（如分子中的原子）全面且对称表示这一挑战，该问题在物理学和理论化学中至关重要。随着机器学习技术在科学领域的广泛采用，这一问题变得更为重要，因为它支撑着模型在遵循基本对称性和守恒定律的同时准确复现物理关系的能力。然而，常用于表示点云的描述符——尤其是那些用于描述原子尺度物质的描述符——无法区分粒子的特殊排列方式。这使得无法通过机器学习预测其性质。存在可证明完备的框架，但仅限于同时描述所有原子之间相互关系的极限情况，而这在实际中并不可行。我们提出了一种基于粒子三重相对排列构建有限相关性描述符的新方法，可用于创建具有通用逼近能力的对称适应模型。我们的策略在一类专门为挑战传统对称描述符而构建的原子排列上进行了验证，展示了其解决这些限制的潜力。