Achieving a complete and symmetric description of a group of point particles, such as atoms in a molecule, is a common problem in physics and theoretical chemistry. The introduction of machine learning to science has made this issue even more critical, as it underpins the ability of a model to reproduce arbitrary physical relationships, and to do so while being consistent with basic 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 introduce, and demonstrate on a particularly insidious class of atomic arrangements, a strategy to build descriptors that rely solely on information on the relative arrangement of triplets of particles, but can be used to construct symmetry-adapted models that have universal approximation power.

翻译：实现一组点粒子（如分子中的原子）的完整且对称的描述，是物理学和理论化学中的常见问题。机器学习的引入使这一问题更加关键，因为它支撑着模型再现任意物理关系的能力，同时保持与基本对称性和守恒律的一致性。然而，常用于表示点云的描述符——尤其是那些描述原子尺度物质的描述符——无法区分粒子的特殊排列，这使得无法通过机器学习其性质。存在可证明完备的框架，但它们仅在同时描述所有原子之间相互关系的极限情况下才完备，而这在实际中并不可行。我们针对一类特别隐蔽的原子排列，引入并展示了一种构建描述符的策略，该策略仅依赖于三元组粒子的相对排列信息，但可用于构建具有通用逼近能力的对称自适应模型。