Completeness of representations in atomistic machine learning

The problem of obtaining a comprehensive and symmetric representation of point particle groups, such as atoms in a molecule, is crucial in machine-learning techniques for physical problems 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 in three dimensions. 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. I will present a class of descriptors based on finite correlations based on the relative arrangement of particle triplets, which can be employed to create symmetry-adapted models with universal approximation capabilities and provide a solution to the problem of complete descriptors for machine learning