Supplement Number 1

The electronic supporting material for this publi cation include further details on the implementation, different models and datasets, and additional results on benchmarks. The source code torch-pme and jax-pme packages are publicly available at https://github.com/lab-cosmo/torch-pme and https://github.com/lab-cosmo/jax-pme. Some results are compared with the generalized LODE descriptors introduced in previ- ous work by the authors, which have been computed using the featomic (formerly rascaline) package, available at https://github.com/metatensor/featomic.