Atomistic machine learning with irreducible Cartesian natural tensors

Atomistic machine learning is a powerful tool for accurate and efficient investigation of material behavior at the atomic scale. While attempts have been made to construct models directly within Cartesian space, they face challenges in providing a systematic framework based on irreducible representations—a core feature of widely used spherical models. Here we propose Cartesian Natural Tensor Networks to overcome these limitations and thus offer a general, symmetry-preserving framework for atomistic machine learning. We present a theory of irreducible representations using Cartesian natural tensors, comprising their construction, their products, and a systematic scheme to decompose and reconstruct high-rank physical tensors. Leveraging this machinery, we develop equivariant machine learning interatomic potentials for materials and molecular systems with performance on par with leading spherical models. It further captures accurate structure–property relationships for tensorial quantities ranging from low-rank dipole moments to high-rank tensors with complex symmetries, such as the elastic constant tensor.