pyeCE: A Python Implementation of the Embedded Cluster Expansion

Abstract

The cluster expansion is a widely used approach for predicting the finite-temperature thermodynamics of alloys from zero-kelvin first-principles calculations, but its conventional formulation becomes intractable for materials with more than three or four chemical species. High-entropy alloys have therefore remained largely out of reach. We present pyeCE, an open-source Python library that implements the embedded cluster expansion (eCE), in which machine learning maps many chemical species onto a smaller set of effective species and thereby limits the growth in the number of cluster functions. pyeCE provides the complete modeling workflow, including the construction of symmetry-adapted descriptors with a learnable per-sublattice chemical embedding, a neural-network energy model, ladder-based training, uncertainty quantification, and finite-temperature simulations through Monte Carlo sampling. Built on the PyTorch and pymatgen libraries, it supports systems with multiple species on multiple sublattices and runs on graphics processing units. We demonstrate the package on two material systems. In the first, a single model spanning the full composition space of a 9-component refractory alloy resolves short-range order and order--disorder behavior. This model enables rapid screening for compositions with strong Cr clustering, a feature linked to the formation of a continuous, corrosion-resistant oxide scale. In the second, a model of hydrogen dissolution in a Mo--Nb--W alloy reproduces the composition dependence of hydrogen uptake and resolves the interstitial environments that hydrogen occupies. The modular design of pyeCE allows it to be extended to problems beyond alloy thermodynamics, including kinetics, defect energetics, and the coupling of chemical order to magnetic and vibrational degrees of freedom.