Quantum mechanics

Algorithm

Cluster expansion

Philosophy

Mathematical physics

Applied mathematics

Theoretical computer science

Basis (linear algebra)

Convergence (economics)

Stability (learning theory)

Computation

Economics

Market economy

Computer science

Mathematical optimization

Acoustics

A priori and a posteriori

Polynomial

Completeness (order theory)

Physics

Mathematics

Invariant (physics)

Epistemology

Numerical stability

Mathematical analysis

Geometry

Precomputation

Numerical analysis

Closure (psychology)

Economic growth

Permutation (music)

Machine learning

Basis function

Journal of Computational Physics

The Atomic Cluster Expansion (Drautz (2019) [21]) provides a framework to systematically derive polynomial basis functions for approximating isometry and permutation invariant functions, particularly with an eye to modelling properties of atomistic systems. Our presentation extends the derivation by proposing a precomputation algorithm that yields immediate guarantees that a complete basis is obtained. We provide a fast recursive algorithm for efficient evaluation and illustrate its performance in numerical tests. Finally, we discuss generalisations and open challenges, particularly from a numerical stability perspective, around basis optimisation and parameter estimation, paving the way towards a comprehensive analysis of the convergence to a high-fidelity reference model.

Atomic cluster expansion: Completeness, efficiency and stability