Complete and Efficient Covariants for Three-Dimensional Point Configurations with Application to Learning Molecular...

Area of Science:

• Computational chemistry
• Machine learning
• Quantum mechanics

Background:

• Incorporating SO(3)-covariance is crucial for machine learning models of molecular physical properties.
• Existing low-body-order feature models lack completeness.
• Higher-order methods are needed for comprehensive modeling.

Purpose of the Study:

• To formulate and prove general completeness properties for higher-order SO(3)-equivariant machine learning models.
• To determine the minimum number of features required for completeness up to k atoms.
• To improve the computational efficiency of these models.

Main Methods:

• Development of higher-order feature formulations for SO(3)-covariance.
• Proof of general completeness properties for these formulations.
• Replacement of Clebsch-Gordan operations with matrix multiplications.

Main Results:

• 6k - 5 features are sufficient for completeness with up to k atoms.
• Matrix multiplications achieve completeness, reducing computational scaling from O(l^6) to O(l^3).
• The methods are applicable to quantum chemistry and general 3D point configuration problems.

Conclusions:

• The proposed higher-order methods ensure completeness in SO(3)-equivariant machine learning for molecular properties.
• Optimization through matrix multiplication significantly enhances computational efficiency.
• These advancements offer broader applicability in computational science.