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Cormorant: Covariant Molecular Neural Networks

Brandon Anderson, Truong-Son Hy, Risi Kondor

TL;DR

Cormorant introduces a rotationally covariant neural network for modeling molecular forces and properties by tying each neuron to a physically meaningful atom subset and enforcing SO(3) covariance through Clebsch–Gordan-based nonlinearities. The architecture combines input featurization, covariant CG layers, and a scalar regression head, achieving strong performance on MD-17 and QM-9 while preserving translation invariance. Key contributions include the explicit use of Clebsch–Gordan nonlinearities to couple tensor orders, and a detailed scheme for one- and two-body interactions via covariant vertex and edge networks. The approach offers a principled, physics-informed alternative to traditional force fields and other ML potentials, with promising accuracy and interpretability for learning molecular energetics and properties.

Abstract

We propose Cormorant, a rotationally covariant neural network architecture for learning the behavior and properties of complex many-body physical systems. We apply these networks to molecular systems with two goals: learning atomic potential energy surfaces for use in Molecular Dynamics simulations, and learning ground state properties of molecules calculated by Density Functional Theory. Some of the key features of our network are that (a) each neuron explicitly corresponds to a subset of atoms; (b) the activation of each neuron is covariant to rotations, ensuring that overall the network is fully rotationally invariant. Furthermore, the non-linearity in our network is based upon tensor products and the Clebsch-Gordan decomposition, allowing the network to operate entirely in Fourier space. Cormorant significantly outperforms competing algorithms in learning molecular Potential Energy Surfaces from conformational geometries in the MD-17 dataset, and is competitive with other methods at learning geometric, energetic, electronic, and thermodynamic properties of molecules on the GDB-9 dataset.

Cormorant: Covariant Molecular Neural Networks

TL;DR

Cormorant introduces a rotationally covariant neural network for modeling molecular forces and properties by tying each neuron to a physically meaningful atom subset and enforcing SO(3) covariance through Clebsch–Gordan-based nonlinearities. The architecture combines input featurization, covariant CG layers, and a scalar regression head, achieving strong performance on MD-17 and QM-9 while preserving translation invariance. Key contributions include the explicit use of Clebsch–Gordan nonlinearities to couple tensor orders, and a detailed scheme for one- and two-body interactions via covariant vertex and edge networks. The approach offers a principled, physics-informed alternative to traditional force fields and other ML potentials, with promising accuracy and interpretability for learning molecular energetics and properties.

Abstract

We propose Cormorant, a rotationally covariant neural network architecture for learning the behavior and properties of complex many-body physical systems. We apply these networks to molecular systems with two goals: learning atomic potential energy surfaces for use in Molecular Dynamics simulations, and learning ground state properties of molecules calculated by Density Functional Theory. Some of the key features of our network are that (a) each neuron explicitly corresponds to a subset of atoms; (b) the activation of each neuron is covariant to rotations, ensuring that overall the network is fully rotationally invariant. Furthermore, the non-linearity in our network is based upon tensor products and the Clebsch-Gordan decomposition, allowing the network to operate entirely in Fourier space. Cormorant significantly outperforms competing algorithms in learning molecular Potential Energy Surfaces from conformational geometries in the MD-17 dataset, and is competitive with other methods at learning geometric, energetic, electronic, and thermodynamic properties of molecules on the GDB-9 dataset.

Paper Structure

This paper contains 30 sections, 24 equations, 1 table.

Table of Contents

  1. Introduction
  2. The nature of physical interactions in molecules
  3. Scalar interactions.
  4. Dipole/dipole interactions.
  5. Quadropole/quadropole interactions.
  6. Spherical tensors and representation theory
  7. The general form of interactions
  8. CORMORANT:$\:$ COvaRiant MOleculaR Artificial Neural neTworks
  9. Covariant neurons
  10. One-body and two-body interactions
  11. Overall structure and comparison with other architectures
  12. Experiments
  13. Conclusions
  14. Architecture
  15. Notation
  16. ...and 15 more sections

Theorems & Definitions (2)

  • Definition 1
  • Definition 2