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.

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.

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.

AUTHOR RESPONSE TO THE REVIEWS OF "CORMORANT: COVARIANT MOLECULAR NEURAL NETWORKS" We thank all three reviewers for their insightful comments and positive evaluations of our manuscript. We will update the paper to reflect their suggestions by the camera ready deadline. We will shortly release a Python library that implements the Cormorant architecture. Currently we are just cleaning up and documenting the code. As for the other points brought up by the reviewers we have the following comments: 1. Structure and supplement: As suggested by Reviewer 2, we will move some details of the technical implementation in Section 4.4 to the supplement.

The paper is well-written and clearly draws the connection between physical interactions, tensors and the proposed neural network Cormorant. The proposed network is related to earlier work on tensor field networks [Thomas et al] and covariant compositional networks, but presents architectural changes that lead to improved results on the QM9 and MD-17 benchmarks. Confusingly, while the introduction motivates the work by prediction of atomic force fields, only scalar values are predicted in the experiments. This is also part of the definition of Cormorant: "C3. The type of each output neuron is [...] a scalar". This seems not to be compatible with force field predictions, and also some other important chemical properties are vectors (e.g.

All reviewers agree that this is a refreshing and interesting paper.

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.

Equivariant neural networks (ENNs) are graph neural networks embedded in $\mathbb{R}^3$ and are well suited for predicting molecular properties. The ENN library e3nn has customizable convolutions, which can be designed to depend only on distances between points, or also on angular features, making them rotationally invariant, or equivariant, respectively. This paper studies the practical value of including angular dependencies for molecular property prediction using the QM9 data set. We find that for fixed network depth, adding angular features improves the accuracy on most targets. For most, but not all, molecular properties, distance-only e3nns (L0Nets) can compensate by increasing convolutional layer depth. Our angular-feature e3nn (L1Net) architecture beats previous state-of-the-art results on the global electronic properties dipole moment, isotropic polarizability, and electronic spatial extent.

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.