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We describe N-body networks, a neural network architecture for learning thebehavior and properties of complex many body physical systems. Our specificapplication is to learn atomic potential energy surfaces for use in moleculardynamics simulations. Our architecture is novel in that (a) it is based on ahierarchical decomposition of the many body system into subsytems, (b) theactivations of the network correspond to the internal state of each subsystem,(c) the "neurons" in the network are constructed explicitly so as to guaranteethat each of the activations is covariant to rotations, (d) the neurons operateentirely in Fourier space, and the nonlinearities are realized by tensorproducts followed by Clebsch-Gordan decompositions. As part of the descriptionof our network, we give a characterization of what way the weights of thenetwork may interact with the activations so as to ensure that the covarianceproperty is maintained.

N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentials