Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities

Normalizing flows are exact-likelihood generative neural networks which approximately transform samples from a simple prior distribution to samples of the probability distribution of interest. Recent work showed that such generative models can be utilized in statistical mechanics to sample equilibrium states of many-body systems in physics and chemistry. To scale and generalize these results, it is essential that the natural symmetries in the probability density -- in physics defined by the invariances of the target potential -- are built into the flow. We provide a theoretical sufficient criterion showing that the distribution generated by \textit{equivariant} normalizing flows is invariant with respect to these symmetries by design. Furthermore, we propose building blocks for flows which preserve symmetries which are usually found in physical/chemical many-body particle systems. Using benchmark systems motivated from molecular physics, we demonstrate that those symmetry preserving flows can provide better generalization capabilities and sampling efficiency.