Neural Network Renormalization Group

We present a variational renormalization group approach using deep generative model composed of bijectors. The model can learn hierarchical transformations between physical variables and renormalized collective variables. It can directly generate statistically independent physical configurations by iterative refinement at various length scales. The generative model has an exact and tractable likelihood, which provides renormalized energy function of the collective variables and supports unbiased rejection sampling of the physical variables. To train the neural network, we employ probability density distillation, in which the training loss is a variational upper bound of the physical free energy. The approach could be useful for automatically identifying collective variables and effective field theories.