Interpretable Hierarchical Normalizing Flows via a Gauge-Fixed Neural Renormalization Group

Physical systems with structure at every scale, such as critical phenomena, are hard to simulate and to interpret. A hierarchical generative model has the potential to address both, especially if its layers can be read as a coarse graining and reused at sizes never trained on. The renormalization group (RG) formalizes such a coarse graining, but many learned hierarchies realize the same density, so reading their layers as an RG is not well posed. We construct a neural RG for the two-dimensional Ising model with this gauge freedom fixed: an exact-likelihood hierarchical normalizing flow built from continuous normalizing flows on 2×2 blocks with a fixed Haar split, an exactly symmetric vector field, kinetic-energy and marginal-Gaussianity regularizers, and a normalization of the hidden variables fitted after training. With the gauge fixed, we observe that independently trained models learn the same block maps, and the maps of a temperature ladder order themselves along the RG flow between its three fixed points. Because the critical map is the same at every scale, a model trained at L = 64 extends to L = 128 by repeating one level, without new training, and reproduces the two-point function and magnetization distribution as well as a model trained at L = 128, whereas a parameter-matched flow-matching model transfers poorly.