where state i is specified by the identity of the metastable state (Form) and the physical conditions of the given Canonical ensemble (N, V, T ). Table , based on raw data reported in SI Figure , reports the lattice Helmholtz free energy differences (in units of k B T ) between the two forms of ice, computed using all of the methods (i.e., from models H and C, and from the ECM). These comparisons show a significant level of agreement between the different methods, indicating that the results from the PGMs are accurate. We note, however, that the error bars are higher in larger systems and at higher temperatures because such ensembles occupy a larger volume of configurational space. That being said, we consistently observe smaller error bars from results of model H, compared to model C. This indicates that model H (based on the decoupled, quaternion-based representation of the Cartesian blocks) is better suited for efficiently learning configurational distributions of ice crystals. Moreover, since both models we compare used the same architecture for the trainable part of the model (spline coupling layers; see Fig. ), we expect that the computational gains from using the decoupled representation of the Cartesian block can be generalized to other types of crystals. Figure reports per-molecule, temperature-reduced entropy difference (in units of k B ) between the two forms of ice (Ic and XI), estimated using the different The reason for the lower variance in the BAR V estimates from model H, compared to model C, can be linked to the differences in the amount of overlap observed between the potential energies of the data and the potential energies of the samples drawn from the models during training (shown in SI figures 2 and 3). When taken together, the current results show that in the large systems containing 64 molecules and at the higher temperature (50 o C) (green error bars in Fig. A and B, ii), both models tend to approach a bottleneck in accuracy.