Periodic Bayesian Flow Networks with Additive Accuracy

Generating periodic data---such as fractional atomic coordinates in crystal structures and phase patterns in compressive light-field (CLF) displays---is challenging because wrap-around boundaries complicate probabilistic modeling and learning. While Bayesian Flow Networks (BFNs) offer a powerful generative framework with strictly additive accuracy in Euclidean space, existing periodic adaptations typically sacrifice additivity and become sensitive to schedule heuristics. We introduce \emph{PeriodicBFN}, which embeds each periodic scalar into a two-dimensional unit-circle representation and performs Gaussian Bayesian updates in the resulting Cartesian space, thereby restoring strictly additive accuracy. To address invariance in periodic generative modeling, we further derive a Rao--Blackwellized objective that analytically marginalizes global periodic translations, producing a translation-invariant target with reduced gradient variance. Experiments on crystal structure prediction and multi-layer phase synthesis for CLF displays demonstrate improved training stability and strong performance. To our knowledge, this is the first work to extend periodic-data generative modeling to phase synthesis for modern glasses-free 3D display systems.