Normalizing flows have been successfully modeling a complex probability distribution as an invertible transformation of simple base distribution. However, there are often applications that require more than invertibility. For instance, the computation energies and forces in physics requires second derivatives to be well-defined continuous. Smooth normalizing employ infinitely differentiable tra...

In this paper, we investigate non-uniform elementary cellular automata (i.e., one-dimensional whose cells can use different Wolfram rules to update their states) in the context of number conservation. As a result, obtain an exhaustive characterization such number-conserving on all finite grids both with periodic and null boundary conditions. The obtained allows, inter alia, enumerate automata, ...