Analytic Bijections for Smooth and Interpretable Normalizing Flows

A key challenge in normalizing flows is finding expressive invertible scalar bijections. Existing approaches face trade-offs: affine transformations are smooth and analytically invertible but lack expressivity; monotonic splines offer local control but are only piecewise smooth and act on bounded domains; residual flows achieve smoothness but need numerical inversion. We introduce three families of *analytic bijections* that are globally smooth ($C^\infty$), defined on all of $\mathbb{R}$, and analytically invertible in closed form, combining the favorable properties of prior approaches. Beyond serving as drop-in replacements in coupling flows, where they match or exceed spline performance, we develop *radial flows*: a novel architecture using direct parametrization that transforms the radial coordinate while preserving angular direction. Radial flows exhibit exceptional training stability, produce geometrically interpretable transformations, and on targets with radial structure can achieve comparable quality to coupling flows with $1000\times$ fewer parameters. We provide comprehensive evaluation on 1D and 2D benchmarks, and demonstrate applicability to higher-dimensional physics problems through experiments on $\phi^4$ lattice field theory, where our bijections outperform affine baselines and enable problem-specific designs that address mode collapse.

Many AI systems learn to generate realistic data, from images to scientific measurements, by gradually reshaping a simple starting distribution (like random noise) into a complex one that matches reality. The reshaping is built from small mathematical "building-block" functions, and a key requirement is that every step can be run backward exactly. Unlike the diffusion models behind popular image generators, this exact reversibility lets the model compute precise probabilities for the data it sees, which is essential in some scientific applications. Existing building blocks force an awkward trade-off: some are smooth but too simple to capture rich structure, while more flexible ones are not smooth, only work over a limited range, or can only be reversed approximately. We introduce a new family of building blocks that avoid this trade-off: they are smooth everywhere, flexible enough to capture intricate shapes, and can be reversed with an exact formula. Building on them, we design "radial flows" that reshape data outward or inward from a center point, like ripples on a pond (although not necessarily as symmetric). These are easy to interpret, stable to train, and can match the quality of much larger models while using many fewer parameters.