Neural Diffeomorphic Non-uniform B-spline Flows

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 transformation, but with price slow non-analytic inverse transforms. In this work, we propose diffeomorphic non-uniform B-spline at least twice continuously while bi-Lipschitz continuous, enabling efficient parametrization retaining analytic transforms based on sufficient condition for diffeomorphism. Firstly, investigate C(k-2)-diffeomorphic kth-order transformations. Then, derive cubic neural flows. Lastly, performed experiments solving force matching problem Boltzmann generators, demonstrating our C2-diffeomorphic yielded solutions better previous spline faster smooth Our source code is publicly available https://github.com/smhongok/Non-uniform-B-spline-Flow.