Normalizing flows have been successfully modeling a complex probability distribution as an invertible transformation of a simple base distribution. However, there are often applications that require more than invertibility. For instance, the computation of energies and forces in physics requires the second derivatives of the transformation to be well-defined and continuous. Smooth normalizing flows employ infinitely differentiable transformation, but with the price of slow non-analytic inverse transforms. In this work, we propose diffeomorphic non-uniform B-spline flows that are at least twice continuously differentiable while bi-Lipschitz continuous, enabling efficient parametrization while retaining analytic inverse transforms based on a sufficient condition for diffeomorphism. Firstly, we investigate the sufficient condition for Ck-2-diffeomorphic non-uniform kth-order B-spline transformations. Then, we derive an analytic inverse transformation of the non-uniform cubic B-spline transformation for neural diffeomorphic non-uniform B-spline flows. Lastly, we performed experiments on solving the force matching problem in Boltzmann generators, demonstrating that our C2-diffeomorphic non-uniform B-spline flows yielded solutions better than previous spline flows and faster than smooth normalizing flows. Our source code is publicly available at https://github.com/smhongok/Non-uniform-B-spline-Flow.

翻译：归一化流成功地将复杂概率分布建模为简单基分布的可逆变换。然而，许多应用对变换的要求往往不止于可逆性。例如，物理学中能量与力的计算要求变换的二阶导数定义良好且连续。平滑归一化流采用无限可微变换，但代价是反变换缓慢且非解析。本文提出至少二阶连续可微且同时满足双Lipschitz连续的微分同胚非均匀B样条流，该流基于微分同胚的充分条件实现高效参数化，同时保持解析反变换。首先，我们研究Ck-2-微分同胚非均匀k阶B样条变换的充分条件；其次，推导非均匀三次B样条变换的解析反变换以构建神经微分同胚非均匀B样条流；最后，通过玻尔兹曼生成器中的力匹配问题实验证明，所提出的C2-微分同胚非均匀B样条流在解的质量上优于现有样条流，且计算速度比平滑归一化流更快。源代码已开源：https://github.com/smhongok/Non-uniform-B-spline-Flow。