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:标准化流已成功地将复杂的概率分布建模为简单基本分布的可逆变换。然而,往往需要更多的东西。例如,在物理学中计算能量和力要求将变换的二阶导数定义良好且连续。平滑的标准化流采用无限可微变换,但付出了慢速的非解析逆变换的代价。在这项工作中,我们提出了可微形变非均匀B样条流,其至少连续两次可微,同时是双Lipschitz连续的,从而实现了高效参数化,同时保持了基于形变的解析逆变换的足够条件。首先,我们研究了Ck-2-可微形变非均匀k阶B样条变换的足够条件。然后,我们推导出了非均匀立方B样条变换的解析逆变换,用于神经形变非均匀B样条流。最后,我们进行了求解玻尔兹曼发生器的力匹配问题的实验,证明了我们的C2-可微形变非均匀B样条流产生的解决方案比以前的样条流更好,比平滑的标准化流更快。我们的源代码公开在 https://github.com/smhongok/Non-uniform-B-spline-Flow。