神经微分同胚非均匀B样条流
Neural Diffeomorphic Non-uniform B-spline Flows
AI总结：
本文提出至少二次连续可微且双Lipschitz连续的微分同胚非均匀B样条流，推导其解析逆变换，并在Boltzmann生成器力匹配任务中取得优于已有样条流且快于光滑归一化流的结果。
摘要：
归一化流已成功地将复杂概率分布建模为简单基础分布的可逆变换。然而，许多应用往往需要的不仅仅是可逆性。例如，物理学中能量和力的计算要求变换的二阶导数良定义且连续。光滑归一化流采用无限可微的变换，但代价是逆变换非解析且速度缓慢。在本工作中，我们提出了微分同胚非均匀B样条流，其至少二次连续可微且双Lipschitz连续，在保留基于微分同胚充分条件的解析逆变换的同时，实现了高效参数化。首先，我们研究了Ck-2微分同胚非均匀k阶B样条变换的充分条件。然后，我们为神经微分同胚非均匀B样条流推导了非均匀三次B样条变换的解析逆变换。最后，我们针对Boltzmann生成器中的力匹配问题进行了实验，证明我们的C2微分同胚非均匀B样条流得到的解优于先前的样条流，并且比光滑归一化流更快。我们的源代码已在https://github.com/smhongok/Non-uniform-B-spline-Flow公开提供。
英文摘要：
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.
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