Normalizing flows (NF) are a class of powerful generative models that have gained popularity in recent years due to their ability to model complex distributions with high flexibility and expressiveness. In this work, we introduce a new type of normalizing flow that is tailored for modeling positions and orientations of multiple objects in three-dimensional space, such as molecules in a crystal. Our approach is based on two key ideas: first, we define smooth and expressive flows on the group of unit quaternions, which allows us to capture the continuous rotational motion of rigid bodies; second, we use the double cover property of unit quaternions to define a proper density on the rotation group. This ensures that our model can be trained using standard likelihood-based methods or variational inference with respect to a thermodynamic target density. We evaluate the method by training Boltzmann generators for two molecular examples, namely the multi-modal density of a tetrahedral system in an external field and the ice XI phase in the TIP4P-Ew water model. Our flows can be combined with flows operating on the internal degrees of freedom of molecules, and constitute an important step towards the modeling of distributions of many interacting molecules.

翻译：归一化流（Normalizing Flows, NF）是一类强大的生成模型，近年来因其能够以高灵活性和表现力建模复杂分布而广受关注。本文提出一种新型归一化流，专为建模三维空间中多个对象（如晶体中的分子）的位置和朝向而设计。该方法基于两个关键思想：首先，在单位四元数群上定义平滑且具表现力的流，从而捕捉刚体的连续旋转运动；其次，利用单位四元数的双重覆盖性质，在旋转群上定义合适的概率密度。这确保我们的模型能够通过标准似然方法或相对于热力学目标密度的变分推断进行训练。我们通过训练两个分子示例的玻尔兹曼生成器来评估该方法：即外场中四面体系统的多模态密度以及TIP4P-Ew水模型中的冰XI相。该流可与作用于分子内部自由度的流相结合，为建模多相互作用分子分布迈出了重要一步。