In this work, we study discrete morphological symmetries of dynamical systems, a predominant feature in animal biology and robotic systems, expressed when the system's morphology has one or more planes of symmetry describing the duplication and balanced distribution of body parts. These morphological symmetries imply that the system's dynamics are symmetric (or approximately symmetric), which in turn imprints symmetries in optimal control policies and in all proprioceptive and exteroceptive measurements related to the evolution of the system's dynamics. For data-driven methods, symmetry represents an inductive bias that justifies data augmentation and the construction of symmetric function approximators. To this end, we use group theory to present a theoretical and practical framework allowing for (1) the identification of the system's morphological symmetry group $\G$, (2) data-augmentation of proprioceptive and exteroceptive measurements, and (3) the exploitation of data symmetries through the use of $\G$-equivariant/invariant neural networks, for which we present experimental results on synthetic and real-world applications, demonstrating how symmetry constraints lead to better sample efficiency and generalization while reducing the number of trainable parameters.

翻译：本文研究动力学系统的离散形态对称性，这是动物生物学与机器人系统中的显著特征，当系统的形态具有一个或多个对称平面时得以体现，这些对称平面描述了身体部位的复制与平衡分布。此类形态对称性意味着系统动力学具有对称性（或近似对称性），进而将对称性映射至最优控制策略以及与系统动力学演化相关的所有本体感觉与外感受测量中。对于数据驱动方法而言，对称性构成了归纳偏置，为数据增强及对称函数逼近器的构建提供了依据。为此，我们利用群论提出了一套理论与实践的框架，该框架能够实现：(1) 识别系统的形态对称群 $\G$；(2) 对本體感觉与外感受测量进行数据增强；(3) 通过使用 $\G$ 等变/不变神经网络来利用数据对称性。我们针对合成数据与真实世界应用展示了实验结果，证明了对称性约束如何提升样本效率与泛化能力，同时减少可训练参数的数量。