We present a comprehensive study on discrete morphological symmetries of dynamical systems, which are commonly observed in biological and artificial locomoting systems, such as legged, swimming, and flying animals/robots/virtual characters. These symmetries arise from the presence of one or more planes/axis of symmetry in the system's morphology, resulting in harmonious duplication and distribution of body parts. Significantly, we characterize how morphological symmetries extend to symmetries in the system's dynamics, optimal control policies, and in all proprioceptive and exteroceptive measurements related to the system's dynamics evolution. In the context of data-driven methods, symmetry represents an inductive bias that justifies the use of data augmentation or symmetric function approximators. To tackle this, we present a theoretical and practical framework for identifying the system's morphological symmetry group $\G$ and characterizing the symmetries in proprioceptive and exteroceptive data measurements. We then exploit these symmetries using data augmentation and $\G$-equivariant neural networks. Our experiments on both synthetic and real-world applications provide empirical evidence of the advantageous outcomes resulting from the exploitation of these symmetries, including improved sample efficiency, enhanced generalization, and reduction of trainable parameters.

翻译：我们针对动态系统中常见的离散形态对称性开展了系统研究，这类对称性广泛存在于生物与人工运动系统（如腿足、游动及飞行类动物/机器人/虚拟角色）中。这些对称性源于系统形态结构中一个或多个对称面/轴的存在，导致肢体部件的和谐复制与分布。重要的是，我们揭示了形态对称性如何延伸至系统动力学、最优控制策略，以及与系统动态演化相关的所有本体感知和外部感知测量中的对称性。在数据驱动方法背景下，对称性构成了一种归纳偏置，为数据增强或对称函数逼近器的使用提供了合理性依据。为此，我们提出了一套理论与实践的框架，用于识别系统的形态对称群$\G$，并刻画本体感知与外部感知数据测量中的对称性。进而通过数据增强和$\G$-等变神经网络对这些对称性加以利用。我们在合成数据与真实世界应用上的实验提供了经验证据，证明了利用这些对称性所带来的优势成果，包括样本效率提升、泛化能力增强以及可训练参数数量的减少。