Title: Intelligent mechanical metamaterials towards learning static and dynamic behaviors

Abstract: The exploration of intelligent machines has recently spurred the development of physical neural networks, a class of intelligent metamaterials capable of learning, whether in silico or in situ, from observed data. In this study, we introduce a back-propagation framework for lattice-based mechanical neural networks (MNNs) to achieve prescribed static and dynamic performance. This approach leverages the steady states of nodes for back-propagation, efficiently updating the learning degrees of freedom without prior knowledge of input loading. One-dimensional MNNs, trained with back-propagation in silico, can exhibit the desired behaviors on demand function as intelligent mechanical machines. The framework is then employed for the precise morphing control of the two-dimensional MNNs subjected to different static loads. Moreover, the intelligent MNNs are trained to execute classical machine learning tasks such as regression to tackle various deformation control tasks. Finally, the disordered MNNs are constructed and trained to demonstrate pre-programmed wave bandgap control ability, illustrating the versatility of the proposed approach as a platform for physical learning. Our approach presents an efficient pathway for the design of intelligent mechanical metamaterials for a wide range of static and dynamic target functionalities, positioning them as powerful engines for physical learning.