  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.
