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Physical systems exhibiting neuromechanical functions promise to enable structures with directly encoded autonomy and intelligence. We report on a class of neuromorphic metamaterials embodying bioinspired mechanosensing, memory, and learning functionalities obtained by leveraging mechanical instabilities and flexible memristive materials. Our prototype system comprises a multistable metamaterial whose bistable units filter, amplify, and transduce external mechanical inputs over large areas into simple electrical signals using piezoresistivity. We record these mechanically transduced signals using non-volatile flexible memristors that remember sequences of mechanical inputs, providing a means to store spatially distributed mechanical signals in measurable material states. The accumulated memristance changes resulting from the sequential mechanical inputs allow us to physically encode a Hopfield network into our neuromorphic metamaterials. This physical network learns a series of external spatially distributed input patterns. Crucially, the learned patterns input into our neuromorphic metamaterials can be retrieved from the final accumulated state of our memristors. Therefore, our system exhibits the ability to learn without supervised training and retain spatially distributed inputs with minimal external overhead. Our system's embodied mechanosensing, memory, and learning capabilities establish an avenue for synthetic neuromorphic metamaterials enabling the learning of touch-like sensations covering large areas for robotics, autonomous systems, wearables, and morphing structures., Comment: Manuscript: 13 pages, 4 figures, 1 table Supplementary Information: 11 pages, 17 figures, 2 tables

Affine matrix rank minimization problem is a fundamental problem with a lot of important applications in many fields. It is well known that this problem is combinatorial and NP-hard in general. In this paper, a continuous promoting low rank non-convex fraction function is studied to replace the rank function in this NP-hard problem. Inspired by our former work in compressed sensing, an iterative singular value thresholding algorithm is proposed to solve the regularization transformed affine matrix rank minimization problem. For different $a>0$, we could get a much better result by adjusting the different value of $a$, which is one of the advantages for the iterative singular value thresholding algorithm compared with some state-of-art methods. Some convergence results are established and numerical experiments show that this thresholding algorithm is feasible for solving the regularization transformed affine matrix rank minimization problem. Moreover, we proved that the value of the regularization parameter $\lambda>0$ can not be chosen too large. Indeed, there exists $\bar{\lambda}>0$ such that the optimal solution of the regularization transformed affine matrix rank minimization problem is equal to zero for any $\lambda>\bar{\lambda}$. Numerical experiments on matrix completion problems show that our method performs powerful in finding a low-rank matrix and the numerical experiments about image inpainting problems show that our algorithm has better performances than some state-of-art methods.