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Recent advancements in topological metamaterials have unveiled fruitful physics and numerous applications. Whereas initial efforts focus on achieving topologically protected edge states through principles of structural symmetry, the burgeoning field now aspires to customize topological states, tailoring their emergence and frequency. Here, our study presents the realization of topological phase transitions utilizing compliant mechanisms on the facets of Miura-folded metamaterials. This approach induces two opposite topological phases, leading to topological states at the interface. Moreover, we exploit the unique folding behavior of Miura-folded metamaterials to tune the frequency of topological states and dynamically toggle their presence. Our research not only paves the way for inducing topological phase transitions in Miura-folded structures but also enables the on-demand control of topological states, with promising applications in wave manipulation and vibration isolation., Comment: 22 pages, 7 figures

Recent advances unveiled physical neural networks as promising machine learning platforms, offering faster and more energy-efficient information processing. Compared with extensively-studied optical neural networks, the development of mechanical neural networks (MNNs) remains nascent and faces significant challenges, including heavy computational demands and learning with approximate gradients. Here, we introduce the mechanical analogue of in situ backpropagation to enable highly efficient training of MNNs. We demonstrate that the exact gradient can be obtained locally in MNNs, enabling learning through their immediate vicinity. With the gradient information, we showcase the successful training of MNNs for behavior learning and machine learning tasks, achieving high accuracy in regression and classification. Furthermore, we present the retrainability of MNNs involving task-switching and damage, demonstrating the resilience. Our findings, which integrate the theory for training MNNs and experimental and numerical validations, pave the way for mechanical machine learning hardware and autonomous self-learning material systems., Comment: 25 pages, 5 figures