Disordered mechanical systems can deform along a network of pathways that branch and recombine at special configurations called bifurcation points. Multiple pathways are accessible from these bifurcation points; consequently, computer-aided design algorithms have been sought to achieve a specific structure of pathways at bifurcations by rationally designing the geometry and material properties of these systems. Here, we explore an alternative physical training framework in which the topology of folding pathways in a disordered sheet is changed in a desired manner due to changes in crease stiffnesses induced by prior folding. We study the quality and robustness of such training for different `learning rules', that is, different quantitative ways in which local strain changes the local folding stiffness. We experimentally demonstrate these ideas using sheets with epoxy-filled creases whose stiffnesses change due to folding before the epoxy sets. Our work shows how specific forms of plasticity in materials enable them to learn non-linear behaviors through their prior deformation history in a robust manner.

Mechanical metamaterials are usually designed to show desired responses to prescribed forces. In some applications, the desired force-response relationship is hard to specify exactly, but examples of forces and desired responses are easily available. Here, we propose a framework for supervised learning in thin, creased sheets that learn the desired force-response behavior by physically experiencing training examples and then, crucially, respond correctly (generalize) to previously unseen test forces. During training, we fold the sheet using training forces, prompting local crease stiffnesses to change in proportion to their experienced strain. We find that this learning process reshapes nonlinearities inherent in folding a sheet so as to show the correct response for previously unseen test forces. We show the relationship between training error, test error, and sheet size (model complexity) in learning sheets and compare them to counterparts in machine-learning algorithms. Our framework shows how the rugged energy landscape of disordered mechanical materials can be sculpted to show desired force-response behaviors by a local physical learning process.