Kirigami, the art of cutting paper to create three-dimensional shapes, has emerged as a powerful technique for designing multifunctional metamaterials. The introduction of cuts into thin sheets under tension leads to nonlinear responses and complex three-dimensional deformations. However, designing kirigami structures to achieve specific functionalities remains challenging, often relying on time-consuming trial-and-error methods. Optimization methods have been explored but can be computationally expensive. This research proposes a data-driven approach using machine learning to address this inverse design problem. Existing machine learning applications in Kirigami have primarily focused on mechanical properties, neglecting the topological reconfigurability aspect and experimental validation. This study aims to fill this gap by developing a comprehensive workflow comprising four modules: (1) exploration of the design space, (2) inverse design, (3) investigation of deployment conditions and tunability, and (4) utility for achieving configurations beyond the initial design space. The study focuses on a specific kirigami motif consisting of two inner panels with internal line and U-shaped cuts, known for its diverse out-of-plane deformations. The meta-atom geometry is defined by several parameters (length, width, thickness, hinge lengths, cut dimensions, and distances to boundaries), along with elastic properties and uniaxial stretch. Finite element analysis (FEA) is used to simulate the kirigami's behavior, providing data for the machine learning framework.

Prior research on Kirigami has demonstrated its potential in various applications, including soft robotics, stretchable electronics, optics, and textiles. Numerical and experimental investigations have shown the variability of 3D shapes achievable through slight geometric variations. However, systematic methods for inverse design—determining the cut layout for a desired functionality—have been limited. Previous work has utilized optimization methods, but these can be computationally expensive for highly nonlinear problems. Data-driven methods offer an alternative, but most prior machine learning efforts have focused on predicting mechanical properties rather than topological reconfigurability. This study distinguishes itself by addressing the inverse design problem directly and incorporating experimental validation.

The research methodology involves four main stages. First, a space-filling Sobol sequence is used to sample the kirigami design space, varying the ten geometric parameters defining the cut layout. Linear and geometrically nonlinear finite element analyses are conducted to obtain imperfection modes and deformation metrics. These metrics (deformed coordinates, 3D rotations, and force-displacement curves) form the dataset. Second, a clustering approach (k-means) is employed to explore the space of achievable deformed configurations, identifying representative clusters of shapes. Third, a tandem deep neural network (T-DNN) architecture is used for inverse design—predicting the cut layout needed to achieve a specific deformed configuration. The T-DNN consists of two sub-networks: one predicting the linear buckling behavior and another predicting the nonlinear deformations. Fourth, symbolic regression is used to develop surrogate models that predict parameters relevant to the tunable actuation of the designed kirigami meta-atom, such as the relationship between force and deformation angle. The entire framework is presented as a concatenated workflow, providing a systematic approach to kirigami design and control.

The k-means clustering analysis identified five distinct clusters of kirigami deformations, including symmetric and asymmetric linear and nonlinear shapes. The T-DNN successfully predicted kirigami cut layouts corresponding to desired target deformations, demonstrating the feasibility of inverse design. Symbolic regression generated surrogate models for predicting actuation parameters, providing insights into the tunability of the kirigami designs. Experimental validation confirmed the accuracy of the predicted kirigami configurations. The framework was successfully applied to design bioinspired metamaterials at different hierarchical levels, including a pollen-trapping structure, a droplet-transporting structure, a solar tracker, and a light-focusing reflector. The experimental results show good agreement with the predictions, but some discrepancies arise due to factors such as higher-order imperfection modes, material plasticity, and manufacturing imperfections. These discrepancies highlight the importance of incorporating uncertainties into the data-based design frameworks in future work.

The study successfully demonstrated a comprehensive data-driven framework for the design and control of kirigami metamaterials. The framework addresses the inverse design problem, a key challenge in kirigami engineering. The use of machine learning significantly accelerates the design process compared to traditional trial-and-error methods. The experimental validation confirms the framework's effectiveness and provides confidence in its applicability. The bioinspired examples showcase the potential of this approach for creating complex and functional metamaterials. Future work should focus on improving the framework's accuracy by incorporating higher-order imperfection modes, material plasticity, and manufacturing uncertainties.

This research presents a novel data-driven framework for the design and control of shape-programmable 3D kirigami metamaterials. The integrated approach of clustering, tandem neural networks, and symbolic regression enables efficient inverse design and facilitates the exploration of the vast kirigami design space. Successful experimental validation and the demonstration of bioinspired metamaterial designs highlight the framework's practicality and potential. Future work can extend the framework to broader design spaces, incorporate advanced machine learning techniques, and address the influence of material properties and manufacturing uncertainties.

The current study explored a relatively limited design space for the kirigami motif. The accuracy of the predictions could be improved by incorporating higher-order imperfection modes, material plasticity, and manufacturing imperfections into the model. The experimental validation, while successful, was limited to a specific set of applications. Further experimental validation with a wider range of kirigami designs and applications is needed to fully assess the framework’s robustness.