Data-free physics-informed inverse programming of bistable kirigami energy landscapes

Kirigami metamaterials, formed by introducing cuts into planar sheets, can exhibit bistability through geometry-dependent energy landscapes. Designing structures with prescribed energy barriers and target deformation states is therefore essential for programmable mechanical functionality. Here, we present a physics-informed neural network (PINN) framework that does not require pre-collected labeled training datasets and unifies forward prediction and inverse programming of bistable kirigami energy landscapes by embedding equilibrium conditions, energy formulations, and geometric compatibility directly into the learning objective. In the forward setting, the framework predicts continuous energy landscapes with coefficients of determination above 0.99 and barrier errors below 0.1%. In the inverse setting, it identifies kirigami geometries from fully prescribed energy curves with barrier errors below 5%, and further extends to a minimally specified setting in which only the target energy barrier and zero-energy stable states are given. In this underdetermined case, the framework autonomously infers the full energy landscape while achieving a mean barrier error of 0.4%. Finite element analysis and experiments on 3D-printed prototypes confirm that the programmed ordering of energy barriers is preserved. By assembling unit cells with different programmed barriers, we further demonstrate sequential actuation, in which units with lower barriers transform first during tensile loading. This work establishes an efficient physics-informed approach for programming bistable energy landscapes in planar kirigami systems, with potential applications in deployable structures, soft robotic actuators, and impact mitigation devices.