Physics-Informed Neural Network for Inverse Design of Cylindrical Kresling Origami with Discrete Side Count Optimization

Origami mechanisms excel in space and material utilization, but their design variables involve the coupling of continuous and discrete variables, complicating reverse engineering—especially for polygons with dis crete side counts. Traditional numerical optimization methods struggle with discrete variables and lack automation. This paper proposes a Physics-Informed Neural Network (PINN) framework for the reverse engineering of Kresling origami. In this framework, the discrete side count *m* is first adjusted by a dis tance penalty to shorten the distance between the optimized and target values. This ensures contin uously differentiable gradients and smooth transitions, avoiding loss bounces. Then, a soft projection function (a softmax function with a learnable temperature) strictly constrains the output value to integers, ultimately achieving continuous differentiability. Physical constraints (potential energy difference and near zero torque, achieved through ∂U/∂β ≈ 0) are embedded in the loss function, enabling label-free training. Novel engineering constraints ensure manufacturability. Numerical examples show MSE of the energy curve < 1e-5 in all tested cases and convergence to integer m in >95% of runs.