A centerline-based energy model of elastic ribbons using implicit neural ODE*

Ribbons are slender structures with length » width & width » thickness. We present a simulation framework based on implicit neural ordinary differential equation (iNODE) designed for modeling behavior of elastic ribbons. Our approach uses a 1D energy model based on physics-informed neural networks (PINNs) in the Discrete Elastic Rods (DER) algorithm, originally devised for 1D rods. Adapting DER for ribbons posed challenges in capturing bending and twisting interactions. While the Discrete Elastic Plates (DEP) captures ribbon mechanics precisely, it is computationally expensive compared to rod-based methods. To strike a balance between computational efficiency and physical accuracy, we employ a NN-based energy model of ribbon's central axis within a rod-like framework. Using DEP simulations, we collect time-series data of curvatures and twist of ribbon's central axis across various bending and twisting configurations. Next, we deploy our iNODE to train a PINN-based energy model. It employs implicit Euler integrator on top of NODE, enabling efficient simulation of quasi-static dynamic systems. It outperforms the standard NODE in capturing ribbon mechanics with reduced data and faster training. We perform ablation study on our energy model, comparing state-of-the-art ribbon analytical models. Our energy model stands as a potential benchmark for forthcoming analytical models in the ribbon domain. Furthermore, iNODE offers a powerful tool for non-linear dynamics of systems where analytical expressions proves challenging, enabling efficient training for quasi-static systems and improved generalization.