Deep Spring: A Neural Network that Learns Physics Through Differentiable Simulation for the Inverse Design of Suspended Elastic Rods*

Predicting the geometrically nonlinear deformation of slender structures has been extensively studied in various scientific fields, including mechanics and computer graphics. We are interested in rods, which are unique structures that undergo large deformation even under minimal loads, such as their own weight. While the forward problem of predicting a rod's deformation can be easily addressed with efficient frameworks such as Discrete Elastic Rods (DER), the inverse problem of finding the initial geometry that deforms into the target shape is a challenging nonlinear optimization problem. Previously, we have proposed a propagative method to solve the inverse problem for an entire rod. Leveraging the DeepSpringNet, a neural network designed to determine the initial geometry based on the deformed shape of a rod segment, we decompose the problem into manageable subproblems, starting from the fixed end and progressing towards the free end. However, the iterative nature of this method makes it susceptible to error propagation, where the error at a later step reflects the cumulative error from all prior steps. In this work, we address this problem through two solutions: 1) We enforce DeepSpringNet to learn the underlying physics by including the forward simulation in the training loop, which increases the accuracy of the neural network, 2) We compute the loss after linking multiple segments together, and backpropagate through the propagated error. This work brings us one step closer to real-time prediction of undeformed shapes of rods given their deformed shapes under gravity.