Rapidly encoding generalizable dynamics in a Euclidean symmetric neural network: a Slinky case study

Abstract

Slinky, a helical elastic rod, is a seemingly simple structure with unusual mechanical behavior; for example, it can walk down a flight of stairs under its own weight. Taking the Slinky as a test-case, we propose a physics-informed deep learning approach for building reduced-order models of physical systems. The approach introduces a Euclidean symmetric neural network (ESNN) architecture that is trained under the neural ordinary differential equation framework to learn the 2D latent dynamics from the motion trajectory of a reduced-order representation of the 3D Slinky. The ESNN implements a physics-guided architecture that simultaneously preserves energy invariance and force equivariance on Euclidean transformations of the input, including translation, rotation, and reflection. The embedded Euclidean symmetry provides physics-guided interpretability and generalizability, while preserving the full expressive power of the neural network. We demonstrate that the ESNN approach is able to accelerate simulation by one to two orders of magnitude compared to traditional numerical methods and achieve a superior generalization performance, i.e., the neural network, trained on a single demonstration case, predicts accurately on unseen cases with different Slinky configurations and boundary conditions.

Dynamic modeling of a sliding ring on an elastic rod with incremental potential formulation

“…The frictional force, similarly, is given by maximal dissipation principle with a velocity-related normalized parameter to increase functional continuous. The incremental potential model and its extension have already shown powerful performance when dealing with complex rod-to-rod interaction problems, e.g., knot tying [57], slinky toy [58], and flagella bundling [59]. However, if both the ring and the rod are discretized into nodes and edges and they are simulated using an established rod-to-rod interaction model, the computational time is still significant due to the additional degrees of freedom.…”