  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 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 under 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
while classic neural networks fail to learn the Slinky dynamics, i.e., the
ESNN, trained on a single demonstration case, predicts the motions accurately
for unseen cases of different Slinky configurations and boundary conditions.
Further investigation into the ESNN reveals that it explicitly learns the
nonlinear coupling between stretching and bending of the Slinky.
