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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.