applications such as climate science and transportation require learning complex dynamics from large-scale spatiotemporal data. existing machine learning frameworks are still insufficient to learn spatiotemporal dynamics as they often fail to exploit the underlying physics principles. representation theory can be used to describe and exploit the symmetry of the dynamical system. we will show how to design neural networks that are equivariant to various symmetries for learning spatiotemporal dynamics. our methods demonstrate significant improvement in prediction accuracy, generalization, and sample efficiency in forecasting turbulent flows and predicting real-world trajectories. this is joint work with robin walters, rui wang, and jinxi li.