Geometric deep neural networks learn representations that transform in a predictable way in response to particular transformations of the input. Conventional (i.e., planar) convolutional neural network (CNN) models are among the best known geometric neural network models. These networks process input images with a highly nonlinear and cascaded convolutional processing scheme, generating highly expressive representations that are equivariant to input translations. This is particularly useful in the context of 2D image prediction problems where 2D translations are a relevant symmetry (i.e., a transformation that preserves attributes of interest in the data). However, when the data change, the relevant symmetry group may also change. In robotics, for example, SE(3) transformations are likely more relevant than 2D or 3D translations. In our work, we develop and apply geometric deep neural network models in varied contexts for improved training efficiency and generalization performance.

While system identification techniques are useful when system measurements are low-dimensional, computational cost precludes their use with high-dimensional and entangled representations such as images. In contrast, an unstructured “black-box” neural network may successfully predict the time evolution of a system directly from images during training, but due to its opacity, it is difficult to understand the solution or know how well it will generalize. Physics-guided deep learning for dynamical systems offers a path toward interpretable reduced representation learning from high-dimensional data. In our work, we show that constraining the flexibility of neural network models with the relative rigidity of scientific theory (e.g., physics-based constraints) leads to solutions with greater interpretability and generalizability than their unstructured “black-box” counterparts.