In this paper we develop a manifestly geometric framework for equivariant manifold neural ordinary differential equations (NODEs), and use it to analyse their modelling capabilities for symmetric data. First, we consider the action of a Lie group $G$ on a smooth manifold $M$ and establish the equivalence between equivariance of vector fields, symmetries of the corresponding Cauchy problems, and equivariance of the associated NODEs. We also propose a novel formulation of the equivariant NODEs in terms of the differential invariants of the action of $G$ on $M$, based on Lie theory for symmetries of differential equations, which provides an efficient parameterisation of the space of equivariant vector fields in a way that is agnostic to both the manifold $M$ and the symmetry group $G$. Second, we construct augmented manifold NODEs, through embeddings into equivariant flows, and show that they are universal approximators of equivariant diffeomorphisms on any path-connected $M$. Furthermore, we show that the augmented NODEs can be incorporated in the geometric framework and parameterised using higher order differential invariants. Finally, we consider the induced action of $G$ on different fields on $M$ and show how it can be used to generalise previous work, on, e.g., continuous normalizing flows, to equivariant models in any geometry.

We investigate the optimization of multilayer perceptrons on symmetric data. We compare the strategy of constraining the architecture to be equivariant to that of using augmentation. We show that, under natural assumptions on the loss and non-linearities, the sets of equivariant stationary points are identical for the two strategies, and that the set of equivariant layers is invariant under the gradient flow for augmented models. Finally, we show that stationary points may be unstable for augmented training although they are stable for the equivariant models.

High-resolution wide-angle fisheye images are becoming more and more important for robotics applications such as autonomous driving. However, using ordinary convolutional neural networks or vision transformers on this data is problematic due to projection and distortion losses introduced when projecting to a rectangular grid on the plane. We introduce the HEAL-SWIN transformer, which combines the highly uniform Hierarchical Equal Area iso-Latitude Pixelation (HEALPix) grid used in astrophysics and cosmology with the Hierarchical Shifted-Window (SWIN) transformer to yield an efficient and flexible model capable of training on high-resolution, distortion-free spherical data. In HEAL-SWIN, the nested structure of the HEALPix grid is used to perform the patching and windowing operations of the SWIN transformer, resulting in a one-dimensional representation of the spherical data with minimal computational overhead. We demonstrate the superior performance of our model for semantic segmentation and depth regression tasks on both synthetic and real automotive datasets.