Seminar Geometry, Algebra and Physics in Deep Neural Networks (GAPinDNNs)

Abstract: Neural ODEs are neural network models where the network is not specified by a discrete sequence of hidden layers. Instead, the network is defined by a vector field describing how the data evolves continuously over time governed by an ordinary differential equation (ODE). These models can be generalized for data living on non-Euclidean manifolds, a concept known as manifold neural ODEs. In our paper, we develop a geometric framework for equivariant manifold neural ODEs. Our work includes a novel formulation of equivariant neural ODEs in terms of differential invariants, based on Lie theory for symmetries of differential equations. We also construct augmented manifold neural ODEs and show that they are universal approximators of equivariant diffeomorphisms on any path-connected manifold.