  Neural ODEs and i-ResNet are recently proposed methods for enforcing
invertibility of residual neural models. Having a generic technique for
constructing invertible models can open new avenues for advances in learning
systems, but so far the question of whether Neural ODEs and i-ResNets can model
any continuous invertible function remained unresolved. Here, we show that both
of these models are limited in their approximation capabilities. We then prove
that any homeomorphism on a $p$-dimensional Euclidean space can be approximated
by a Neural ODE operating on a $2p$-dimensional Euclidean space, and a similar
result for i-ResNets. We conclude by showing that capping a Neural ODE or an
i-ResNet with a single linear layer is sufficient to turn the model into a
universal approximator for non-invertible continuous functions.
