  We introduce a provably stable variant of neural ordinary differential
equations (neural ODEs) whose trajectories evolve on an energy functional
parametrised by a neural network. Stable neural flows provide an implicit
guarantee on asymptotic stability of the depth-flows, leading to robustness
against input perturbations and low computational burden for the numerical
solver. The learning procedure is cast as an optimal control problem, and an
approximate solution is proposed based on adjoint sensivity analysis. We
further introduce novel regularizers designed to ease the optimization process
and speed up convergence. The proposed model class is evaluated on non-linear
classification and function approximation tasks.
