  We introduce new methods for integrating nonlinear differential equations on
low-rank manifolds. The methods are based on oblique projectors onto the
tangent spaces of low-rank manifolds with an interpolation property that
collocates the differential equation on the manifold. These projections yield
evolution equations that enable low-rank time integration of any vectorfield
that can be evaluated entry-wise. The proposed methods do not require the
vectorfield to have a low-rank structure, thus overcoming significant
limitations of dynamical low-rank approximation based on orthogonal projection.
To construct the oblique projectors, we devise a sparse tensor sampling
algorithm based on the discrete empirical interpolation method (DEIM) that
parameterizes tensor train manifolds and their tangent spaces with cross
interpolation. Using these projectors we propose two time integration schemes
on low-rank tensor train manifolds. The first is based on a time-dependent
tensor cross interpolation representation of the solution and the second is a
direct generalization of the well-known orthogonal projector-splitting
integrator to a oblique projectors. We demonstrate the proposed methods with
applications to several tensor differential equations arising from the
discretization of partial differential equations.
