A collocation method for nonlinear tensor differential equations on low-rank manifolds

We present a new method to compute the solution to a nonlinear tensor differential equation with dynamical low-rank approximation. The idea of dynamical low-rank approximation is to project the differential equation onto the tangent space of a low-rank tensor manifold at each time. Traditionally, an orthogonal projection onto the tangent space is employed, which is challenging to compute for nonlinear differential equations. We introduce a novel interpolatory projection onto the tangent space that is easily computed for many nonlinear differential equations and satisfies the differential equation at a set of carefully selected indices. To select these indices, we devise a new algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes any tensor train and its tangent space with tensor cross interpolants. We demonstrate the proposed method with applications to tensor differential equations arising from the discretization of partial differential equations.