  We present a new method based on functional tensor decomposition and dynamic
tensor approximation to compute the solution of a high-dimensional
time-dependent nonlinear partial differential equation (PDE). The idea of
dynamic approximation is to project the time derivative of the PDE solution
onto the tangent space of a low-rank functional tensor manifold at each time.
Such a projection can be computed by minimizing a convex energy functional over
the tangent space. This minimization problem yields the unique optimal velocity
vector that allows us to integrate the PDE forward in time on a tensor manifold
of constant rank. In the case of initial/boundary value problems defined in
real separable Hilbert spaces, this procedure yields evolution equations for
the tensor modes in the form of a coupled system of one-dimensional
time-dependent PDEs. We apply the dynamic tensor approximation to a
four-dimensional Fokker-Planck equation with non-constant drift and diffusion
coefficients, and demonstrate its accuracy in predicting relaxation to
statistical equilibrium.
