We present a new method based on functional tensor train 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 tensor manifold at each time. Such a projection can be computed by minimizing a convex energy functional over the tangent space. This yields a unique optimal velocity vector that allows us to integrate the PDE forward in time on the tensor manifold. 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 method to a four-dimensional Fokker-Planck equation with non-constant drift and diffusion coefficients, and demonstrate its accuracy in predicting relaxation to statistical equilibrium.