  Emerging tensor network techniques for solutions of Partial Differential
Equations (PDEs), known for their ability to break the curse of dimensionality,
deliver new mathematical methods for ultrafast numerical solutions of
high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev
spectral collocation method, in both space and time, for solution of the time
dependent convection-diffusion-reaction (CDR) equation with inhomogeneous
boundary conditions, in Cartesian geometry. Previous methods for numerical
solution of time dependent PDEs often use finite difference for time, and a
spectral scheme for the spatial dimensions, which leads to slow linear
convergence. Spectral collocation space-time methods show exponential
convergence, however, for realistic problems they need to solve large
four-dimensional systems. We overcome this difficulty by using a TT approach as
its complexity only grows linearly with the number of dimensions. We show that
our TT space-time Chebyshev spectral collocation method converges
exponentially, when the solution of the CDR is smooth, and demonstrate that it
leads to very high compression of linear operators from terabytes to kilobytes
in TT-format, and tens of thousands times speedup when compared to full grid
space-time spectral method. These advantages allow us to obtain the solutions
at much higher resolutions.
