  We present a novel tensor interpolation algorithm for the time integration of
nonlinear tensor differential equations (TDEs) on the tensor train and Tucker
tensor low-rank manifolds, which are the building blocks of many tensor network
decompositions. This paper builds upon our previous work (Donello et al.,
Proceedings of the Royal Society A, Vol. 479, 2023) on solving nonlinear matrix
differential equations on low-rank matrix manifolds using CUR decompositions.
The methodology we present offers multiple advantages: (i) It delivers
near-optimal computational savings both in terms of memory and floating-point
operations by leveraging cross algorithms based on the discrete empirical
interpolation method to strategically sample sparse entries of the
time-discrete TDEs to advance the solution in low-rank form. (ii) Numerical
demonstrations show that the time integration is robust in the presence of
small singular values. (iii) High-order explicit Runge-Kutta time integration
schemes are developed. (iv) The algorithm is easy to implement, as it requires
the evaluation of the full-order model at strategically selected entries and
does not use tangent space projections, whose efficient implementation is
intrusive. We demonstrate the efficiency of the presented algorithm for several
test cases, including a nonlinear 100-dimensional TDE for the evolution of a
tensor of size $70^{100} \approx 3.2 \times 10^{184}$ and a stochastic
advection-diffusion-reaction equation with a tensor of size $4.7 \times 10^9$.
