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 $4.7 \times 10^9$.

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