Some of the most computationally demanding problems in science and engineering involve high-dimensional matrix and tensor differential equations (MDEs and TDEs). These arise in a range of applications, including the Schrödinger equation, probability density function transport in turbulent combustion, the Fokker-Planck equation, the Boltzmann transport equation, and the Hamilton-Jacobi-Bellman equation, among others. However, performing computations—or even storing data—for such high-dimensional problems is notoriously difficult due to the curse of dimensionality: the number of degrees of freedom grows exponentially with the dimension. Tensor network low-rank approximations offer a promising approach to mitigate this challenge by exploiting multi-dimensional correlations. In this presentation, we focus on the computational cost associated with solving nonlinear MDEs and TDEs in low-rank forms. We present a novel formulation based on adaptive cross interpolation for solving such equations efficiently. Specifically, we develop CUR decomposition algorithms grounded in the Discrete Empirical Interpolation Method (DEIM) for matrix, tensor train, and Tucker tensor formats. We demonstrate that these DEIM-CUR algorithms can achieve near-optimal accuracy for a broad class of nonlinear MDEs and TDEs. We illustrate the effectiveness of our approach across several application domains, including turbulent combustion, sensitivity analysis, uncertainty quantification, and high-dimensional partial differential equations.