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 $4.7 \times 10^9$.

翻译：本文提出了一种新颖的张量插值算法，用于在张量链和Tucker张量低秩流形上对非线性张量微分方程进行时间积分，这两种流形是许多张量网络分解的基本构件。本工作基于我们先前利用CUR分解在低秩矩阵流形上求解非线性矩阵微分方程的研究（Donello等人，《皇家学会会刊A辑》，第479卷，2023年）。所提出的方法具有多重优势：（i）通过利用基于离散经验插值法的交叉算法，策略性地对时间离散化张量微分方程的稀疏条目进行采样，以低秩形式推进解，从而在内存和浮点运算方面实现接近最优的计算节省。（ii）数值实验表明，该时间积分算法在存在小奇异值的情况下具有鲁棒性。（iii）发展了高阶显式Runge-Kutta时间积分格式。（iv）算法易于实现，因为它仅需在全阶模型上对策略性选择的条目进行求值，且不使用其高效实现具有侵入性的切空间投影。我们通过多个测试案例验证了所提算法的效率，包括一个描述尺寸为$70^{100} \approx 3.2 \times 10^{184}$的张量演化的非线性100维张量微分方程，以及一个涉及尺寸为$4.7 \times 10^9$的张量的随机平流-扩散-反应方程。