Dynamical low-rank approximation (DLRA) provides a rigorous, cost-effective mathematical framework for solving high-dimensional tensor differential equations (TDEs) on low-rank tensor manifolds. Despite their effectiveness, DLRA-based low-rank approximations lose their computational efficiency when applied to nonlinear TDEs, particularly those exhibiting non-polynomial nonlinearity. In this paper, we present a novel algorithm for the time integration of TDEs on the tensor train and Tucker tensor low-rank manifolds, which are the building blocks of many tensor network decompositions. This paper builds on 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 leverages 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. As a result, it offers near-optimal computational savings both in terms of memory and floating-point operations. (ii) The time integration is robust in the presence of small or zero singular values. (iii) The algorithm is remarkably easy to implement, as it requires the evaluation of the full-order model TDE at strategically selected entries and it does not use tangent space projections, whose efficient implementation is intrusive and time-consuming. (iv) We develop high-order explicit Runge-Kutta schemes for the time integration of TDEs on low-rank manifolds. We demonstrate the efficiency of the presented algorithm for several test cases, including a 100-dimensional TDE with non-polynomial nonlinearity.

翻译：动态低秩近似为在低秩张量流形上求解高维张量微分方程提供了严谨且高性价比的数学框架。尽管有效，基于动态低秩近似的低秩逼近在处理非线性张量微分方程时，特别是涉及非多项式非线性的情况，会丧失计算效率。本文提出了一种新颖算法，用于在张量列和Tucker张量低秩流形（许多张量网络分解的基础构件）上实现张量微分方程的时间积分。本工作基于我们先前在利用CUR分解求解低秩矩阵流形上的非线性矩阵微分方程的研究。该方法具有多重优势：(i) 基于离散经验插值方法的交叉算法，通过策略性采样时间离散张量微分方程的稀疏项，以低秩形式推进解的发展，从而在内存和浮点运算方面实现近最优的计算节省；(ii) 在奇异值较小或为零时仍保持时间积分的鲁棒性；(iii) 算法实现极为简便，仅需评估全阶模型张量微分方程在策略性选取的条目，无需使用切空间投影（其高效实现具有侵入性且耗时）；(iv) 我们开发了高阶显式龙格-库塔格式用于低秩流形上张量微分方程的时间积分。我们通过多个测试案例验证了所提算法的效率，其中包括一个含非多项式非线性的100维张量微分方程。