Cross Algorithms for Cost-Effective Time Integration of Nonlinear Tensor Differential Equations on Low-Rank Tucker Tensor and Tensor Train Manifolds

Abstract

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.