We present a new method to compute the solution to a nonlinear tensor differential equation with dynamical low-rank approximation. The idea of dynamical low-rank approximation is to project the differential equation onto the tangent space of a low-rank tensor manifold at each time. Traditionally, an orthogonal projection onto the tangent space is employed, which is challenging to compute for nonlinear differential equations. We introduce a novel interpolatory projection onto the tangent space that is easily computed for many nonlinear differential equations and satisfies the differential equation at a set of carefully selected indices. To select these indices, we devise a new algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes any tensor train and its tangent space with tensor cross interpolants. We demonstrate the proposed method with applications to tensor differential equations arising from the discretization of partial differential equations.

翻译：我们提出了一种新方法，利用动态低秩逼近计算非线性张量微分方程的解。动态低秩逼近的核心思想是在每个时刻将微分方程投影到低秩张量流形的切空间上。传统上采用正交投影到切空间的方法，但这对非线性微分方程的计算具有挑战性。我们引入了一种新颖的插值投影到切空间的方法，该方法可轻松应用于许多非线性微分方程，并在精心选择的索引集合上满足微分方程。为了选择这些索引，我们基于离散经验插值法（DEIM）设计了一种新算法，该算法通过张量交叉插值来参数化任意张量列车及其切空间。我们通过偏微分方程离散化产生的张量微分方程实例，验证了所提方法的有效性。