We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.

翻译：本文提出了在低秩流形上积分非线性微分方程的新方法。这些方法基于切空间的插值投影，能够对可按分量求值的向量场进行低秩时间积分。本方法的关键优势在于不要求向量场具有低秩结构，从而克服了传统基于正交投影的动态低秩方法的显著局限性。为构建插值投影算子，我们基于离散经验插值方法（DEIM）开发了一种稀疏张量采样算法，该算法通过交叉插值参数化张量链流形及其切空间。利用这些投影算子，我们提出了两种在低秩张量链流形上的时间积分格式：第一种格式在选定的插值指标处积分解，并通过交叉插值构建解；第二种格式将著名的正交投影算子分裂积分器推广至插值投影算子。我们通过偏微分方程离散化产生的若干张量微分方程应用案例，验证了所提方法的有效性。