We present a new method based on functional tensor decomposition and dynamic tensor approximation to compute the solution of a high-dimensional time-dependent nonlinear partial differential equation (PDE). The idea of dynamic approximation is to project the time derivative of the PDE solution onto the tangent space of a low-rank functional tensor manifold at each time. Such a projection can be computed by minimizing a convex energy functional over the tangent space. This minimization problem yields the unique optimal velocity vector that allows us to integrate the PDE forward in time on a tensor manifold of constant rank. In the case of initial/boundary value problems defined in real separable Hilbert spaces, this procedure yields evolution equations for the tensor modes in the form of a coupled system of one-dimensional time-dependent PDEs. We apply the dynamic tensor approximation to a four-dimensional Fokker–Planck equation with non-constant drift and diffusion coefficients, and demonstrate its accuracy in predicting relaxation to statistical equilibrium.

我们提出了一种基于函数张量分解和动态张量逼近的新方法，用于计算高维时变非线性偏微分方程（PDE）的解。动态逼近的想法是每次将PDE解的时间导数投影到低阶功能张量流形的切线空间上。可以通过最小化切空间上的凸能量函数来计算这样的投影。这个最小化问题产生了唯一的最佳速度矢量，该矢量使我们能够将PDE及时向前积分到恒定秩的张量流形上。在实数可分离的希尔伯特空间中定义的初值/边值问题的情况下，此过程以一维时间相关的PDE耦合系统的形式生成张量模式的演化方程。