Emerging tensor network techniques for solutions of Partial Differential Equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultrafast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for solution of the time dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time dependent PDEs often use finite difference for time, and a spectral scheme for the spatial dimensions, which leads to slow linear convergence. Spectral collocation space-time methods show exponential convergence, however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to very high compression of linear operators from terabytes to kilobytes in TT-format, and tens of thousands times speedup when compared to full grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

翻译：新兴的张量网络技术因能突破维数灾难，为高维问题的超快数值求解提供了新的数学方法。本文提出一种基于张量列（TT）的切比雪夫谱配点法，在空间和时间维度上同时求解具有非齐次边界条件的时间依赖对流-扩散-反应（CDR）方程，采用笛卡尔几何。以往求解时间依赖偏微分方程的数值方法通常对时间采用有限差分、对空间维度采用谱格式，这导致线性收敛速度较慢。时空谱配点法虽能实现指数收敛，但处理实际问题时需要求解大型四维系统。我们通过引入TT方法克服这一难题，因为其复杂度仅随维度数量线性增长。研究表明，当CDR方程解光滑时，所提出的TT时空切比雪夫谱配点法可实现指数收敛，并且能将线性算子从太字节量级压缩至千字节量级的TT格式，与全网格时空谱方法相比实现数万倍加速。这些优势使我们能够在更高分辨率下获得数值解。