Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial Differential Equations (PDEs). Here, we present a mixed Tensor Train (TT)/Quantized Tensor Train (QTT) approach for the numerical solution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian geometry. Discretizing a realistic three-dimensional (3D) BNTE by (i) diamond differencing, (ii) multigroup-in-energy, and (iii) discrete ordinate collocation leads to huge generalized eigenvalue problems that generally require a matrix-free approach and large computer clusters. Starting from this discretization, we construct a TT representation of the PDE fields and discrete operators, followed by a QTT representation of the TT cores and solving the tensorized generalized eigenvalue problem in a fixed-point scheme with tensor network optimization techniques. We validate our approach by applying it to two realistic examples of 3D neutron transport problems, currently solved by the PARallel TIme-dependent SN (PARTISN) solver. We demonstrate that our TT/QTT method, executed on a standard desktop computer, leads to a yottabyte compression of the memory storage, and more than 7500 times speedup with a discrepancy of less than 1e-5 when compared to the PARTISN solution.

翻译：张量网络技术以其能够突破维数诅咒的低秩近似能力，正成为新数学方法的基础，用于高维偏微分方程的超快速数值求解。本文提出一种混合张量列/量化张量列方法，用于在笛卡尔几何中数值求解含时无关玻尔兹曼中子输运方程。通过（i）菱形差分、（ii）多群能量离散和（iii）离散纵标配置对三维实用BNTE进行离散化处理，会产生规模巨大的广义特征值问题，通常需要采用无矩阵方法和大型计算机集群。基于该离散化过程，我们首先构建偏微分方程场和离散算子的TT表示，随后对TT核实施QTT表示，并采用张量网络优化技术在定点迭代方案中求解张量化广义特征值问题。通过应用于两个当前由PARTISN求解器处理的三维中子输运实用算例，我们验证了所提方法的有效性。实验表明，在标准桌面计算机上执行的TT/QTT方法，相较于PARTISN解，实现了存储器存储的尧字节级压缩，速度提升超过7500倍，且偏差小于1e-5。