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：张量网络技术以其低秩近似能力打破维度灾难，正在成为高维偏微分方程超快数值求解的新数学方法基础。我们针对笛卡尔几何下的稳态玻尔兹曼中子输运方程数值解，提出一种混合张量列/量化张量列方法。通过（i）菱形差分法、（ii）多群能量离散和（iii）离散纵标配点法离散化真实三维玻尔兹曼中子输运方程，会产生大型广义特征值问题，通常需要无矩阵方法和大型计算机集群。基于该离散化方案，我们构建了偏微分方程场和离散算子的张量列表示，随后对张量列核心进行量化张量列表示，并采用张量网络优化技术以不动点方案求解张量化广义特征值问题。通过两个真实三维中子输运问题实例验证该方法，这些实例目前由并行时间相关SN求解器求解。我们证明了在标准台式计算机上执行的张量列/量化张量列方法，相比并行时间相关SN求解器可实现约字节级内存压缩，在偏差小于1e-5的前提下，计算加速超过7500倍。