求解现实稳态Boltzmann中子输运方程的张量网络方法
Tensor Networks for Solving Realistic Time-independent Boltzmann Neutron Transport Equation
AI总结：
提出混合TT/QTT方法求解稳态Boltzmann中子输运方程，在标准台式机上实现YB级存储压缩和超7500倍加速，且误差小于1e-5。
摘要：
张量网络技术以其打破维度灾难的低秩近似能力而闻名，正成为高维偏微分方程（PDE）超快数值求解新数学方法的基础。在此，我们提出一种混合张量序列（TT）/量子化张量序列（QTT）方法，用于数值求解笛卡尔几何下的稳态Boltzmann中子输运方程（BNTEs）。通过(i)菱形差分、(ii)能量多群和(iii)离散纵迹配置对现实的3D BNTE进行离散化，会产生庞大的广义特征值问题，通常需要无矩阵方法和大型计算机集群。从该离散化出发，我们构建了PDE场和离散算子的TT表示，随后构建了TT核的QTT表示，并在采用张量网络优化技术的定点方案中求解张量化的广义特征值问题。我们将该方法应用于目前由PARallel TIme-dependent SN（PARTISN）求解器求解的3D中子输运问题的两个现实示例来验证。我们证明，在标准台式计算机上执行的TT/QTT方法实现了存储的YB级压缩，且与PARTISN解相比，加速超过7500倍，差异小于1e-5。
英文摘要：
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.
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