发表机构
University of Pittsburgh; Los Alamos National Laboratory; University of Oxford; University of Hamburg（匹兹堡大学; 洛斯阿拉莫斯国家实验室; 牛津大学; 汉堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究评估了用矩阵乘积态（MPS）编码各向同性湍流数据的效果，发现其可在低内存下高保真重构流场，为湍流降阶分析提供了可行方法。
AI 中文摘要
张量网络（Tensor Networks, TNs）最初为模拟多体量子系统开发，提供了逼近高维场的系统框架，该框架通过将场分解为具有小键维度的相互连接张量实现，从而限制了跨场二分体捕获的相关性。属于TNs家族的矩阵乘积态（Matrix Product State, MPS）拟设在此被用作降阶建模框架，以构建各向同性湍流数据的截断表示。考虑两组直接数值模拟（Direct Numerical Simulation, DNS）数据集：不可压缩三维流动的流体动力学场，以及类似流动中的守恒菲克标量。每个场通过一系列奇异值分解（Singular Value Decompositions, SVD）编码为MPS，其中小奇异值被舍弃。将截断表示收缩回完整网格，并将所得重构场与DNS结果进行比较。在分解前，对输运变量的空间张量指数应用交错排序，以定位主导的张量间相关性。速度重构仅使用原始DNS内存的5%即可达到99.8%的保真度，而标量场在15%的内存使用量下达到相同保真度。系统检查了包括速度梯度、耗散和结构函数在内的广泛低阶和高阶统计量。在这些压缩水平下，总动能和标量能量的相对误差均在0.2%以内，而平均耗散和平均标量耗散仍在DNS生成值的约10%以内。这些发现支持MPS适用于复杂湍流数据集的可扩展降阶分析，并推动在计算湍流中进一步探索基于TN的方法。
英文摘要
Tensor networks (TNs), originally developed for simulating many-body quantum systems, provide a systematic framework for approximating high-dimensional fields. This is achieved by factorizing the field into interconnected tensors with small bond dimensions, thereby restricting the correlations captured across field bipartitions. Belonging to the family of TNs, the matrix product state (MPS) ansatz is utilized here as a reduced-order modeling framework to construct truncated representations of isotropic turbulent flow data. Two direct numerical simulation (DNS) datasets are considered: the hydrodynamic field of an incompressible three-dimensional flow, and a conserved Fickian scalar in a similar flow. Each field is encoded as an MPS through a sequence of singular value decompositions (SVDs) in which small singular values are discarded. The truncated representation is contracted back to the full grid, and the resulting reconstructed field is compared against DNS. An interleaved ordering of the spatial tensor indices of the transport variables is applied prior to decomposition in order to localize the dominant inter-tensor correlations. Velocity reconstructions achieve $99.8\%$ fidelity using only $5\%$ of the original DNS memory, while the scalar field reaches the same fidelity at $15\%$ memory usage. A wide range of lower- and higher-order statistics, including velocity gradients, dissipation, and structure functions, are systematically examined. At these compression levels, the total kinetic energy and the scalar energy are both recovered within $0.2\%$ relative error, while the mean dissipation and mean scalar dissipation remain within approximately $10\%$ of the DNS generated values. These findings support the suitability of MPS for scalable reduced-order analysis of complex turbulent datasets and motivate further exploration of TN-based methods in computational turbulence.
Comments23 pages, 17 figures, 5 tables
