Problem
Research questions and friction points this paper is trying to address.
Tensor Networks
Matrix Product State
Isotropic Turbulent Flows
Data Compression
Reduced-Order Modeling
A priori Assessment of Tensor-Network Encoding for Isotropic Turbulent Flows
🤖 AI Summary
本文使用矩阵乘积态（MPS）方法对各向同性湍流数据进行降阶建模，通过奇异值分解实现高效压缩与重构，验证了该方法在处理复杂湍流数据中的有效性。
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Tensor Networks
Matrix Product State (MPS)
Singular Value Decomposition (SVD)
Reduced-Order Modeling
Isotropic Turbulence
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