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A priori Assessment of Tensor-Network Encoding for Isotropic Turbulent Flows

This paper demonstrates that the matrix product state (MPS) tensor network ansatz can effectively compress isotropic turbulent flow data from direct numerical simulations, achieving 99.8% reconstruction fidelity with only 5–15% of the original memory while accurately preserving key statistical properties like kinetic energy and dissipation rates.

Original authors: Massen Esmaeili, Hirad Alipanah, Robert Pinkston, Peyman Givi, Daniel Livescu, Andrew J. Daley, Dieter Jaksch, Juan José Mendoza-Arenas

Published 2026-09-01

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CC BY 4.0

Original authors: Massen Esmaeili, Hirad Alipanah, Robert Pinkston, Peyman Givi, Daniel Livescu, Andrew J. Daley, Dieter Jaksch, Juan José Mendoza-Arenas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Turbulence is the chaotic, swirling motion of fluids that surrounds us, from the smoke rising from a cigarette to the air rushing over an airplane wing. Predicting how these flows behave is one of the most difficult challenges in physics because the motion happens on so many different scales at once. The largest swirls contain most of the energy, while the tiniest eddies are where that energy finally dissipates into heat. To simulate this accurately on a computer, scientists must track every single swirl, from the massive ones down to the microscopic ones. This requirement creates a data problem of staggering proportions. A single, high-quality computer simulation of such a flow can generate so much information that it would fill the memory of the world's most powerful supercomputers, making it nearly impossible to store, analyze, or share the results. For decades, researchers have sought ways to compress this data without losing the essential physics that govern the flow.

A team of researchers has now tested a new approach to this problem, borrowing a tool originally designed for a completely different field of science: quantum physics. In the quantum world, scientists use a mathematical framework called tensor networks to describe systems with many interacting parts, finding a way to represent complex states without needing to list every single possibility. The researchers applied this same framework to the study of turbulent fluids. They took two sets of detailed computer-generated data—one representing the swirling motion of a fluid and another representing a passive substance, like a dye, being mixed into that fluid—and attempted to compress them using a specific type of tensor network known as a matrix product state. The goal was to see if they could shrink the data down to a fraction of its original size while still preserving the accurate statistics of the flow, such as how fast the fluid moves or how quickly the dye mixes.

The process worked by breaking the massive three-dimensional grid of data into a chain of smaller, interconnected pieces. The researchers then analyzed the connections between these pieces to identify which parts of the data were truly essential and which were redundant. By discarding the least significant connections, they created a much smaller, compressed version of the original flow. When they expanded this compressed data back to its full size to compare it with the original, the results were strikingly accurate. For the fluid velocity, the compressed version retained 99.8% of the original information while using only 5% of the memory space. For the mixing scalar, the compressed version reached the same level of accuracy using 15% of the memory. This means that a dataset that would normally require a massive amount of storage could be reduced to the size of a small file without losing the ability to see the flow's behavior.

The researchers did not just check if the pictures looked similar; they rigorously tested the physics hidden within the data. They examined the energy of the flow, the rate at which it dissipated, and the statistical patterns of how the fluid moved at different speeds. Even at high levels of compression, the total energy of the flow was recovered with an error of less than 0.2%. The average rate at which energy was lost to friction was also very close to the original, staying within about 10% of the true value. This is a significant achievement because the rate of energy loss depends on the tiniest, most chaotic details of the flow, which are usually the first things to disappear when data is compressed. The researchers found that the compressed data successfully preserved the complex, intermittent bursts of motion that characterize turbulence, proving that the method did not just smooth out the rough edges but kept the essential, jagged nature of the flow intact.

However, the study also revealed where the limits of this compression lie. While the large-scale structures of the flow were preserved beautifully, the most extreme, fine-scale events required the highest level of detail to be captured accurately. When the researchers used a more aggressive compression that removed more data, the statistics related to the smallest scales, such as the sharp gradients where the fluid changes direction rapidly, began to show larger errors. This suggests that while the method is incredibly efficient for storing and analyzing the bulk of the flow, capturing the very rarest and most intense moments of turbulence requires keeping more of the original data. The researchers found that the way they arranged the data before compressing it mattered greatly; by interleaving the information from different directions, they were able to capture the correlations between the swirling motions more effectively than by grouping them separately.

This work demonstrates that tensor networks, a tool from quantum mechanics, can serve as a powerful new way to handle the massive datasets generated by fluid dynamics. It offers a path forward for storing and analyzing complex turbulent flows without needing the immense storage capacity currently required. The findings suggest that this method can be used to create reduced-order models that are both compact and physically faithful, allowing scientists to study complex flows more efficiently. While the current tests were performed on idealized, uniform flows, the success of the method provides a strong foundation for exploring its use in more complex, real-world scenarios, such as flows with walls, chemical reactions, or multiple phases. The study confirms that the essential physics of turbulence can be encoded in a surprisingly small amount of information, opening the door to a new era of computational fluid dynamics where the data is manageable, yet the science remains precise.

Technical Summary: A Priori Assessment of Tensor-Network Encoding for Isotropic Turbulent Flows

Problem Statement
Direct numerical simulation (DNS) of turbulent flows is computationally prohibitive due to the exponential growth in memory and processing requirements as the Reynolds number increases. Resolving the full spectrum of scales, from energy-containing eddies to dissipative scales, necessitates extremely fine grids and small time steps, often exceeding the capabilities of current supercomputers. While reduced-order modeling (ROM) techniques exist, they typically rely on modal decompositions (e.g., POD) or projection-based methods that may not fully preserve the multiscale correlations inherent in isotropic turbulence. Concurrently, Tensor Network (TN) methods, specifically Matrix Product States (MPS), have proven highly effective in compressing high-dimensional data in quantum many-body physics and other fields, yet their applicability to characterizing fully developed turbulent flows remains largely unexplored. There is a lack of systematic a priori assessments quantifying how MPS-based compression impacts turbulence statistics across spatial scales.

Methodology
This study employs the Matrix Product State (MPS) ansatz as a reduced-order modeling framework to construct truncated representations of isotropic turbulent flow data. The methodology involves:

Data Encoding: Two DNS datasets are utilized: an incompressible 3D hydrodynamic velocity field ( $Re_\lambda \approx 433$ ) and a conserved passive Fickian scalar field transported by a similar flow ( $Re_\lambda \approx 180$ ). Both fields are defined on a $1024^3$ grid.
Tensor Reshaping: The 3D fields are reshaped into unstructured tensors of order $N = \log_2(n)$ using a qubit-based encoding ( $d=2$ ), where each physical index corresponds to a specific length scale along a spatial direction.
Index Ordering: To optimize correlation capture, the spatial tensor indices are reordered. The study compares "split," "staircase," and "interleaved" orderings. The interleaved ordering ( $x_1, y_1, z_1, x_2, y_2, z_2, \dots$ ) is selected because it places indices of comparable length scales from different spatial directions adjacent to one another, localizing dominant inter-tensor correlations within short-range bond indices.
Compression via SVD: The tensor is factorized into a chain of site tensors connected by bond indices through a sequence of Singular Value Decompositions (SVDs). Small singular values are discarded based on a user-defined cutoff threshold ( $10^{-C}$ , where $C \in \{2, 3, 4, 5\}$ ). The truncation is optimal in the $L_2$ (Frobenius) sense, bounding the error by the sum of squared discarded singular values.
Reconstruction and Validation: The truncated MPS is contracted back to the full $1024^3$ grid. The reconstructed fields are compared against the original DNS data using infidelity metrics, compression ratios, and a comprehensive suite of statistical measures.

Key Contributions

Systematic A Priori Assessment: The paper provides a rigorous evaluation of MPS encoding on high-fidelity DNS data, moving beyond previous limited studies to examine a wide range of low- and high-order statistics.
Optimization of Index Ordering: It demonstrates that interleaved ordering significantly outperforms split ordering for isotropic turbulence by aligning the MPS chain structure with the physical correlation structure of the flow (where scales of similar size across $x, y, z$ are statistically related).
Quantification of Trade-offs: The study systematically maps the trade-off between compression ratio (storage reduction) and accuracy across various statistical quantities, including energy spectra, probability density functions (PDFs), structure functions, and gradient-dependent metrics.

Results

Compression Efficiency: The MPS representation achieves substantial storage reductions. For the velocity field, a cutoff of $C=4$ yields a compression ratio (CR) of approximately 22 (retaining ~5% of original memory) with an infidelity of $0.18\%$ . For the scalar field, a cutoff of $C=4$ yields a CR of approximately 6.5 (retaining ~15% of original memory) with an infidelity of $0.18\%$ . At the highest fidelity setting for the scalar field ( $C=5$ ), the CR decreases to approximately 1.7, retaining roughly 59% of the original memory, with an infidelity of $0.02\%$ . The scalar field generally requires higher memory usage than the velocity field to achieve comparable fidelity due to its sharper gradients.
Statistical Fidelity:
- Low-Order Statistics: Total kinetic energy and scalar energy are recovered with relative errors below $0.2\%$ at $C=4$ .
- Gradient-Dependent Statistics: Quantities relying on spatial derivatives, such as mean dissipation and scalar dissipation, show higher sensitivity to truncation. At $C=4$ , mean dissipation errors are approximately $4\%$ for velocity and $10\%$ for the scalar. These errors decrease to $\sim 2.6\%$ and $\sim 1.8\%$ respectively at $C=5$ .
- Spectral and Structural Features: The reconstructed energy spectra follow the Kolmogorov $k^{-5/3}$ inertial scaling accurately up to the cutoff-dependent dissipative range. The characteristic "teardrop" distribution of the $Q-R$ invariants (velocity gradient tensor invariants) and the Kolmogorov 4/5-law for third-order structure functions are well-preserved at $C=4$ and $C=5$ .
- PDFs: The MPS captures the near-Gaussian distribution of velocity components and the heavy-tailed distributions of velocity gradients and dissipation. However, extreme events (tails of the PDFs) are under-represented at lower cutoffs ( $C=3$ ) and require $C=4$ or $C=5$ for accurate reproduction.
Schmidt Spectra Analysis: The study reveals that the Schmidt spectrum exhibits a pyramidal structure with asymmetry; larger bond dimensions are required at the "fine-scale" end of the MPS chain to capture the wide-range correlations of small-scale fluctuations, while large-scale structures are captured with fewer singular values.

Significance and Claims
The authors claim that these findings support the suitability of MPS for scalable reduced-order analysis of complex turbulent datasets. The work demonstrates that TN-based methods can preserve not only large-scale energy-containing structures but also the finer scales associated with turbulent mixing and intermittency, provided the truncation threshold is sufficiently low. The adaptive nature of the MPS construction—retaining more singular values where inter-tensor correlations are stronger—allows for a balanced preservation of flow physics across scales.

The paper concludes that while MPS is effective for a priori compression and analysis, its application to time-evolving simulations and more complex flows (e.g., anisotropic, wall-bounded, or reacting flows) requires further exploration. It suggests that future work should investigate other TN architectures (e.g., Tree Tensor Networks, PEPS) that might better capture the specific correlation structures of 3D turbulence and explore the parallel between TN truncation and subgrid-scale modeling in Large Eddy Simulation (LES).

Technical Summary: A Priori Assessment of Tensor-Network Encoding for Isotropic Turbulent Flows

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