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A priori Assessment of Tensor-Network Encoding for Isotropic Turbulent Flows
Massen Esmaeili, Hirad Alipanah, Robert Pinkston, Peyman Givi, Daniel Livescu, Andrew J. Daley, Dieter Jaksch, Juan José Mendoza-Arenas
TL;DR
The paper asks whether MPS tensor-network encoding can compress fully developed isotropic turbulent-flow data while preserving turbulence statistics. It applies sequential-SVD MPS compression to DNS velocity and passive-scalar fields, reconstructs them on the full grid, and compares them with DNS. At C=4, reconstructions reach about 0.2% infidelity while compressing velocity and scalar storage by factors of about 22 and 7, respectively, with gradient-sensitive statistics remaining less accurate.
Problem
The applicability of tensor-network methods to fully developed turbulent flows and their effects on multiscale turbulence statistics remain largely unexplored.
Method
The study encodes DNS velocity and passive-scalar fields as truncated MPS representations using sequential SVDs, interleaved spatial ordering, full-grid reconstruction, and DNS comparisons.
Results
At C=4, reconstructed velocity and scalar fields have about 0.2% infidelity while compressing storage by factors of about 22 and 7, respectively.
Takeaways & Limitations
MPS can provide storage-efficient reduced-order representations that preserve a wide range of velocity and scalar statistics, including higher-order and gradient-sensitive quantities with differing accuracy.
Takeaways & Limitations
The authors suggest that alternative tensor-network architectures may better capture the fine-scale and multiscale correlations limiting MPS efficiency in three-dimensional isotropic turbulence.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper addresses the largely unexplored use of tensor-network methods for characterizing fully developed turbulent flows. It proposes a systematic assessment of MPS compression for velocity and passive-scalar DNS data across turbulence statistics and spatial scales.
- DNS of turbulent flows is computationally expensive because resolving all scales requires extremely fine grids, small time steps, long simulations, and frequent sampling.
- MPS methods are investigated as a way to represent high-dimensional turbulent fields through tensor factorizations that restrict correlations across field bipartitions.
- The work extends prior MPS analysis by assessing two turbulent flows and quantifying compression-accuracy trade-offs across low- and high-order statistics.The examined statistics include large-scale means and higher-order structure functions probing dynamics and intermittency.
- The study targets three-dimensional incompressible homogeneous isotropic turbulence and passive-scalar mixing as baseline and more demanding tests, respectively.The scalar field is expected to develop sharp gradients and fine structures.
- The manuscript introduces MPS formalism and reconstruction procedures, then evaluates velocity and scalar truncations before summarizing findings and future directions.
2. MPS Formalism
The paper presents MPS as a one-dimensional tensor-network factorization whose bond dimensions adapt to retained singular values and inter-tensor correlations. Fields are reshaped, decomposed by sequential SVDs, truncated, reconstructed, and evaluated using DNS-based compression and accuracy metrics.
- MPS Formalism: MPS represents an order-N tensor with d^N elements as a one-dimensional chain of site tensors connected by auxiliary bond indices.For the qubit encoding used here, each physical index is binary and d=2.
- MPS Formalism: Bond dimensions χ_k control the inter-tensor correlations captured between neighboring MPS sites and may vary along the chain.The bond index α_k connects sites k and k+1.
- MPS Formalism: The construction reshapes the tensor into a matrix, performs sequential SVDs, converts left singular-vector factors into core tensors, and passes the remaining product forward.The procedure repeats until the final physical index is isolated.
- MPS Formalism: Truncating singular values produces an approximate MPS, while retaining all singular values gives an exact factorization.The number retained at each bond determines χ_k.
- MPS Formalism: The truncation error equals the sum of squared discarded singular values and is optimal in the Frobenius-norm sense.Lower cutoffs retain more singular values but increase memory and computational cost; larger cutoffs improve compression while risking fine-scale loss.
- MPS Formalism: Schmidt spectra interpret singular values as correlations across MPS bipartitions, with interleaved ordering associating earlier bonds with larger scales and later bonds with finer scales.The ordering is chosen to localize dominant correlations, whereas alternative orderings may suit anisotropic flows.
- 2.1. Implementation: The study applies MPS encoding to 1024^3 DNS velocity and passive-scalar datasets, including stationary isotropic turbulence and a zero-mean scalar field.Multiple snapshots are examined; the velocity dataset has Re_λ≈433, while the scalar develops an approximately Gaussian distribution with variance 6 × 10^-4.
- 2.1. Implementation: Compression ratio measures storage reduction, while infidelity compares reconstructed fields contracted to the full grid against DNS using a normalized inner-product measure.Velocity components are separately truncated and reconstructed for multiple cutoff values.
3. MPS representation of the hydrodynamic field
The hydrodynamic field is compressed with truncated MPS representations and evaluated through visual, spectral, structural, and turbulence-statistics comparisons against DNS. Increasing the cutoff precision progressively improves reconstruction, while fine-scale and gradient-sensitive quantities remain the most demanding.
- MPS compression and reconstruction: Cutoff 10^-C is applied to each velocity component, whose truncated MPS is contracted back to the full grid for comparison with DNS.The tested cutoffs are C∈{2, 3, 4, 5}, with singular values below the threshold removed.
- Field reconstruction: At C=2, reconstructions are visibly blurred with blocking artifacts, whereas differences from DNS become minimal at C=4 and C=5.The visual comparison uses a u-velocity slice in the xy-plane at z=π.
- Inter-tensor correlations: As C increases, the Schmidt-spectrum contours retain more singular values, with larger-scale correlations near the left bonds and finer scales toward the right.The maximum retained bond dimension χmax is obtained from the largest contour height across bonds.
- Compression and fidelity: At C=4, the average compression corresponds to approximately 3673 equivalent grid points instead of the original 1024^3 DNS grid, while infidelities remain on the order of 10^-3.The infidelity decreases by roughly an order of magnitude when C increases by one, and the same monotonic behavior occurs for all velocity components.
- Turbulence statistics: At C=5, mean dissipation error falls to approximately 2.6%, while total kinetic-energy error decreases from approximately 14.9% at C=2 to much smaller values at higher cutoffs.Gradient-based statistics are more sensitive because truncation first removes correlations associated with fine scales.
- Higher-order and gradient-sensitive statistics: At C=4 and C=5, MPS reproduces velocity-gradient distributions, strain statistics, and third-order structure-function scaling more closely than at lower cutoffs.At C=3, strain-eigenvalue tails and the rise toward s*=+1 are underestimated, whereas C=5 distributions nearly overlap DNS and highly truncated fields still capture inertial-range scaling to a similar degree as DNS.
4. MPS representation of the scalar field
MPS reconstructions preserve scalar structures and scalar energy well at moderate cutoffs, while scalar dissipation remains more sensitive to truncation because it depends on fine-scale gradients.
- Scalar-field reconstruction: At C=4 and C=5, reconstructed scalar contours recover small-scale texture with only mild smoothing and structures nearly indistinguishable from DNS.At C=3, filaments and contrast are attenuated.
- Scalar-field reconstruction: The scalar field has finer-scale structure than the velocity field, so identical cutoffs yield lower compression ratios.Storage reduction nevertheless remains substantial, and infidelity decreases rapidly as the cutoff is decreased.
- Scalar statistics: Scalar energy is recovered with relative errors below 0.2% for C=4.This statistic remains accurate at moderate truncation levels.
- Scalar statistics: Scalar dissipation is more sensitive to truncation than scalar energy because it depends directly on scalar gradients.Fine-scale-dependent quantities require smaller truncation errors for accurate reconstruction.
- Statistical comparisons: Pointwise scalar comparisons and scalar-dissipation PDFs assess agreement against DNS across cutoffs C=3 to 5.The visualizations include scatter plots and reference distributions for the reconstructed fields.
5. Conclusions
MPS compresses isotropic turbulent velocity and scalar DNS fields while retaining low infidelity and broad statistical agreement. Adaptive truncation helps preserve information across energy-containing and dissipative scales, but the study motivates testing other TN architectures and more complex flows.
- Main findings: At C=4, velocity and scalar fields reach about 0.2% infidelity with compression factors of about 22 and 7, respectively.The representation reduces storage by factors ranging from a few to the order 10^4 depending on cutoff.
- Interpretation: Adaptive bond-wise singular-value selection retains more information where inter-tensor correlations are stronger.The authors connect this adaptivity with preservation of large-scale energy-containing structures and small-scale dissipative motions.
- Interpretation: Reconstructed statistics align with Kolmogorov laws, supporting retention of essential turbulent-flow features at large compression ratios.This conclusion is stated within the isotropic-turbulence assessment.
- Future directions: Future work should test alternative TN architectures, temporal Navier–Stokes evolution, LES comparisons, and chemically reacting, multiphase, wall-bounded, and anisotropic flows.These directions address fine-scale and multiscale correlations that can limit MPS efficiency and assess generality beyond the present setting.
Appendix A. Definitions of turbulence statistics
The appendix defines the turbulence statistics used to evaluate reconstructed fields and specifies the physical parameters for the hydrodynamic and scalar datasets.
- Definitions: The turbulence statistics considered in this work are defined following Pope.These definitions support the reported comparisons between MPS reconstructions and DNS.
- Physical parameters: For the hydrodynamic dataset, the kinematic viscosity is ν=1.85 × 10^-4.This parameter is listed alongside the scalar-dataset diffusivity information.
- Physical parameters: For the scalar dataset, ν=5 × 10^-4 and Sc=1 implies molecular diffusivity D=ν.The passage identifies ν and D as the relevant transport parameters.
- Notation: The definitions include velocity components, kinematic viscosity, energy spectrum, and molecular diffusivity.These symbols are explicitly identified in the appendix text.