Compression, simulation, and synthesis of turbulent flows with tensor trains

TL;DR

The paper tackles the high-dimensional challenge of turbulence simulations by employing tensor-train (TT/MPS) representations to compress, evolve, and synthesize turbulent velocity fields. It benchmarks TT encodings on isotropic turbulence data at $Re_\lambda=315$, showing that a TT with $\chi$ around 1000 can capture the inertial-range energy spectrum $E(k) \propto k^{-5/3}$ and intermittency with only about $2.2\%$ of the full parameter count. It extends a 2D TT Navier–Stokes solver to 3D with a divergence-free projection and explicit time stepping, achieving stability and a memory footprint of $\approx 0.03\%$ of the dense representation over $9$–$10$ turnover times. Finally, it introduces a TT-based multiscale turbulence synthesis method that achieves linear growth of bond dimension with the number of scales and reproduces key turbulence signatures, highlighting TT as a quantum-inspired toolkit for efficient turbulence treatment and real-time generation of turbulent-like flows.

Abstract

Numerical simulations of turbulent fluids are paramount to real-life applications, from predicting and modeling flows to diagnostic purposes in engineering. However, they are also computationally challenging due to their intrinsically non-linear dynamics, which require a very high spatial resolution to accurately describe them. A promising idea is to represent flows on a discrete mesh using tensor trains (TTs), featuring a convenient scaling of the number of parameters with the mesh size. However, it is unclear how the compression power of TTs is affected by the complexity of the flows, as measured by the Reynolds number. In fact, no comprehensive analysis of how the TT representation affects the turbulent properties has yet been carried out. We fill this gap by analyzing TTs as an Ansatz to compress, simulate, and generate 3D snapshots with turbulent-like features. Specifically, we first investigate the effect of TT compression on key turbulence signatures, such as the energy spectrum, the PDF of velocity increments, and flatness. Second, we extend the 2D TT-solver introduced in [1] to a 3D cubic domain with periodic boundary conditions. We use it to simulate the incompressible Navier-Stokes dynamics at $Re_λ=315$ for a total of 9-10 Kolmogorov turnover times, showcasing the numerical stability of the TT-solver in fully developed turbulent regimes. Third, we develop a TT algorithm to synthesize artificial snapshots that exhibit turbulent-like features, with a logarithmic cost in the mesh size. Our analysis demonstrates the ability of the TT representation to capture the characteristic features of turbulence. This offers a powerful quantum-inspired toolkit for the computational treatment of turbulent flows.

Figures (6)

• Figure 1: Different TT encodings of a velocity field. The discretized velocity component $u(x_{\textbf{i}},y_{\textbf{j}},z_{\textbf{k}})$ can be decomposed into two different types of TTs: stacked and interleaved. In both cases, the TT consists of a 1D chain of tensors connected over their virtual (horizontal) indices. Each tensor has a physical (vertical) index labeled by a bit $i_l$, $j_l$, or $k_l$ in the binary representation of $(\textbf{i},\textbf{j},\textbf{k})$. These binary indices naturally define a notion of spatial scales, with $(i_m,j_m,k_m)$ signifying the $m$-$th$ subdivision of the dyadic grid. The maximal cardinality over all virtual indices is called the bond dimension of the TT, which captures the amount of inter-scale correlations. The stacked and interleaved encodings differ in the ordering of the binary indices, as described in Eqs. \ref{['eq:MPS_sta']} and \ref{['eq:MPS_inter']}. The three components of the velocity field ($u,v,w$) are encoded into three individual TTs. These can, in turn, be represented by a single TT using an additional tensor with a 3-dimensional physical index $p$, defined as the concatenated TT representation of the full velocity field. An example of this is shown at the bottom for the stacked encoding.
• Figure 2: Schematic representation of the three investigated aspects of turbulence. We numerically investigate three different settings to analyze the TT encoding of turbulent flows. A. Single snapshot compression. Here, we encode the velocity field at a given time, called a snapshot, into its corresponding TT representation. We use the well-known turbulence DNS dataset Cardesa_2017 and compare key statistical metrics of the compressed TT snapshots with those of the original dataset for increasing bond dimensions $\chi$. We perform this analysis for both stacked and interleaved encoding. The results are reported in Sec. \ref{['subsection:Single snapshot compression']}. B. 3D TT-based solver. Using the TT snapshot obtained from the DNS solution as the initial condition, we simulate the time evolution of the flow completely within the TT representation. We first project onto the divergence-free manifold of the velocity field and perform the time stepping using an explicit Euler scheme. We compute the energy spectrum of the time-series of solutions obtained. We report the results for the interleaved encoding in Sec. \ref{['subsection:3D MPS-based solver']}. C. TT snapshots synthesis. Here, we construct a TT field that exhibits some key turbulent features: the divergence-free condition, the Kolmogorov energy spectrum, and intermittency, quantified by the flatness. The algorithm generates random low-rank TTs ($\chi=10$) at each spatial scale $m$, then interpolates them to the desired resolution $M$. The final snapshot is the summation of these TTs weighted by the appropriate weights $\omega_m$. We compute and verify these properties for an ensemble of 20 snapshots. In this instance, we restrict ourselves to the interleaved encoding. The detailed explanation with results is outlined in Sec. \ref{['subsection:Synthetic_turbulence']}.
• Figure 3: Single snapshot compression: Turbulent statistical metrics for the compressed TT snapshots. The snapshots have been compressed and projected to satisfy the divergence-free condition. This is achieved through an iterative scheme that alternates between the compression and projection steps, for a total of 10 iterations. We end with a compression step, such that the resulting TTs have exactly the reported $\chi$. a. Kinetic turbulent energy spectrum as a function of the wave-number magnitude for various values of $\chi$ and the two possible encodings. The plots are on a log-log scale, and the approximately linear region corresponds to the inertial range, with a power-law decay given by Eq. \ref{['eq:E_k']}. We note that $\chi = 1000$ already reproduces entirely the inertial range for both the encodings, with the compressed spectra detaching from the original one (dotted line) around $k=70$. The inset shows a zoomed version of the main plot around that region. $\chi = 1000$ corresponds to a TT with only the $2.2\%$ of the total number of parameters required for its dense-vector representation. b. The main plot reports the flatness, as per Eq. \ref{['eq:flatness']}, for the interleaved encoding and various bond dimensions. The inset shows the flatness for both the encodings at $\chi = 1000$. We see that for very small separation distances the curves deviate from the ground truth, either by overshooting (for low $\chi$s) or by undershooting (for large $\chi$s). We explain this behavior in the main text. c. PDF of the longitudinal velocity increments with $\chi = 1000$ for various separation distances. Here, we report the separation distances in terms of computational unit cells for clarity. The curves are shifted for the sake of clarity. We note that the interleaved encoding captures the non-Gaussian statistics that emerge for small separations better than the stacked one. d. PDF of the velocity increments for a given separation distance ($r=5$) and various bond dimensions. We again observe that the interleaved encoding captures the non-Gaussian statistics better than the stacked one, especially for moderate bond dimensions. We also observe, as expected, that we converge to the ground-truth statistics as $\chi$ increases.
• Figure 4: 3D TT-based solver:$E(K)$ and divergence $L^{\infty}$-norm during the 3D TT time evolution. We show the energy spectrum $E(k)$ as a function of the wave number $k$ (main figure) and the $L^{\infty}$-norm of the divergence (inset) over time. We performed a total of 1200 computational time steps, with the following parameters: kinematic viscosity $\nu = 0.00067$; time step $\Delta t = 0.0002$; number of tensor cores per spatial dimension $N_x = N_y = N_z = 10$; Reynolds number at the Taylor micro scale $\mathrm{Re}_{\lambda}=315$. These simulations use only the $0.03\%$ of the total number of parameters needed for the full vector representation. The number of time steps simulated is equivalent to 120 time steps in the original dataset presented in the main text Cardesa_2017. This evolution time roughly corresponds to 9-10 Kolmogorov turnover times. The TTs used have a low bond dimension, $\chi = 100$. The low bond dimension, and the absence of a forcing term in the Eqs. \ref{['eq:NS']}, explains the small deviations from the expected shape for $E(k)$.
• Figure 5: TT snapshots synthesis: Turbulence metrics for the synthetic generated snapshots with the proposed TT algorithm.a. shows the energy spectrum $E(k)$ plotted against wavenumber $k = |\mathbf{k}|$ on a log–log scale, with the dashed red line indicating $k^{-5/3}$ power law decay. We also plot a 3D snapshot of the velocity magnitude of the synthetic turbulent field. b. presents the flatness (kurtosis) of velocity increments (Eq. \ref{['eq:flatness']}) as a function of physical separation $r$, on a log-log scale; The deviation from the flat line indicates the presence of intermittency, or non-Gaussianity, in the velocity fluctuations. c. displays the TT bond dimension $\chi$ versus the number of scales $M$ (one third of the total TT tensors), together with the dashed linear fit $\chi \propto 201.8 M$, showing a linear growth. Moreover, we show the compression rates achieved by the TT-generated snapshots with respect to the full vector representations as a function of $M$. We average over 20 synthetic snapshots with random seeds, and the shaded regions denote one standard deviation.