State estimation in homogeneous isotropic turbulence using super-resolution with a 4DVar training algorithm

TL;DR

The paper tackles state estimation in turbulent flows using coarse observations by marrying variational data assimilation with trajectory-based super-resolution trained in a differentiable DNS. It introduces SRdyn, a trajectory-consistent SR model, and 4DVarSR, a hybrid that initialises 4DVar with SR outputs; both operate without high-resolution training data. SRdyn excels at reconstructing the initial high-resolution field, while 4DVar provides stronger long-time fidelity; together they achieve robust predictions beyond the assimilation window. The approach demonstrates that embedding physical constraints and trajectory information into learning can yield accurate, data-efficient high-resolution reconstructions and effective 4DVar initializations, with implications for higher-Re studies and irregular observation setups.

Abstract

Variational data assimilation and machine-learning based super-resolution are two alternative approaches to state estimation in turbulent flows. The former is an optimisation problem featuring a time series of coarse observations, the latter usually requires a library of high-resolution 'ground truth' data. We show that the classic '4DVar' data assimilation algorithm can be used to train neural networks for super-resolution in three-dimensional isotropic turbulence without the need for high-resolution reference data. To do this, we adapt a pseudo-spectral version of the fully-differentiable JAX-CFD solver (Kochkov et al, Proc. Nat. Acad. Sci. 118, 2021) to three-dimensional flows and combine it with a convolutional neural network for super-resolution. As a result we are able to include entire trajectories in our loss function which is minimised with gradient-based optimisation to define the neural network weights. We show that the resulting neural networks outperform 4DVar for state estimation at initial time over a wide variety of metrics, though 4DVar leads to more robust predictions towards the end of its assimilation window. We also present a hybrid approach in which the trained neural network output is used to initialise 4DVar. The resulting performance is more than twice as accurate as other state estimation strategies for all times and performs well even beyond known limiting lengthscales, all without requiring access to high-resolution measurements at any point.

Figures (17)

• Figure 1: Sample slices of the vorticity field $\omega_{y}/\omega_{rms}$ at $z=0$. Black and red lines indicate the Taylor microscale $\lambda_T$ and the critical length scale $l_C=5\pi\eta_K$, respectively. The white grid lines in the lower left corners of the first three frames visualise the coarsened grids for coarsening factors $M\in\{4,8,16\}$ relative to the DNS grid.
• Figure 2: Neural network architecture used to train the SRdyn model with 'U.S.' indicating an upsampling layer. The field $\boldsymbol{u}^{p}_{0}$ that the network reconstructs from the coarse-grained DNS snapshot $\mathcal{C}(\boldsymbol{u}_{0})$ is subsequently advanced in time to compute the loss function \ref{['eq:LossfctTC']}.
• Figure 3: Reconstruction error $\epsilon_{\boldsymbol{u}}$ (equation \ref{['eq:errL2uvw']}) under time advancement for coarsening factors $M=4$, $M=8$ and $M=16$. Reconstructions shown are tri-cubic interpolation (blue), SRdyn (black), 4DVar (orange, solid/dashed/dotted lines for $T_{DA}/T_e =\TunrollNormA,\TunrollNormB,\TunrollNormC$ respectively). The grey (SRdyn) and orange (4DVar, $T_{DA}$=0.9$T_e$) shaded regions indicate the ensemble mean $\pm$ one standard deviation over an extended set of trajectories with different, statistically independent initial conditions, while the solid lines are the representative initial condition discussed in much of §\ref{['sec:results']}.
• Figure 4: Comparison of the reconstructed velocity fields at $t=0$ with the high-resolution DNS snapshot that is to be reproduced, alongside the respective coarse-grained fields. Shown are slices of $v/u_{rms}$ at $z=0$ for the different reconstruction procedures and coarsening factors $M\in\{4,8,16\}$, with colours ranging from blue (dark) to yellow (bright) in the interval $[-3,3]$. For the 4DVar runs, only the best performing case (in terms of the reconstruction error $\epsilon_{\boldsymbol{u}}$, cf. equation \ref{['eq:errL2uvw']}) is shown for each coarsening factor. Red boxes highlight individual high-speed regions in the ground truth state and its reconstructions.
• Figure 5: Standard probability density functions of (left) the velocity fluctuations $u^{\prime}$ ($\sigma=u_{rms}$), (middle) the vorticity fluctuations $\omega^{\prime}$ ($\sigma=\omega_{rms}$) and (right) the local dissipation rate $\varepsilon$ ($\sigma=\overline{\varepsilon}$) for the reconstructed fields at $t=0$ (rows one, three and five) and $t=T_e$ (rows two, four and six): (a) $M=4$, (b) $M=8$, (c) $M=16$. Variables shown are the DNS ground truth (thick dark grey), tri-cubic interpolation (blue), SRdyn (black), 4DVar (orange, line styles as before) and the standard normal distribution (light blue, dashed, shown for $u^{\prime}$ only).