Latent-space variational data assimilation in two-dimensional turbulence

TL;DR

This work tackles the challenge of estimating a turbulent flow state from limited measurements by shifting the data assimilation problem from the full state space to a learned latent space. By encoding the vorticity field with an implicit rank-minimising autoencoder (IRMAE) and decoding to the physical state, the authors perform variational assimilation in latent space, linking latent updates to the state-space adjoint via the decoder Jacobian. Across two-dimensional Kolmogorov flow at moderate Reynolds numbers, latent-space DA yields up to ~2 orders of magnitude improvement in relative error over traditional state-space DA and SR-informed initializations, while better preserving small-scale features and enabling longer predictive horizons. The results demonstrate that observability can be substantially enhanced by choosing the right latent coordinates, suggesting a powerful integration of data-driven latent representations with physics-based adjoint-variational methods for turbulence state estimation.

Abstract

Starting from limited measurements of a turbulent flow, data assimilation (DA) attempts to estimate all the spatio-temporal scales of motion. Success is dependent on whether the system is observable from the measurements, or how much of the initial turbulent field is encoded in the available measurements. Adjoint-variational DA minimises the discrepancy between the true and estimated measurements by optimising the initial velocity or vorticity field (the `state space'). Here we propose to instead optimise in a lower-dimensional latent space which is learned by implicit rank minimising autoencoders. Assimilating in latent space, rather than state space, redefines the observability of the measurements and identifies the physically meaningful perturbation directions which matter most for accurate prediction of the flow evolution. When observing coarse-grained measurements of two-dimensional Kolmogorov flow at moderate Reynolds numbers, the proposed latent-space DA approach estimates the full turbulent state with a relative error improvement of two orders of magnitude over the standard state-space DA approach. The small scales of the estimated turbulent field are predicted more faithfully with latent-space DA, greatly reducing erroneous small-scale velocities typically introduced by state-space DA. These findings demonstrate that the observability of the system from available data can be greatly improved when turbulent measurements are assimilated in the right space, or coordinates.

Figures (5)

• Figure 1: Schematic of latent-space data assimilation. The latent representation $\eta$ is mapped to state space $\omega$ by the decoder $\mathcal{F}_{\mathcal{D}}$, where the adjoint field $\omega^\dagger$ is computed. The grey arrow closes the standard, state-space 'loop'. In latent-space assimilation, the latent state is updated by the transformed adjoint field $\eta^\dagger$. Sensor resolution is indicated by the lattice of black dots in the lower left corner of $\omega$.
• Figure 2: Relative error $\varepsilon$ of the estimated turbulent field by (blue) InterpDA, (orange) SR, (green) SR-DA and (red) LatentDA at (left) $Re = 40$ with $M = 16$, (middle) $Re = 100$ with $M = 16$ and (right) $Re = 400$ with $M = 32$ as a function of time normalised by the Lyapunov timescale. Bold lines: ensemble average of ten independent trajectories; Coloured regions: max/min values attained by the ensemble. Grey region marks the data-assimilation time horizon.
• Figure 3: Comparison of (a) reference $\omega^R_0$ at $t=0$ and the estimated field $\omega^*_0$ from (i-iv) InterpDA, SR, SR-DA, LatentDA at $Re = 400$. (b) Contours of the out-of-plane vorticity, (c) enstrophy spectra $\varOmega(k)$ as a function of wavenumber $k$. The black spectra and the black contours in (biv) denote the reference data.
• Figure 4: (a) The spectra $\varOmega_{adj}$ of the (green) adjoint field in state space $\omega^{\dagger}$ and (red) latent adjoint field decoded to state space $\mathcal{F}_{\mathcal{D}}(\eta + \alpha \eta^{\dagger}) - \mathcal{F}_{\mathcal{D}}(\eta)$. Contours of the (bi) state-space adjoint and (bii) decoded latent adjoint fields.
• Figure 5: (a) The relative error $\varepsilon$ of the instantaneous reconstruction of the reference turbulent field $\omega^R_{0,i}$ (lines) and the time-averaged relative error over the DA time horizon (circles) using the first $i$ adjoint POD modes in (green) state and (red) latent space at $Re = 100$. (b) Contours of the out-of-plane vorticity of the reconstructed field with 500 adjoint POD modes in (i) state and (ii) latent space, and (iii) the reference turbulent field.