LEVDA: Latent Ensemble Variational Data Assimilation via Differentiable Dynamics

Phillip Si; Peng Chen

cs.LGFeb 23, 2026

LEVDA: Latent Ensemble Variational Data Assimilation via Differentiable Dynamics

Authors: Phillip Si, Peng Chen

Organizations: School of Computational Sciences and Engineering, Georgia Institute of Technology

Abstract

Long-range geophysical forecasts are fundamentally limited by chaotic dynamics and numerical errors. While data assimilation can mitigate these issues, classical variational smoothers require computationally expensive tangent-linear and adjoint models. Conversely, recent efficient latent filtering methods often enforce weak trajectory-level constraints and assume fixed observation grids. To bridge this gap, we propose Latent Ensemble Variational Data Assimilation (LEVDA), an ensemble-space variational smoother that operates in the low-dimensional latent space of a pretrained differentiable neural dynamics surrogate. By performing four-dimensional ensemble-variational (4DEnVar) optimization within an ensemble subspace, LEVDA jointly assimilates states and unknown parameters without the need for adjoint code or auxiliary observation-to-latent encoders. Leveraging the fully differentiable, continuous-in-time-and-space nature of the surrogate, LEVDA naturally accommodates highly irregular sampling at arbitrary spatiotemporal locations. Across three challenging geophysical benchmarks, LEVDA matches or outperforms state-of-the-art latent filtering baselines under severe observational sparsity while providing more reliable uncertainty quantification. Simultaneously, it achieves substantially improved assimilation accuracy and computational efficiency compared to full-state 4DEnVar.

Figures & tables

Setting	Grid	Simulation steps	Obs. times (interval)	$\Delta t$	Obs. grid
Kolmogorov flow	$150\times 150$	200	40 (5)	0.04	$10\times 10$
Tsunami modeling	$150\times 150$	2000	50 (40)	$\approx 21$ s	$10\times 10$
Atmospheric modeling	$512\times 256$	30240	63 (480)	$\approx 60$ s	$16\times 8$

Table 1: Summary of experimental setups (adapted from Xiao et al. (2026) ). Obs. times represent the total number of observation times while interval in parentheses denotes the spacing (in simulation steps) between consecutive observations. Obs. grid indicates the spatial subsampling resolution.

Panel of Figure 1 — Figure 1: Relative RMSE of the ensemble-mean trajectory over time for Kolmogorov flow (left), tsunami modeling (middle), and atmospheric modeling (right); shaded regions show the standard deviation of member-wise relative RMSE.

Figure 2: Atmospheric benchmark: final-time zonal wind $u$ at 500 hPa. The visualization compares the ground truth (with sparse observations) to the assimilated states and errors for different methods.

Panel of Figure 3 — Figure 3: Effect of irregular sampling on tsunami modeling: (a) moving observation locations (randomized at each observed time), (b) irregular observation times with fixed spatial locations, and (c) joint spatiotemporal irregularity. Curves depict the relative RMSE of the ensemble-mean trajectory; shaded regions indicate the standard deviation of member-wise relative RMSE.

Example	Kolmogorov Flow				Tsunami Modeling				Atmospheric Modeling
Metric	4DEnVar	Latent-EnSF	LD-EnSF	LEVDA	4DEnVar	Latent-EnSF	LD-EnSF	LEVDA	4DEnVar	Latent-EnSF	LD-EnSF	LEVDA
Encoder Training (s)	$0$	$43,156$	$298$	$0$	$0$	$27,718$	$312$	$0$	$0$	$58,231$	$1429$	$0$
Assimilation (s)	$-$	$7.57$	$0.35$	$13.91$	$24,639$	$5.77$	$0.37$	$11.32$	$-$	$37.5$	$1.42$	$19.47$
Assimilation dim.	$D_{x}{=}45{,}000$	$D_{s}{=}499$	$D_{s}{=}10$	$D_{s}{=}10$	$D_{x}{=}67{,}500$	$D_{s}{=}400$	$D_{s}{=}12$	$D_{s}{=}12$	$D_{x}{=}393{,}216$	$D_{s}{=}512$	$D_{s}{=}52$	$D_{s}{=}52$

Table 2: Wall-clock runtime comparison. We report the latent assimilation dimension

D_{s}

for latent methods (Latent-EnSF, LD-EnSF, LEVDA) and the full-state dimension

D_{x}

for 4DEnVar. Assimilation runtimes correspond to full-trajectory assimilation with an ensemble size of

K=100

for latent methods, whereas 4DEnVar is restricted to

K=40

due to the high computational cost of full-state propagation. 4DEnVar simulations use an AMD 7543 CPU (64 threads), while data-driven and latent methods utilize an Nvidia A6000 GPU.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Tsunami benchmark sensitivity analysis: LEVDA Relative RMSE as a function of (a) smoothing time horizon $\tau$ , (b) ensemble size $K$ , and (c) spatial observation stride (sparsity).

Panel of Figure 5 — Figure 5: Additional Tsunami benchmark ablations: (a) Impact of temporal sparsity on LEVDA Relative RMSE (using extended smoothing window $\tau=10$ ). (b) Robustness to observation noise (5%, 10%, 20% noise-to-signal ratios) under fixed likelihood weighting. (c) Parameter estimation accuracy: Relative RMSE of the parameter $u$ over the assimilation window.

Panel of Figure 6 — Figure 6: Spread-Error Ratio (SER) over time for Kolmogorov flow, tsunami modeling, and atmospheric modeling (left-to-right). Values closer to $1$ indicate better calibration of the ensemble spread.

Panel of Figure 7 — Figure 7: Continuous Ranked Probability Score (CRPS) over time for Kolmogorov flow, tsunami modeling, and atmospheric modeling (left-to-right). Lower values represent higher probabilistic accuracy.

Capability	Latent-EnSF	LD-EnSF	SDA	FlowDAS	APPA	LEVDA
Smoothing	X	X	✓	✓	✓	✓
Differentiable Dynamics	X	✓	X	X	X	✓
Moving observations	X	X	✓	✓	✓	✓
Irregular-time observations	X	X	X	X	X	✓
Parameter estimation	X	✓	X	X	X	✓

Appendix

Table 3: Feature comparison across latent and score-based DA methods. Here, “Differentiable Dynamics” refers to the use of an explicit learned surrogate for state evolution (as opposed to implicit trajectory priors), while “Irregular-time Observations” denotes native support for data at arbitrary time stamps without resampling to a fixed temporal grid.

Figure 8: Visualization of surface elevation $\eta$ in tsunami dynamics. The rows display: (1) ground truth; (2) LDNet predictions from a known initial condition; (3) prediction errors; and (4) sparse observations ( $10\times 10$ subsampled from a $150\times 150$ grid). The remaining rows show assimilation results starting from a deviated initial condition: (5) a single assimilated trajectory from the LD-EnSF ensemble ( $N=20$ ); (6) LD-EnSF error; (7) a single assimilated trajectory from the LEVDA ensemble ( $N=20$ ); and (8) LEVDA error.

Figure 9: Kolmogorov flow benchmark: vorticity field, comparing ground truth and assimilated reconstruction. The rows display: (1) ground truth; (2) perturbed trajectory with different Reynolds number; (3) error of perturbed trajectory; (4) sparse observations; ( $10\times 10$ subsampled from a $150\times 150$ grid (5) assimilated results of a single trajectory from the LD-EnSF ensemble ( $N=20$ ); (6) LD-EnSF error; (7) a single assimilated trajectory from the LEVDA ensemble ( $N=20)$ ; (8) LEVDA error.

Data Assimilation

Data assimilation combines observations with dynamical models to estimate hidden system states and their uncertainty over time.

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LD-EnFF: Latent-Dynamics Ensemble Flow Filtering for Data Assimilation with Sparse Observations

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School of Mathematics, Georgia Institute of Technology · School of Computational Science and Engineering, Georgia Institute of Technology

Data Assimilation Bayesian Filtering Latent Dynamics Modeling

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Data assimilation combines model forecasts with noisy, incomplete observations to estimate the evolving state of a dynamical system. Existing methods face two compounding challenges: high-dimensional nonlinear dynamics make repeated forward simulation computationally expensive, while sparse observations provide limited direct information about the full state. To address these challenges, we propose the Latent-Dynamics Ensemble Flow Filter (LD-EnFF), a sequential Bayesian filtering framework that performs both forecast propagation and filtering updates in a compact latent space. LD-EnFF combines a latent dynamics surrogate for ensemble propagation with a variational autoencoder (VAE)-based observation model that evaluates a state-dependent observation likelihood in latent space. At each assimilation step, an ensemble filtering update based on flow matching uses the forecast ensemble and this likelihood to generate posterior samples, jointly updating latent states and uncertain parameters. This design avoids repeated full-state simulation during forecasting and full-field reconstruction during likelihood evaluation. LD-EnFF substantially outperforms a broad range of data assimilation algorithms on benchmarks spanning Kolmogorov flow, tsunami propagation, and atmospheric modeling, all featuring complex dynamics and sparse, noisy observations.

Abstract

Data Assimilation

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