DAWIS: Data Assimilation with Windowed Inverse Sampling via Multitask Interpolants

Erik Wikingsson; Martin Andrae; Tomas Landelius; Fredrik Lindsten

stat.MLOct 2, 2026

DAWIS: Data Assimilation with Windowed Inverse Sampling via Multitask Interpolants

Authors: Erik Wikingsson, Martin Andrae, Tomas Landelius, Fredrik Lindsten

Organizations: Linköping University, Sweden · Swedish Meteorological and Hydrological Institute

Abstract

Flow- and diffusion-based generative models have recently emerged as flexible and highly efficient forecasting models for dynamical systems. When combined with inference-time guidance, they offer a promising route to high-dimensional non-Gaussian data assimilation (DA), the problem of combining forecasts with observations to estimate latent system states. Existing filters, however, condition on a fixed history and assimilate only the most recent observation, leaving them unable to revise past states when new observations arrive. Estimates then stay tethered to a history that later observations may contradict, and errors accumulate over the assimilation run. To this end, we introduce DAWIS, a unified DA method covering filtering, fixed-lag smoothing, and block smoothing within a single framework. DAWIS replaces the single flow time of a state-level prior with a multitask stochastic interpolant over a window of consecutive states, assigning a separate flow time to each. An assimilation cycle inverts the window to a vector of per-state turning points and regenerates it under observation guidance, with the turning points controlling how strongly each state is held fixed, revised, or generated from scratch. The same construction can also absorb the forecast into the assimilation cycle, removing the need for a separate forecasting model. Experiments on challenging nonlinear systems show that DAWIS improves on both filtering and smoothing baselines under sparse, noisy, and nonlinear observations. The code for DAWIS is available at https://github.com/Erik-Wikingsson/DAWIS

Figures & tables

Figure 1: The DAWIS assimilation cycle, here visualized for DAWIS-Joint with a prior of $w+1=3$ states. (i) Initialization: The samples $\hat{{\bm{x}}}_{n-w:n-1}$ from the previous cycle start at flow time $1$ and the new state ${\bm{z}}_{\bm{0}}\sim\rho_{\bm{0}}$ . (ii) Inversion: the unconditional SDE moves each slot $k$ to its turning point $\tau_{\min}^{k}$ , partially noising the old states and running the new one forward from noise. (iii) Guidance: the guided SDE regenerates the window from ${\bm{z}}_{{\bm{\tau}}_{\min}}$ , conditioned on observations ${\bm{y}}_{n-w:n}$ .

Method	${\bm{x}}_{\mathcal{W}}$	${\bm{\tau}}_{0}$	${\bm{\tau}}_{\min}$	${\bm{y}}_{\mathcal{W}}$
DAWIS Filter / Lagged Sm.	$[\hat{{\bm{x}}}_{n-w:n-1},{\bm{x}}^{\rm f}_{n}]$	$(1,\dots,1)$	${\bm{\tau}}_{\min}$	${\bm{y}}_{n-w:n}$
DAWIS-Joint	$[\hat{{\bm{x}}}_{n-w:n-1},\ast]$	$(1,\dots,1,0)$	${\bm{\tau}}_{\min}$	${\bm{y}}_{n-w:n}$
DAWIS Block Smoother	$[{\bm{x}}_{n:n+w}]$	$(1,\tau_{\min},\dots,\tau_{\min},1)$	${\bm{\tau}}_{0}$	${\bm{y}}_{\mathcal{B}}$
Block Gibbs Smoother	$[{\bm{x}}_{n:n+w}]$	$(1,0,\dots,0,1)$	${\bm{\tau}}_{0}$	${\bm{y}}_{\mathcal{B}}$
Joint AR $w\|1$	$[\hat{{\bm{x}}}_{n-w},\ast,\dots,\ast]$	$(1,0,\dots,0)$	${\bm{\tau}}_{0}$	${\bm{y}}_{n-w+1:n}$
Joint AR $1\|w$	$[\hat{{\bm{x}}}_{n-w:n-1},\ast]$	$(1,\dots,1,0)$	${\bm{\tau}}_{0}$	${\bm{y}}_{n}$

Table 1: Entries marked

\ast

are irrelevant since

\beta_{0}=0

{\bm{\tau}}_{\min}={\bm{\tau}}_{0}

means the inversion step is empty.

Filtering
Method	Noisy	Sparse	Multimodal	Saturating	SEVIR
EnSF ( Bao et al., 2024 )	$4.95{\scriptstyle\pm 0.83}$	$4.11{\scriptstyle\pm 0.61}$	$1.79{\scriptstyle\pm 1.38}$	$5.79{\scriptstyle\pm 1.72}$	$21.92{\scriptstyle\pm 20.35}$
FlowDAS ( Chen et al., 2025 )	$2.73{\scriptstyle\pm 0.50}$	$3.22{\scriptstyle\pm 0.56}$	$2.50{\scriptstyle\pm 0.48}$	$3.66{\scriptstyle\pm 0.41}$	$15.66{\scriptstyle\pm 7.17}$
LETKF ( Liang et al., 2025 )	$1.32{\scriptstyle\pm 0.14}$	$2.22{\scriptstyle\pm 0.25}$	$4.86{\scriptstyle\pm 1.11}$	$2.28{\scriptstyle\pm 1.06}$	$4.67{\scriptstyle\pm 3.17}$
DAISI ( Andrae et al., 2026 )	$1.30{\scriptstyle\pm 0.14}$	$1.69{\scriptstyle\pm 0.20}$	$1.55{\scriptstyle\pm 0.09}$	$1.79{\scriptstyle\pm 0.45}$	$3.97{\scriptstyle\pm 2.65}$
SDA-Filter ( Andrae et al., 2026 )	$1.32{\scriptstyle\pm 0.13}$	$1.75{\scriptstyle\pm 0.25}$	$2.03{\scriptstyle\pm 0.32}$	$4.26{\scriptstyle\pm 0.70}$	$\underline{3.79}{\scriptstyle\pm 2.25}$

Table 2: The CRPS for experiments on SQG and SEVIR . We display the mean and standard deviation across 10 independent trajectories, averaged over the last 20 (10 for SEVIR ) steps. The best score for each experiment is highlighted in bold and the second best with an underline . Smoothing methods calculate the metrics over the whole trajectory.

Panel of Figure 2 — Figure 2: Ensemble mean and a single member for all SQG experiments. Here visualized for the endpoint for filtering and the midpoint for smoothing. DAWIS exhibits robust performance for all settings, including the challenging multimodal and saturating observations where DAISI and SDA struggle in comparison.

Appendix figures & tables46 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameter	Noisy	Sparse	Multimodal	Saturating	SEVIR
Guide Method	MMPS	MMPS	MMPS	MMPS	MMPS
Solver Steps	100	100	100	100	100
$\bm{\varepsilon}$	0.03	0.03	0.03	0.03	2.5/1.0
$\bm{\tau_{\min}^{\mathrm{start}}}$	0.4	0.4	0.4	0.4	0.4
$\bm{\tau_{\min}^{\mathrm{end}}}$	0.3	0.1	0.2	0.4	0.1
$\bm{\tau_{\min}^{\mathrm{init}}}$	0.0	0.0	0.0	0.0	1.0

Appendix

Table 3: Hyperparameter configurations for DAWIS for all SQG and SEVIR experiments.

Method	Noisy	Sparse	Multimodal	Saturating	SEVIR
Block Gibbs Smoother	0	0	0	0	0
DAWIS Block Smoother	0.9	0.7	0.7	0.9	0.9
DAWIS Soft Block Smoother	0.9	0.7	0.7	0.9	0.9

Appendix

Table 4: Interior turning point

\tau_{\min}

of the block smoothers. The boundary turning point is

1

for the Block Gibbs and DAWIS Block Smoothers and

0.9

for the DAWIS Soft Block Smoother.

Optimizer hyperparameters
Optimiser	AdamW ( Loshchilov and Hutter, 2017a )
Initialization	Xavier Uniform ( Glorot and Bengio, 2010 )
LR decay schedule	Cosine ( Loshchilov and Hutter, 2017b )
Peak LR	1e-3
Minimum LR	1e-5
Weight decay	1e-2

Appendix

Table 5: Optimizer hyperparameters, shared across all datasets and window sizes.

Hyperparameter	Noisy	Sparse	Multimodal	Saturating	SEVIR
Localization scale	$2500\text{\,}\mathrm{km}$	$1500\text{\,}\mathrm{km}$	$1500\text{\,}\mathrm{km}$	$4500\text{\,}\mathrm{km}$	$3\text{\,}\mathrm{p}\mathrm{x}$
Relaxation to prior spread	0.5	0.5	0.5	0.9	0.0

Appendix

Table 6: Hyperparameter configurations for LETKF.

Experiment	Solver Steps	MC Samples	Window	Guidance Strength $\zeta$
Noisy	500	25	6	1
Sparse	500	25	6	3
Multimodal	500	25	6	0.5
Saturating	500	25	6	5
SEVIR	500	25	6	0.1

Appendix

Table 7: Hyperparameter configurations for FlowDAS for all SQG and SEVIR experiments.

Experiment	Solver Steps	$\bm{\epsilon_{\alpha}}$
Noisy	1000	0.05
Sparse	1000	0.05
Multimodal	1000	0.05
Saturating	1000	0.05
SEVIR	1000	0.05

Appendix

Table 8: Hyperparameter configurations for EnSF for all SQG and SEVIR experiments.

Experiment	Guide Method	Solver Steps	$\bm{\varepsilon}$	Corrections	$\bm{\tau}$	Guidance Strength $\zeta$
Noisy	MMPS	100	1	1	0.3	2
Sparse	MMPS	100	1	1	0.3	2
Multimodal	MMPS	100	1	1	0.5	2
Saturating	MMPS	100	1	1	0.3	20
SEVIR	MMPS	100	1	1	0.3	10

Appendix

Table 9: Hyperparameter configurations for SDA Filter for all SQG and SEVIR experiments.

Experiment	Guide Method	Solver Steps	$\bm{\varepsilon}$	$\bm{t_{\min}}$	Guidance Strength $\zeta$
Noisy	DPS	100	0.03	0.4	10
Sparse	MMPS	100	0.03	0.3	1
Multimodal	MMPS	100	0.03	0.3	1
Saturating	MMPS	100	0.03	0.3	20
SEVIR	MMPS	100	0.03	0.3	1

Appendix

Table 10: Hyperparameter configurations for DAISI for all SQG and SEVIR experiments.

	Joint AR 1 $\|$ W		Joint AR W $\|$ 1
Experiment	Solver Steps	$\zeta$	Solver Steps	$\zeta$
Noisy	100	5	100	4
Sparse	100	5	100	7
Multimodal	100	5	25	5
Saturating	100	20	100	150
SEVIR	100	3	100	1

Appendix

Table 11: Hyperparameter configurations for Joint AR for all SQG and SEVIR experiments. Both variants use MMPS and

\varepsilon=1

Experiment	Guide Method	Solver Steps	$\bm{\varepsilon}$	Corrections	$\bm{\tau}$	Guidance Strength $\zeta$
Noisy	MMPS	100	1	1	0.5	2
Sparse	MMPS	100	1	1	0.5	5
Multimodal	MMPS	200	1	3	1.0	3
Saturating	MMPS	100	1	2	0.5	30
SEVIR	MMPS	100	1	1	0.5	3

Appendix

Table 12: Hyperparameter configurations for SDA for all SQG and SEVIR experiments.

Experiment	Data	Obs operator	$\sigma_{\text{obs}}$	Sparsity
Noisy	SQG- $64$	$x$	$5$	$25\%$
Sparse	SQG- $64$	$x$	$1$	$5\%$
Multimodal	SQG- $64$	$(x/7)^{2}$	$1$	$25\%$
Saturating	SQG- $64$	$\arctan(x)$	$0.01$	$25\%$
SEVIR	SEVIR	$x$	$0.001$	$10\%$

Appendix

Table 13: Experiment configurations.

Figure 3: Visualization of the observations for each configuration before sparsity is applied.

Filtering
Method	Noisy	Sparse	Multimodal	Saturating	SEVIR
EnSF Bao et al. (2024)	$7.80{\scriptstyle\pm 1.07}$	$6.95{\scriptstyle\pm 0.82}$	$3.31{\scriptstyle\pm 1.95}$	$8.80{\scriptstyle\pm 2.39}$	$35.12{\scriptstyle\pm 26.68}$
FlowDAS Chen et al. (2025)	$4.87{\scriptstyle\pm 0.79}$	$6.09{\scriptstyle\pm 0.94}$	$4.84{\scriptstyle\pm 0.81}$	$6.77{\scriptstyle\pm 0.61}$	$27.31{\scriptstyle\pm 13.16}$
LETKF Liang et al. (2025)	$2.49{\scriptstyle\pm 0.26}$	$4.31{\scriptstyle\pm 0.47}$	$8.97{\scriptstyle\pm 2.04}$	$4.19{\scriptstyle\pm 1.86}$	$12.19{\scriptstyle\pm 7.04}$
DAISI Andrae et al. (2026)	$2.42{\scriptstyle\pm 0.25}$	$3.31{\scriptstyle\pm 0.40}$	$2.99{\scriptstyle\pm 0.19}$	$3.65{\scriptstyle\pm 0.99}$	$11.02{\scriptstyle\pm 6.10}$
SDA-Filter	$2.36{\scriptstyle\pm 0.22}$	$3.16{\scriptstyle\pm 0.40}$	$3.69{\scriptstyle\pm 0.57}$	$5.98{\scriptstyle\pm 0.77}$	$\underline{10.05}{\scriptstyle\pm 5.00}$

Appendix

Table 14: The RMSE for experiments on SQG and SEVIR . We display the mean and standard deviation across 10 independent trajectories, averaged over the last 20 (10 for SEVIR ) steps. The best score for each experiment is highlighted in bold and the second best with an underline . Smoothing methods calculate the metrics over the whole trajectory.

Filtering
Method	Noisy	Sparse	Multimodal	Saturating	SEVIR
EnSF Bao et al. (2024)	$0.43{\scriptstyle\pm 0.05}$	$0.49{\scriptstyle\pm 0.05}$	$1.24{\scriptstyle\pm 0.33}$	$0.48{\scriptstyle\pm 0.39}$	$0.35{\scriptstyle\pm 0.21}$
FlowDAS Chen et al. (2025)	$0.66{\scriptstyle\pm 0.07}$	$\underline{0.96}{\scriptstyle\pm 0.06}$	$0.87{\scriptstyle\pm 0.11}$	$\mathbf{1.00}{\scriptstyle\pm 0.09}$	$0.53{\scriptstyle\pm 0.23}$
LETKF Liang et al. (2025)	$\mathbf{1.00}{\scriptstyle\pm 0.05}$	$1.05{\scriptstyle\pm 0.07}$	$0.78{\scriptstyle\pm 0.07}$	$0.63{\scriptstyle\pm 0.12}$	$1.16{\scriptstyle\pm 0.40}$
DAISI Andrae et al. (2026)	$0.89{\scriptstyle\pm 0.03}$	$\underline{1.04}{\scriptstyle\pm 0.04}$	$1.04{\scriptstyle\pm 0.19}$	$1.34{\scriptstyle\pm 0.18}$	$\underline{0.92}{\scriptstyle\pm 0.19}$
SDA-Filter	$0.72{\scriptstyle\pm 0.02}$	$0.60{\scriptstyle\pm 0.06}$	$0.64{\scriptstyle\pm 0.06}$	$0.43{\scriptstyle\pm 0.07}$	$1.34{\scriptstyle\pm 0.82}$

Appendix

Table 15: The SSR for experiments on SQG and SEVIR . We display the mean and standard deviation across 10 independent trajectories, averaged over the last 20 (10 for SEVIR ) steps. The best score for each experiment is highlighted in bold and the second best with an underline . Smoothing methods calculate the metrics over the whole trajectory.

Figure 6: The CRPS over the assimilation window for the filtering methods.

Figure 7: The RMSE over the assimilation window for the filtering methods.

Figure 8: The CRPS over the assimilation window for all smoothing methods.

Figure 9: The RMSE over the assimilation window for all smoothing methods.

Figure 10: The improvement in RMSE gained by using the lagged smoother for the DAWIS experiments.

Figure 11: Spread-skill ratio of the filtering methods against its ideal value of one.

Figure 12: Rank histograms of the filtering methods for Noisy .

Figure 13: Rank histograms of the filtering methods for Sparse .

Figure 14: Rank histograms of the filtering methods for Multimodal .

Figure 15: Rank histograms of the filtering methods for Saturating .

Figure 16: Rank histograms of the smoothing methods for Noisy .

Figure 17: Rank histograms of the smoothing methods for Sparse .

Figure 18: Rank histograms of the smoothing methods for Multimodal .

Figure 19: Rank histograms of the smoothing methods for Saturating .

Figure 20: Radially averaged power spectra of the filtering ensemble.

Figure 21: Radially averaged power spectra of the lagged smoothing ensemble.

Figure 22: Log spectral distance over the assimilation window for the filtering methods.

CRPS $\downarrow$
Method	Noisy (ds)	Sparse (ds)	Multimodal (ds)	Saturating (ds)
Filtering
DAWIS Filter	$\mathbf{1.12}{\scriptstyle\pm 0.14}$	$\underline{1.22}{\scriptstyle\pm 0.16}$	$\mathbf{1.13}{\scriptstyle\pm 0.13}$	$\underline{1.01}{\scriptstyle\pm 0.15}$
DAWIS-Joint Filter	$\underline{1.13}{\scriptstyle\pm 0.13}$	$\mathbf{1.19}{\scriptstyle\pm 0.17}$	$\underline{1.18}{\scriptstyle\pm 0.13}$	$\mathbf{0.95}{\scriptstyle\pm 0.16}$
Smoothing
En4DVar	$1.35{\scriptstyle\pm 0.04}$	$1.33{\scriptstyle\pm 0.17}$	$1.16{\scriptstyle\pm 0.04}$	$0.94{\scriptstyle\pm 0.08}$

Appendix

Table 16: The CRPS, RMSE and SSR under forward-model error. The truth is a higher-resolution SQG run downsampled onto the

64\times 64

grid, so the model each method integrates is not the model that generated the data; the observation operator, the noise and the trajectories are those of the corresponding experiment above. We display the mean and standard deviation across 10 independent trajectories, averaged over the last 20 steps for filtering methods and over the whole trajectory for smoothing methods. The best score for each experiment is highlighted in bold and the second best with an underline . For SSR, best is closest to 1.

Method	Noisy	Sparse	Multimodal	Saturating	SEVIR
CRPS $\downarrow$
Block Gibbs Smoother	$0.92{\scriptstyle\pm 0.06}$	$0.99{\scriptstyle\pm 0.07}$	$0.90{\scriptstyle\pm 0.05}$	$0.86{\scriptstyle\pm 0.09}$	$3.58{\scriptstyle\pm 2.35}$
DAWIS Block Smoother	$0.90{\scriptstyle\pm 0.05}$	$0.98{\scriptstyle\pm 0.06}$	$0.89{\scriptstyle\pm 0.05}$	$0.68{\scriptstyle\pm 0.06}$	$3.18{\scriptstyle\pm 2.17}$
DAWIS Soft Block Smoother	$0.91{\scriptstyle\pm 0.05}$	$0.97{\scriptstyle\pm 0.06}$	$0.89{\scriptstyle\pm 0.05}$	$0.65{\scriptstyle\pm 0.06}$	$3.15{\scriptstyle\pm 2.19}$
RMSE $\downarrow$
Block Gibbs Smoother	$1.70{\scriptstyle\pm 0.10}$	$1.92{\scriptstyle\pm 0.12}$	$1.68{\scriptstyle\pm 0.09}$	$1.66{\scriptstyle\pm 0.15}$	$9.27{\scriptstyle\pm 5.63}$

Appendix

Table 17: The CRPS, RMSE and SSR for experiments on SQG and SEVIR . We display the mean and standard deviation across 10 independent trajectories, averaged over the whole trajectory. The best score for each experiment is highlighted in bold and the second best with an underline . For SSR, best is closest to 1.

Figure 23: Filtering states at the end of the trajectory for SQG Noisy .

Figure 24: Filtering states at the end of the trajectory for SQG Sparse .

Figure 25: Filtering states at the end of the trajectory for SQG Multimodal .

Figure 26: Filtering states at the end of the trajectory for SQG Saturating .

Figure 27: Filtering states at the end of the trajectory for SEVIR .

Figure 28: Smoothing states halfway through the trajectory for SQG Noisy .

Figure 29: Smoothing states halfway through the trajectory for SQG Sparse .

Figure 30: Smoothing states halfway through the trajectory for SQG Multimodal .

Figure 31: Smoothing states halfway through the trajectory for SQG Saturating .

Figure 32: Smoothing states halfway through the trajectory for SEVIR .

Figure 33: The evolution of the ensemble mean of the filtering distribution over the assimilation window for SQG Multimodal .

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