Probabilistic ocean modeling is essential for risk-aware marine operations, climate analysis, and scientific decision-making, where downstream decisions depend on the conditional distribution of future ocean states rather than a single deterministic trajectory. Recent data-driven ocean models have made deterministic global ocean modeling increasingly accurate and efficient, but probabilistic ocean modeling remains underdeveloped. Specifically, perturbation-driven ensembles can create spread without explicitly optimizing the joint distribution of their members, often yielding poor calibration and physical inconsistency. To address these issues, we propose OceanFlow, a coherent flow matching framework for probabilistic global ocean modeling. OceanFlow first establishes a sharp, stable mean anchor for the conditional ensemble. Flow matching then learns a coherent state-conditioned pathwise law that jointly reshapes the residual distribution around this anchor at each transition, preserving the predictive center while refining state-dependent uncertainty. We further introduce a Factorized Dynamics Operator (FDO), which combines factorized transitions with a learned state-conditioned response to parameterize the mean anchor and residual-flow velocity. Extensive experiments show that OceanFlow improves ensemble-mean accuracy, probabilistic reliability, and physical-consistency diagnostics over competitive baselines, supporting coherent flow matching as an effective route toward data-driven probabilistic ocean modeling. Code is available at https://github.com/YuanGao-YG/OceanFlow.
Figures & tables
Figure 1: Learning coherent ensemble laws for ocean modeling. (a) Deterministic modeling produces a single trajectory. (b) OceanFlow represents ocean uncertainty through a mean-anchored ensemble, jointly refining member residuals under a coherent state-conditioned pathwise law at each transition.
Figure 2: Overview of OceanFlow . The mean anchor controls the predictive center, while coherent state-conditioned pathwise flow jointly transports member residuals. Learned residuals refine the reference ensemble, while final recentering preserves the mean anchor. FDO parameterizes both the deterministic reference and residual velocity through a unified composition of factorized dynamics and learned state-conditioned response.
Method
90-Day
120-Day
RMSE ↓
ACC ↑
CRPS ↓
SSR →1
RMSE ↓
ACC ↑
CRPS ↓
SSR →1
SimVP
0.932 ± 0.005
0.088 ± 0.006
—
—
0.932 ± 0.012
0.058 ± 0.005
—
—
SimVP*
0.811 ± 0.006
0.104 ± 0.005
0.485 ± 0.004
0.582 ± 0.012
0.807 ± 0.010
0.073 ± 0.006
0.489 ± 0.007
0.602 ± 0.015
GraphCast
0.921 ± 0.005
0.198 ± 0.003
—
—
1.031 ± 0.004
0.139 ± 0.003
—
—
GraphCast*
0.854 ± 0.003
0.218 ± 0.002
0.525 ± 0.002
0.452 ± 0.005
0.900 ± 0.006
0.165 ± 0.003
0.547 ± 0.003
0.596 ± 0.007
CirT
2.823 ± 0.022
0.016 ± 0.003
—
—
2.902 ± 0.024
0.009 ± 0.004
—
—
Table 1: Quantitative ocean simulation results over 93 variables. Lower RMSE/CRPS, higher ACC, and SSR closer to 1 are better. Values are mean ± std; red/orange/blue mark the best/second-/third-best results. ∗ denotes Perlin-noise ensemble modeling for deterministic baselines.
Figure 3: 120-day simulation results for several important surface variables. Our method achieves better physical consistency and yields results that are closest to the ground truth.
Figure 4: The RMSE, ACC and CRPS results of several important ocean surface variables.
Figure 5: Rank histograms and calibration curves of ocean simulations.
Method
Simulation
Forecasting
90-Day
120-Day
20-Day
30-Day
RankDev ↓
CalibErr ↓
RankDev ↓
CalibErr ↓
RankDev ↓
CalibErr ↓
RankDev ↓
CalibErr ↓
SimVP*
0.396 ± 0.005
0.355 ± 0.003
0.416 ± 0.006
0.366 ± 0.004
0.328 ± 0.002
0.292 ± 0.002
0.338 ± 0.002
0.307 ± 0.002
GraphCast*
0.309 ± 0.002
0.299 ± 0.001
0.254 ± 0.002
0.261 ± 0.002
0.326 ± 0.002
0.294 ± 0.001
0.352 ± 0.002
0.317 ± 0.001
CirT*
0.368 ± 0.010
0.337 ± 0.008
0.381 ± 0.009
0.343 ± 0.009
0.426 ± 0.004
0.372 ± 0.002
0.390 ± 0.003
0.351 ± 0.002
DSLCast*
0.215 ± 0.002
0.230 ± 0.002
0.163 ± 0.003
0.189 ± 0.002
0.288 ± 0.003
0.261 ± 0.003
0.306 ± 0.002
0.281 ± 0.002
Table 2: Quantitative reliability diagnostics for ocean simulation and forecasting. RankDev is computed as the L2 deviation using all 21 rank bins. Lower values are better.
Method
20-Day
30-Day
RMSE ↓
ACC ↑
CRPS ↓
SSR →1
RMSE ↓
ACC ↑
CRPS ↓
SSR →1
SimVP*
0.667 ± 0.006
0.602 ± 0.004
0.408 ± 0.003
0.221 ± 0.002
0.774 ± 0.008
0.465 ± 0.007
0.475 ± 0.004
0.271 ± 0.005
GraphCast*
0.684 ± 0.005
0.523 ± 0.005
0.414 ± 0.003
0.258 ± 0.002
0.749 ± 0.004
0.395 ± 0.005
0.461 ± 0.002
0.269 ± 0.004
CirT*
1.640 ± 0.020
0.096 ± 0.003
0.787 ± 0.012
0.354 ± 0.008
1.984 ± 0.037
0.044 ± 0.003
0.903 ± 0.016
0.434 ± 0.012
DSLCast*
0.624 ± 0.008
0.641 ± 0.008
0.360 ± 0.004
0.324 ± 0.005
0.712 ± 0.008
0.521 ± 0.007
0.416 ± 0.004
0.357 ± 0.006
OceanFlow
0.589 ± 0.007
0.660 ± 0.006
0.311 ± 0.004
0.514 ± 0.006
0.661 ± 0.008
0.550 ± 0.007
0.357 ± 0.003
0.483 ± 0.007
Table 3: Probabilistic ocean forecasting results for 93 ocean variables.
Figure 6: Rank histograms and calibration curves of probabilistic ocean forecasting.
Lead
Model
Global ocean simulation
Niño-3.4 diagnosis
RMSE ↓
ACC ↑
CRPS ↓
RankDev ↓
SSR →1
CalibErr ↓
RMSE ↓
CRPS ↓
90d
DSLCast*
0.784 ± 0.003
0.399 ± 0.004
0.449 ± 0.001
0.216 ± 0.001
0.665 ± 0.004
0.232 ± 0.001
0.6181
0.5007
90d
OceanFlow
0.698 ± 0.003
0.430 ± 0.003
0.383 ± 0.001
0.095 ± 0.001
0.744 ± 0.001
0.034 ± 0.000
0.1451
0.0901
120d
DSLCast*
0.779 ± 0.003
0.377 ± 0.003
0.435 ± 0.001
0.159 ± 0.001
0.790 ± 0.005
0.188 ± 0.001
0.7977
0.6879
120d
OceanFlow
0.686 ± 0.003
0.416 ± 0.003
0.382 ± 0.001
0.102 ± 0.001
0.836 ± 0.005
0.054 ± 0.001
0.2304
0.1496
Table 4: Out-of-sample 2024 simulation and ENSO diagnosis. The best results are shown in bold .
Figure 7: Spectral fidelity of ensemble mean and member-averaged 120-day ocean simulations
Figure 8: Physical diagnostics of the generated ensemble members and ensemble mean.
Models
SimVP
GraphCast
CirT
OceanFlow
Params (M)
78.54
40.82
61.56
37.45
Table 6: Parameter comparison for retrained models.
Variants
RMSE ↓
ACC ↑
CRPS ↓
RankDev ↓
SSR →1
CalibErr ↓
w/o Factorized Dynamics
1.054 ± 0.008
0.188 ± 0.009
—
—
—
—
w/o State Response
0.934 ± 0.062
0.072 ± 0.090
—
—
—
—
w/o Adaptive State Response
0.844 ± 0.007
0.322 ± 0.010
—
—
—
—
FDO
0.825 ± 0.009
0.352 ± 0.007
—
—
—
—
FDO+Perlin
0.715 ± 0.005
0.406 ± 0.003
0.403 ± 0.002
0.204 ± 0.001
0.625 ± 0.004
0.225 ± 0.001
w/o mean correction/re-centering
0.692 ± 0.005
0.420 ± 0.003
0.376 ± 0.002
0.101 ± 0.001
0.690 ± 0.003
0.059 ± 0.001
Table 7: Ablation results of OceanFlow. The best results are shown in bold .
Appendix figures & tables7 assets
Supplementary material from the paper’s appendix.
Appendix
Type
Full name
Abbreviation
Layers
Time
Dt
Spatial Res.
Atmosphere
10 metre u wind component
U10M
1
1993–2020
24h
1∘
Atmosphere
10 metre v wind component
V10M
1
1993–2020
24h
1∘
Atmosphere
2 metre temperature
T2M
1
1993–2020
24h
1∘
Atmosphere
Mean sea level pressure
MSLP
1
1993–2020
24h
1∘
Ocean
Sea salinity
S
23
1993–2020
24h
1∘
Ocean
Sea stream zonal velocity
U
23
1993–2020
24h
1∘
Appendix
Table 8: The data details used in this work.
Figure 9: Simulation RMSE, ACC, and CRPS over 1–120 days at depths of 109.73 m, 222.48 m, and 453.94 m. Lines and shaded bands show the mean and standard deviation across 20 ICs. Lower RMSE and CRPS and higher ACC are better.
Figure 10: 120-day ensemble-mean and member-level simulation results at representative depth levels for the first IC. Individual members use fixed indices 1, 10, and 20 for every model.
Figure 11: 30-day simulation results of different models at several representative depth levels.
Figure 12: 60-day simulation results of different models at several representative depth levels.
Figure 13: 90-day simulation results of different models at several representative depth levels.
Figure 14: 120-day simulation results of different models at several representative depth levels.
Ocean dynamics are inherently chaotic, yet existing machine learning ocean models produce only deterministic forecasts. We introduce Njord, a probabilistic data-driven model for ocean forecasting, applicable to both global and regional domains. Njord combines a deep latent variable framework with a graph neural network architecture, enabling sampling each forecast step in a single forward pass. We apply Njord globally at 0.25° resolution and regionally to the Baltic Sea at 2 km resolution. To scale to these large ocean grids we introduce K-means cluster meshes that adapt to irregular sea surface geometry. Experiments demonstrate strong performance on both domains compared to deterministic machine learning baselines, while also providing uncertainty estimates from the sampled ensemble forecasts. On the global OceanBench benchmark, Njord achieves the lowest errors on average across upper-ocean variables when evaluated against real-world observations, with the largest improvements in surface temperature prediction.
Daniel Holmberg, Joel Oskarsson, Erik Wikingsson +2
University of Helsinki · ETH AI Center · Linköping University
Accurate uncertainty quantification is critical for many time series forecasting applications, and recent studies have increasingly adopted generative models to capture predictive uncertainty. Existing approaches often combine deterministic forecasting with generative modeling to improve predictive performance. However, their joint training is typically unstable because the deterministic model is embedded within the generative process, resulting in deep computational graphs and changing the probability transport path. In this work, we propose DV-Flow, a unified flow-matching formulation that incorporates the deterministic predictor as a parallel velocity component rather than as the endpoint of the transport path. This design enables stable end-to-end optimization while subsuming several existing formulations as special cases, allowing direct comparisons among them. Extensive experiments on six real-world datasets demonstrate that DV-Flow consistently improves probabilistic forecasting performance over strong baselines. Our code is available at https://github.com/tufts-ml/dv-flow-time-series.
Data assimilation estimates a dynamical state from partial and noisy observations. Classical ensemble filters are efficient but restrict analysis updates through finite sample covariance and affine Gaussian distribution. We introduce the Flow Ensemble Filter (FlowEF), which uses conditional flow matching to transport the forecast ensemble from a classical baseline filter to an analysis ensemble. FlowEF uses a localized Gaussian source during training, transports forecast ensemble members from a baseline filter at deployment, and conditions its velocity field on ensembles from that baseline filter and the observation. The proposed model therefore learns a nonlinear update while mapping each baseline ensemble independently. For sparsely observed dynamical systems, FlowEF improves both deterministic and probabilistic metrics over all four classical ensemble filters. It also achieves the best performance among the state-of-the-art generative data assimilation models.
Haoyuan Chen, Alexandre Thiéry
Department of Statistics and Data Science National University of Singapore 6 Science Drive 2, Singapore 117546