FlowDAS: A Stochastic Interpolant-based Framework for Data Assimilation
Authors: Siyi Chen
Yixuan Jia
Qing Qu
He Sun
Jeffrey A Fessler
Venue: NeurIPS 2025 poster ()
Keywords: Data Assimilation
Stochastic Dynamic System
Generative Models
Stochastic Interpolants
Inverse Problem
Primary area: machine_learning_for_sciences
TL;DR: We introduce a Stochastic Interpolants-based data assimilation framework for stochastic dynamic systems.

Abstract: Data assimilation (DA) integrates observations with a dynamical model to estimate states of PDE-governed systems. Model-driven methods (e.g., Kalman Filter, Particle Filter) presuppose full knowledge of the true dynamics, which is not always satisfied in practice, while purely data-driven solvers learn a deterministic mapping between observations and states and therefore miss the intrinsic stochasticity of real processes. Recently, score-based diffusion models have shown promise for DA by learning a global diffusion prior to represent stochastic dynamics. However, their one-shot generation lacks stepwise physical consistency and struggles with complex stochastic processes. To address these issues, we propose FlowDAS, a generative DA framework that employs stochastic interpolants to learn state transition dynamics through step-by-step stochastic updates. By incorporating observations into each transition, FlowDAS can produce stable, measurement-consistent forecasts. Experiments on Lorenz-63, Navier–Stokes super-resolution/sparse-observation scenarios, and large-scale weather forecasting—where dynamics are partly or wholly unknown—show that FlowDAS surpasses model-driven methods, neural operators, and score-based baselines in accuracy and physical plausibility. Our implementation is available at https://github.com/umjiayx/FlowDAS.

FlowDAS: A Stochastic Interpolant-based Framework for Data Assimilation
Siyi Chen" University of Michigan slyiche@umich edu
Yixuan Jia" University of Michigan jiayxOumi ch, edu
Qing Qu
He Sun
Jeffrey A. Fessler University of Michigan fesslerGumich cetttmmichheee edu
University of Michigan qingqu@umich. ch. edu
Peking University
hesunōpku. edu.cn
Abstract
Data assimilation (DA) integrates observations with a dynamical model to estimate states of PDE-governed systems. Model-driven methods (e.g., Kalman Filter, Parti- cle Filter) presuppose full knowledge of the true dynamics, which is not always satisfied in practice, while purely data-driven solvers learn a deterministic mapping between observations and states and therefore miss the intrinsic stochasticity of real processes. Recently, score-based diffusion models have shown promise for DA by learning global diffusion prior represent stochastic dynamics. However, their one-shot generation lacks stepwise physical consistency and struggles with complex stochastic processes. To address these issues, we propose FlowDAS, a generative DA framework that employs stochastic interpolants to learn state transition dynam- ics through step-by-step stochastic updates. By incorporating observations into each transition, FlowDAS can produce stable, measuremeent-conssstern forecasts. Experiments Lorenz-63, Ravier-Stokes supperresslutto//ppreeeoereeaaa scenarios, and large-scale weather parecasting-wleree dynamics are partly or wholly unknown-show that FlowDAS surpasses model-driven methods, neural operators, and score-based baselines in accuracy and physical plausibility. Our implementation is available https: //te::///tttttt /gtb. htt////jayxx/FOoDAA..
1 Introduction
Recovering state variables in complex dynamical systems is fundamental problem in science and engineering. Accurate state estimation from noisy, incomplete data is critical in weather forecasting [1, oceanography [3, 4), seismology [5, 6), and many other fields [7], where reliable predictions depend on understanding the underlying physics [8-10]. representative example fluid dynamics [11-13], where one aims to reconstruct a continuous velocity field from sparse, noisy observations governed by nonlinear, time-dependent partial differential equations (PDEs)-a task complicated by stochasticity and high dimensionality. To meet these challenges, data assimilation (DA) combines model forecasts with observations to produce physically consistent state estimates; developed first in atmospheric and oceanic forecasting, DA now ubiquitous across many scientific and engineering domains [14-20] 1402
Mathematically, discrete-time stochastic dynamical system be described by:
#k++1 ((xx) + Ex.
y A(xx+1) M++1;
'Equal contribution. Work done while Siyi Chen undergraduate student Peking University. 'Corresponding author. This author is affiliated with the National Biomedical Imaging Center, Peking University, Beijing and Academy for Advanced Interdisciplinary Studies, Peking University. Beijing.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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State Trajectory
M,
SDE
SDE
M2
M
Aud-11
Xx-1
xx
y A(x)
M2
FoL-11
xx-1
Noisy Observations
M;
Rg4+
yk-1
Yx
SDE dynamic
7x, by A(
Figure I: An overview of FlowDAS. We introduce stochastic interpolant-bas framework for data assimilation, named FlowDAS, to estimate states LL:K from the noisy (sparse or low-resolution) observations YL.K- FlowDAS models the stochastic dynamics of the system with flow-bat flow-based based stochastic differential equation (SDE) and incorporates the observations to improve the prediction accuracy. On the right, show conceptual illustration of the geometry of the process estimate # from Øg. .M, denotes the generative manifold at interpolation step s. The gradient guidance V n y A( enforces observation consistency during the generation process.
where Ig E TR. D is the state vector at time step k, (.) is the state transition map, and & E RDD denotes the stochastic force. The observations yı RM are related the state through the measurement map A(-), with observation noise Vh ~ ~ In many practical settings we have an initial window of known states #0.1-1+ after which observations yr.k will only be noisy and partial. The filtering problem in DA seeks to infer the posterior of the state trajectory 8 L.KK given observations YL:K and the initial states L0:L-1, i.e., p(æL.K yL:K Lo:L-1), as shown in Figure 1. Moreover, when the observations are absent, the task reduces to probabilistic forraallstticcrre forecasting [21], where we predict p( L(L L:K Fo.L--1.
Existing DA methods are split into two categories. Model-driven methods-Kalman variants for quasi-innearGGausiia systems [22] and particle filters for fully nonlinear cases [23] accurate physical model and become costly or unstable in high dimensions [24]. Data-driven surrogates learn unknown partially known dynamics directly from observations and embed them within Bayesian inversion frameworks. However, the intrinsic stochasticity of complex systems exposes limitations: neural operators such FNO [25] and Transolver [26] produce only deterministic forecasts and thus cannot quantify uncertainty, while diffusion models [21. 27, 28] struggle to learn physically faithful distributions due to the long Markov chain required to map Gaussian noise into realistic data.
Our contributions. We address the above limitations by building on recent flow-based stochastic interpolant methods [29-31], which learn short-step conditional transition p(æ (xx) instead of a global noise-to-date map. Our main contributions are:
We introduce generative data-assimilation framework that treats a learned stochastic interpolant as the forward surrogate model and assimilates new observations on the fly inference time, without retraining auxiliary filtering steps.
FlowDAS:
Efficient, interpretable learning: By modeling the transition between adjacent states, FlowDAS diffusion bridges, rolls out autoregressively, and
trains faster and more stably than provides clear physical insight into one-step dynamics.
Extensive validation: Across the Lorenz-63 system, incompressible Navier-Stokes super- resolution and sparse-observation tasks, and large-scale SEVIR weather forecasting-wee -here dy- namies range from known partially fully Sawhoowww-FoowDSSSSS consistently outperforms strong baselines in both accuracy and physical plausibility.

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2 Preliminaries
2.1 Related Work
Data assimilation alternates between a forecasting step propagatin the current state forward under dynamical model-and an inverse step that corrects this forecast with new observations.
Model-driven methods Traditional model-driven approaches include the Kalman filter and its many variants [32]. The Kalman filter provides an optimal Hinnimum-variange estimate when the dynamics are linear and all errors are Gaussian, but its performance degrades once these assumptions are not met. Also, there are some variants for nonlinear settings like the Extended Kalman Filter [33, 34], applying the nonlinear transition to predict states and its Jacobian to the covariance, struggling in discontinuity situations and causing computational expense to calculate the Jacobian. The Unscented Kalman Filters [35] bypass the computation of Jacobian by propagating a set of sample points with the nonlinear transformations and deriving the mean and covariance, making it sensitive to sample point parameters and non- -on-Gaassi problems. For fully nonlinear, non- non-Gaussian problems, the bootstrap particle filter (BPF) [23] approximates the posterior with set of weighted particles that are advanced by the dynamics, reweighted by the observation likelihood, and resampled to prevent weight collapse. Although flexible, the BPF requires exponential number of particles the state dimension grows, making it impractical for large physical systems [24]. Besides, both the Kalman family and particle filters rely on having accurate knowledge of the underlying dynamical models, a requirement that is often unmet in real-world applications.
Neural operator-based methods Neural-operator solvers treat forecasting as a deterministic map from the current state to a single future state. The Fourier Neural Operator (FNO; [25]) learns this map in Fourier space, giving fast and accurate predictions, while Transolver ([26)) replaces Fourier layers with a physics-attention Transformer that can effectively capture complex physical correlations. Both models are deterministic and they are not designed for dynamics that are inherently stochastic. Recent work has sought to introduce stochasticity by fitting probabilistic surrogates- as stochastic processes, graphical models, or stochastic Koopman operators [36, 37]- but these efforts typically focus on low-d -iw-immenioonl settings or assume simple Gaussian statistics, limiting their applicability to high-dimensional, strongly nonlinear systems. Researchers have also adapted neural operator-basee models for DA tasks [38-40]. However, these adaptations are typically tailored to narrow use cases-such as arbitrary-resolution data assimilation and have not demonstrated robust generalization across diverse real-world scenarios.
Generative model-based methods Recent works have leveraged score-based methods, in par- ticular diffusion models. for dynamical system modeling. For instance, the Conditional Diffusion Schrödinger Bridge (CDSB) framework extends diffusion models to conditional simulation and demonstrates strong performance on filtering in state-space models [41]. bosse- data assimila- tion (SDA) is a recent data-driven approach that employs score-based diffusion models to estimate state trajectories in dynamical systems [28]. It bypasses explicit physical modeling by learning the joint distribution of short state trajectory segments (c.g., 2k + 1 time steps) via a score network 50( By integrating the score network with the observation model in a diffusion posterior sampling (DPS) framework [42], SDA can generate entire state trajectories in zero-shot and non- autoregressive manners. However, SDA captures state transition implicitly from state concatenation, the learned dynamics of SDA may lack physical interpretability. leading to potential inaccuracies. And its posterior approximations may be less reliable in systems where the physical model is highly sensitive, as illustrated by the double well potential problem in Appendix D.I. Additionally, non- autoregressive diffusion models may struggle with long sequence forecasting in high-dimensional systems, though autoregressive diffusion approaches, like GenCast [43], have shown promising results in such settings. PDEDiff extends SDA by introducing a universal amortized conditional diffusion framework for PDE dynamics, where a conditional score network is trained to predict local state transitions given variable-length histories, enabling both forecasting and data assimilation across different PDE systems [21]. Another closely related line of work is the stochastic interpolant based forecasting framework recently proposed in [31], which formulates the prediction task as conditional sampling of future system states given the current state, using a fictitious, non-physical stochastic process governed by a learned SDE. While this approach enables probabilistic forecasting, it does not incorporate observation- stochastic interpolant dynamics with a seeasurementcconsiiee correction term at each step, enabling integration of observational data and thus unifying generative modeling with data assimilation.
corrections during inference. In contrast, FlowDAS augments the
3

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2.2 Stochastic Interpolants
Stochastic interpolants [29-31] is generative modeling framework that unifies flow-based and diffusion-basee models, providing a smooth, controlled transition between arbitrary probability densities over finite time horizon.
Consider a stochastic process X, defined over the interval s [0, 1], which evolves from an initial state Xo ~ π (Xo) to the target state X1 ~ g(X)). A stochastic interpolant can be described as:
İ, = a, + 3, W.
where (Xo; X1) p( W, is Wiener process for E [0, 10 introduced after Xo and 11 are sampled, ensuring that W. is independent of Xo and X₁. The time-varying coefficients Os: S2, 0, E C* (1. 1J) satisfy boundary conditions og 31 1 and 01 30 on 0, ensuring that To Xo and I Xi, thereby creating smooth interpolation from Xo X₁, Moreover, for all (s, Xo) E [0, 1] R.P. I, Xn has the same distribution as Xs, which is the solution the
following SDE [31]:
dX, (X,,, Xo) ds + o, dW.. where the drift term b,(X. Xo) optimized by minimizing the cost function:
(b.) / [b. Xo) R2ll3] ds.
(5)
The "velocity" of the interpolant path, R2 given by R. - a, X, + B,x, + W., Furthermore, the drift term (X. Xo) is related to the score function V log p(X, Xo):
C. (X,,, Xo)
V Voog((X, Xo)
bg X Xo)
(6)
B,
A.Br
where X2 and c, cx, Xo) are defined to control the score-based dynamics (Appendix A.2).
Building on this framework, stochastic interpolants can estimate the transition p(**+1 3xx by evolving latent path X, smoothly from O (current state) to s = 1 (next state) in dynamical systems. provides a probabilistic yet compact surrogate that is more aligned with the true step-to- step evolution of complex dynamical systems. While stochastic interpolants capture complex system dynamics, they do not yet enforce consistency with measurements.
3 Method
3.1
Stochastic Interpolants for Data Assimilation
Using stochastic interpolants as the engine of data assimilation is attractive but plug-and-play replacement for existing forward models: it raises three fundamental challenges that FlowDAS must overcome. Below we formulate each challenge and describe the corresponding FlowDAS solution.
t
Challenge I: Observation-connistern state generation Stochastic interpolants approximate the state transition p(æ++1 æx3)) by interpolating between the state variables æy and # using the SDE defined in Equation (4) and Equation (6) with boundary conditions Xo, X Eg, 2x+ Moreover, DA forward surrogate must respect the observation 9/++11 while drawing the future state 4k+1. 1. For stochastic interpolants, this requires dbserratioo-ccoddiionee SDE. FlowDAS augments 5, X₂) via Bayes' rule (Appendix A.1):
the original drift b, (X,,
b. Xx, y. Xo) = (X. + logg(y X,, X,)
A₂3,
drift The term V log X,, Xo) captures the observation information, however, it intractable because the observation model only directly links y ys++1 and X, #++1. FlowDAS produces unbiased estimate of it by Monte-Carlo marginalization, i.e., integrating with respect to X1, which can be approximated by J Monte Carlo
Challenge II: Estimating the observation-infone
samples x p(X1 X Xo):
V Vlog(y X,, Xo) VEXIsp(X XX (p X
wi logply x(").
(8)
X1)
j=1
where we apply a softmax function to the J scalars (logr(y) x
د to compute the sample
weights wy = p(v x(())/ Σj=1 p(y x x(1)). We leave detailed derivation in Appendix A.3.

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Challenge III: Efficient Monte Carlo sampling Accurate sampling from p(X; x: Xo) re- quires solving the SDE in Equation (4), which can be computationally intensive. FlowDAS accelerates this step with low-order stochastic integrators:
First-order Milstein method [44, p. 317]:
x = X, + (X,, Xo)(1 -)) d,W,
(9)
and second-order stochastic Runge-
Runge-Kutta Kutta method [44, p. 324]:
X X, (x 8. Xo) + bj (X1, Xo) (1 - 8) d.W.
(10)
2
Both approximations introduce slight numerical bias (i.e., ((11 - ()) and (((1-))())))))) )))), respectively) but significantly accelerated sampling speed. Appendix C.2 further compares these approximations.
Advantages of FlowDAS. Compared with other data-driven surrogates, FlowDAS offers two key
advantages.
Dnsservation-coniise probabilistic forecasts. Neural operators such FNO and Transolver learn a deterministic map +y + +k+i; they yield a single forecast and require a separate optimisa- tion step to reconcile that forecast with the observation Vs++1 FlowDAS learns the full conditional distribution p(z*+1 Øg, y1++1). ). It therefore generates an ensemble of forecasts that are already consistent with the incoming measurement, eliminating the need for any post-hoe update.
Local and physics-aligned transport. FlowDAS directly learns a local bridge between adjacent states, so the diffusion process spans a short distance in state space. Training therefore remains numerically stable and provides clear physical insight into one-step dynamies. By contrast, score- based diffusion models construct global path from Gaussian noise to data; the network must master a far longer transformation and learn the state transition dynamics through channel correlations, which often obscures physical interpretability and can degrade accuracy.
3.2 Implementation Details
Training We train user-defined neural network as the drift model b, Xu), which outputs the velocity given the current interpolant X, and conditioned on the initial state Xo. The network architecture of the drift model may vary across tasks, for example, a Multi-Layer Perceptron (MLP) adopted for the low-dimensional Lorenz-63 system in Section 4. a U-Net for high-dimensional fluid system governed by incompressible Navier-Stokes equations in Section 4.2, and a FNO for the weather forecasting task in Section 4.3. The model is optimized using collected trajectories To:K. minimizing an empirical loss between predicted and target velocities over sampled interpolation times s E [0, 1]. The discrepancy metric (c.g., E or squared Eg norm, etc.) is also task-dependent.
Inference Given trained drift model b.(X... s, Xo). FlowDAS performs inference in an autoregres- sive manner to estimate the state trajectory -1-K- each time step k; we begin from the known state Zt and predict the next state Lx+1 using a discretized stochastic interpolant over a time grid 80 = 0 < 81 < .... < NN = 1. We use N = 500 in our experiments. We set the interpolant state X.. f- Tt and retrieve the next observation y+++1- For each interpolation step 8n, we simulate forward transition using the learned drift and noise. To enforce observation consistency, we generate
J posterior endpoint samples (j) and compute their likelihoods under the measurement model, i.e.,
By A(X ))) A softmax over these values gives importance weights (wi:)). We then apply a correction to the interpolant using weighted gradient -G, V x,, Σj= wy wylly A(X A(X(")))1 ) where Sn denotes the step size. After reaching s = 1, we set Ø +- Xsy and proceed to the next time step. Repeating this procedure over all K yields the full trajectory.
Conditioning While the original stochastic interpolants framework which only models the p(# xx), we take in a sequence previous states to achieve the transition from p(æx xz=-1...k--r to p(2 æt; - 1....x-i)) inspired by [45-48], thus achieving probabilistic prediction conditioned on several previous states. This straightforward extension allows FlowDAS to handle non-Markovian dynamies and empirically leads to markedly improved performance weather-forecasting task (Section 4.3).
5

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Experiments and Results
This section evaluates our proposed framework, FlowDAS, on a range of low- and dimessonnn stochastic dynamical systems, including low-dimensional problems with high-order observation models, such as the double- well potential (Appendix D.1.5) and the chaotic Lorenz 1963 system, as well as nigh-dimensional tasks involving the incompressible Navier-Stokes equations and realistic problem: Particle Image Velocimetry (PIV). Additionally, we demonstrate its applicability on real-world large-scus scale weather forecasting task, where governing dynamics are not available. These results underscore the versatility and robustness of FlowDAS.
4.1 Lorenz 1963
In this experiment, we evaluate the performance of FlowDAS using the Lorenz 1963 system, simplied mathematical model for atmospheric convection that is widely studied in the DA community [28, 49, 50]. The state vector of the Lorenz system, a: = (a, b, c) E TR$,, evolves according to the following nonlinear stochastic ordinary differential equations (ODEs):
da/ dt = μ(b a) + E1,
db/ dt alp c) b E2,
(11)
de/ dt = ab - TC + Es
where jr 10, p 28, and 7 8/33 define the ODE parameters. and & (E1, E3) R is the Brownian process noise, with each component having a standard deviation a = 0.025. This chaotic system poses a significant challenge for numerical methods, so we use the fourth-order Runge-Kutta (RK4) method [51] to simulate its state transition (see Appendix D.2 for details).
We observe only the aretangent- transforrrr value of the first state component a, the observation model of the system is defined as
A(x) + 17 arctan(a) + 17,
(12)
wehre 7) is the observation noise with standard deviation y 0.25.
Dataset and experiments We generate 1,024 independent trajectories, each containing 1,024 states, and split the data into training (80%), validation (10%), and evaluation (10%) sets. Initial states sampled from the statistically stationary regime of the Lorenz system, with additional data generation details provided in Appendix D.2. During inference. we independently estimate 64 trajectories over 15 time steps using FlowDAS and the bascline methods. A total of L = previous state is conditioned for each autoregressive generation. For this low-dimensional problem, we use a fully connected neural network to approximate the drift term in stochastic interpolants; the network architecture is also described in the same appendix.
Baselines and metrics We compare our method against two baselines: the SDA solver with fixed window size of and the classic BPF [23]. Appendix D.2 details the score network architecture for SDA and particle density settings for BPF.
We evaluate the performance of FlowDAS and baselines using four metrics: the expec- tation of log-prior Egg Wi.k) [log æı)]; the expectation of log data likelihood Ey(ari i yr:k) [logp(y) (t:k]]; ; the Wasserstein distance [52] (..-)) between the true trajectory Fi:K and the estimated trajectory #i:k; and the RMSE between the true and estimated states.
FLOWDAS
SDA
BPF
log p(02:K (1) log p(y (i:k) Wi (ZLK (.k)
17.29
-332.7
17.88
-0.228
-6.112
-1.572
0.106
0.528
0.812
RMSE( #1.K)
0.202
1.114
0.270
Table 1: Data assimilation of Lorenz 1963 system. This table summarizes the performance of FlowDAS, SDA, and BPF on the Lorenz 1963 experiment over 15 time steps. FlowDAS outperforms SDA across all evaluation metrics and is competitive with BPF, despite BPF utilizing the true transition equations, which are unknown FlowDAS and SDA. The best results for each metric are highlighted in bold.

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state
Tue siate
state
FlowAS
SDA
10
35
-10
OA Of 14
07 of 18
33
0.0
Figure 2: Data assimilation of Lorenz 1963 system. FlowDAS achieved results comparable to the sfate-of-the-ar model-based BPF method, significantly outperforming the data-driven SDA method in recovering the underlying dynamics of this chaotic system. This highlights the efficiency and robustness of FlowDAS in capturing complex, nonlinear dynamics while maintaining accuracy and stability. The variables 21, 22 and I3 correspond to a, b and in the ODEs of the Lorenz system Equation (11), respectively.
Results As shown in Table I and Figure FlowDAS outperforms SDA across all metrics. FlowDAS is only slightly less effective than BPF in the expected log-prior, as BPF directly incorporates the true system dynamics into its state estimation. Appendix D.2 provides additional comparisons and results.
The success of FlowDAS is primarily due to its accurate mapping from current to future states (ar - 10++)) Despite lacking explicit transition equations, FlowDAS effectively captures the system dynamics through stochastic interpolants, enabling closer approximation of state trajectories compared SIDA, which models joint distributions across sequential states using diffusion models. Additionally, stochastic interpolants allow FlowDAS to produce accurate state estimates while managing inherent variability, avoiding over-concentration on high-probability regions, and effectively dealing with rare events. This advantage is further illustrated in the double well potential experiment (Appendix D.1.5) where FlowDAS outperforms BPF, because BPF tends be trapped by high- probability point estimates.
4.2 Incompressible Navier-Stokes Flow
This section considers a nigh-dimensional dynamical system: incompressible fluid flow governed by the 2D Navier-Stokes (NS) equations with random forcing on the torus T =====p [0, 2x The state transition, Vr is described using the stream function formulation,
dw + U Vw dt vAw dt ow dt + dgg
(13)
where w represents the vorticity field, the state variable in this fluid dynamics system (æ = W). The velocity v -ーッ = (-0, v) expressed in terms of the stream function v0(x, y). which satisfies - Aú = The term de represents white-in-time random forcing acting few Fourier modes, with parameters v, a > 0 specified in Appendix D.
The observation operator A linearly downsamples or selects partial pixels from the simulated vorticity fields (w),
y A(ww) + n)
(14)
where the observation noise 17 has a standard deviation of y = 0.05.
Dataset and experiments In this experiment, system dynamics are simulated by solving Equa- tion (13) using a pseudo-spectral method [53] with a resolution of 256 and a timestep At 10 We simulate 200 flow conditions over t E [0, 100), saving snapshots of fluid vorticity field (w V at the second half of each trajectory (t E [50, 100)) at intervals At = 0.5 with reduced resolution of 128? The data are divided into training (80%), validation (10%), and evaluation (10%) sets.
conduct experiments across different observation resolutions (322, 162) and observation sparsity levels (5%, 1. 1.5625%). For the super-resolution task, the goal is to reconstruct high-resolution vorticity data (1282) from low-resolution observations. the inpainting task, only 5% or 1. 1.5625% of pixel values retained, with the rest set to zero, and attempt recover the complete vorticity field.

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True State
Observation
FlowDAS
FNO-DA
Transolver-
PDEDiff
s
Navier-Stokes Flow: Sparse-observation Task (1.5625%)
Figure 3: Data assimilation of incompressible Navier-Stokes flow. The positive values (red) of the state, i.e., vorticity field, indicate clockwise rotation and negative values (blue) indicate counter- clockwise rotation. FlowDAS achieved results with more accurate details and higher accuracy than all baselines, showing the efficiency of FlowDAS in tackling DA tasks with highly non-linear complex systems. Additionally, FlowDAS is also better at recovering high-frequency information, evidenced by the spectral analysis in Figure S.8.
32 > 128
162 + 1282
5%
1%
FLOWDAS
0.077 + 0.005
0.174 + 0.006
0.156 + 0.005
0.373 + 0.012
PDEDIFF
0.112 : 0.017
0.242
0.035
0.221
+ 0.024
0.575 = 0.068
TRANSOLVER-DA
0.597 + 0.001
0.657 = 0.001
0.660 + 0.001
0.669 = 0.001
FNO-DA
0.653 0.001 0.732 0.001 0.695 = 0.001 0.753 = 0.001
Table 2: RMSE (+ std) of FlowDAS and baselines on incompressible Navier-Stokes super- resolution and sparse-observation tasks. Each value is computed from 50 independently sampled states, with 30 repeated runs per state. All results correspond to single-step step assimilation. Figure S.12 presents the RMSE and standard deviations evaluated along ClowDAS-generatee trajectory.
The model is evaluated four unseen datasets (2x super-resolution, 2x inpainting), with 64 samples for each configuration. A total of = 10 previous states are conditioned for each autoregressive generation.
Baselines and metries We benchmark FlowDAS against three data-driven baselines. First, implement PDEDiff solver with fixed eleven-step window (10 previous states as conditions). Second, we adapt the deterministic neural operators FNO and Transolver the DA setting, namely 'FNO-DA' and "Transolver-D DA'. Each network is trained to minimise RMSE between adjacent states; inference time the raw prediction a'4+1 fx(x) reconciled with the observation yk+1 by solving Øk+1 = arg min, a - x'++1ll 11 + 02 y - A(x)) an optimization step analogous to Kalman update, where 0j and 02 weighting coefficients. The α₁ and a2 are chosen by grid searching, and we report the best results. BPF is not included in our testing, as its particle requirements grow exponentially with system dimensions, making impractical for high-dimensional fluid dynamics systems. Additional details on the model architecture and training for baselines and FlowDAS are provided in Appendix D.3.3.
We evaluate performance using the RMSE between predicted and ground-truth vorticity fields. Additionally, we assess the reconstruction of the kinetic energy spectrum to determine whether the physical characteristies of the fluid are accurately preserved. Appendix D.3.5 and Figure S.8 provide the definitions and results for the kinetic energy spectrum metric.

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2 50
t- 40
t- 30
t- 20
t- 10
+ 10
10
10
True State
Observation
FlowDAS
FNO-DA
Transolver-DA
PDEDiff
Figure Data assimilation weather forecasting on SEVIR Vertical Integrated Liquid dataset under sparse observations. All DA models take previous six states (displayed in the first row; (- 50 min to min) as conditions and estimate the future state at + min. FlowDAS preserves storm cores and spatial texture better than score-based diffusion models and neural operator- baattor--beedddd methods.
Results Figure compares FlowDAS with baselines under the 1.5625% sparse-observation setting, See Appendix D.3 for additional results. Quantitative metrics, including RMSE and kinetic energy spectrum comparisons, are provided Table 2 and Figure S.8. Our method consistently outperforms all baselines in terms of reconstruction accuracy, capturing high-frequency information with greater precision. This advantage is further validated by the kinetic energy spectrum in Figure S.8. The RMSE scores in Table further highlight the effectiveness of FlowDAS in accurately estimating the underlying fluid dynamics from observational data. Appendix D.3.7 provides additional results on more challenging observation cases. Appendix D.3.8 presents the performance of FAS when evaluated along an estimated trajectory.
Particle Image Velocimetry To demonstrate that our framework generalizes beyond the synthetic incompressible NS setting, we also apply FlowDAS to Particle Image Velocimetry, a widely used optical technique that infers sparse planar velocity vectors by tracking tracer particles in sequential images. In this scenario the DA problem to reconstruct the full vorticity (or velocity) field from noisy, sparsely sampled velocities. We generate synthetic PIV frames from the same NS simulations, extract particle displacements with a standard PIV pipeline, and feed the resulting observations into the pretrained FlowDAS model. A comparison with baseline methods confirms that Flow DAS maintains its performance advantage in this practical setting. The full experimental protocol, quantitative metrics, and qualitative results reported in Appendix D.4.
4.3 Weather Forecasting
This experiment evaluates FlowDAS on large-scale, real-world featherffreeasttng problem using the Storm EVent Imagery and Radar (SEVIR) dataset [54]. SEVIR provides multi-moda. observations of severe convective storms across the continental United States. We focus on the Vertically Integrated Liquid (VIL) product, a 2-D proxy for precipitation intensity. Each sample is 128 128 grid covering 384 km x 384 km at km resolution and recorded every 10 min for four hours. Similar to [48], six consecutive VII frames (t ( 50 min to / min) constitute the input, and the task is to predict the next frame at + 10 min. In our DA settings, the state variable is the VIL field æ, and the observation is aquired by randomly sampling 10% of grid cells to emulate sparse radar coverage. A total of L 6 previous states are conditioned for next-frame generation. We split the dataset into 80%/10%//1% for training, validation, and testing.
Baselines and metrics We compare FlowDAS with the PDEDiff solver (fixed window size of 7 with 6 previous states as conditions). Neural-operator baselines, e.g., FNO and Transolver, are similarly obtained as Navier-Stokes experiments. Both models takes concatenation (along channel dimension) of states inputs and output the state at target time step. Performance is measured by RMSE and the Critical Success Index (CST) at thresholds of 720 and T4p dBZ, which are standard verification metrics for precipitation nowcasting.

---

METHOD
RMSE
CSI(720) (0.3) f
CSI(710) (0.5)
FLOWDAS
0.053=0.004
0.746:1.002
0.614=0.044
PDEDIFF TRANSOLVER-DA
0,071#0.007
0.549=0.033
0.387
0.062=0.001
0.663:10.00
0.499
FNO-DA
0.064=0.001
0.641=0.001
0.493=0.002
Table 3: Comparison of FlowDAS and other baselines on RMSE and Critical Success Index
(CSI) of the weather forecasting task. All metrics are averaged over 30 repeated runs with standard deviations.
Inplementation Details Different from [31], we adopted FNO [25] rather than U-Net as the backbone of the stochastic interpolants framework. We provide the comparison between the FNO-based and the U-Net-based FlowDAS in the Appendix D.5, and found that the FNO-based FlowDAS demonstrates better performance.
Results FlowDAS achieves lower RMSE and higher CSI both thresholds, indicating more accurate intensity estimates and better hit rates for heavy precipitation shown in Appendix D.5. Qualitative comparisons Figure (and Figures S.15 to S.17) show that FlowDAS reconstructs coherent precipitation structures and peak intensities that other baselines either smooth out or mis-localize. These results demonstrate that the stochastic- iintepioianntre surrogate scales to high-resolution, real metcorological data, even when underlying governed PDEs are unknown, and retains its advantage over diffusion models and neural operators baselines.
Limitations and Future Work
FlowDAS has so far been validated controlled dynamical systems (e.g., Lorenz-63. 63, Navier-Stokes). and its generalizability to more complex, real- world environments such as numerical weather predic- tion remains to be evaluated. In particular, have not yet examined its performance under Sim2Real conditions or applied it to operational
nigh-dimensional systems such as those targeted by GEN_ BE [55]. Although FlowDASS supports probabilistic inference through sampling, its current sampling speed a limitation. promising direction is to employ post-hoc distillation, training a lightweight surrogate model to approximate the FlowDAS sampler. Another avenue is explore hy brid formulations that combine stochastic interpolants with variational inference techniques to further improve computational efficiency.
6 Conclusion and Future Work
This work introduced FlowDAS, a stochastic interpolant- data assimilation framework designed to address the challenges of high-dimensional, nonlinear dynamical systems. By leveraging stochas- tic interpolants, FlowDAS effectively integrates complex transition dynamics with observational data, enabling accurate state estimation without relying on explicit physical simulations. Through experiments on both low- and high-dimensional -ystems-including the Lorenz 1963 system, incom- pressible Navier-Stoke flow, and weather forecasting Drocasting-FllwwASS Flo demonstrated strong performance in recovering accurate state variables from sparse, noisy observations. These results highlight Flow- DAS as robust alternative to traditional model-driven methods (c.g., particle filters) and data-driven approaches (e.g., ccore-based diffusion models and neural operators) for data assimilation, offering improved accuracy, efficiency, and adaptability.
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14

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A Mathematical Derivation
A.1
Conditional Drift
Stochastic interpolants approximate the state transition p(æ 2xk) by interpolating between the state variables øg and # using the SDE defined in Equation (4) and Equation (6) with boundary conditions Xo, X = Fy. #8+1- The drift term b,(X,. Xo) related to the score function
v loog x
Xo) of state transition distribution (Appendix A.2):
b. (x,, Xn) Cs (x, Xo)
V log p(X, Xo)
(S.15)
B.
ArBs
To generate observation-consiseee states, we modify the SDE to process conditioned on obser- vational data, where the drift term b, (,(X.. y, Xo) incorporates observation information via Bayes' rule:
Ca (X Xo)
V log Dogp(X, y, Xo)
b,(X,: V. Xo)
2g13
Cx Xx, Xo)
V Pogp(y X,, Xo) + V Vlogp(X, Xo)
BB
V Vlogply X Xo)
(X, Xo)
(S.16)
A.2 Relating Drift and Score
By extending Stein's formula [30], we have:
1
V Vlogp(X Xo) VsO. x, - (x)) X,),
|x,),
(S.17)
where X, - a, + 3,X + V80,2. In addition,
6x,,, Xo) à X, 3 X [X| X.] + (xo X,).
(S.18)
X, = a X, + xx [X]
X.] +
X.).
Solve Ex,~r(XX)) (Xo)<= [x,] from Equation (S.18):
E Ex x,) B.b.(X. Xo) - B, (3,dr, Backa)
(S.19)
Vo(0gB. - -.,B))
Define X -a. and
3.X, + (3,dr - ) and insert Equa- tion (S.19) into Equation (S.17), we relate the score function Vlog p(X, Xo) and the drift
(X, Xo) [31] by:
Vog PX, Xx) = A, c. (X,,
(S.20)
A.3 Estimation of the Gradient Log Likelihood of Observation
The term V Plg X,, X,) in Equation (7) captures the observation information. Since the observation model only directly links ys+1 and X, = Øx+1, we compute this term by integrating with respect to Xi:
(1l ()pp Xo) = V Xo) S
X Xo) dX, X p Xo) f X Xo) dX,
V Vlgg( X,
(S.21)
Ex (p(y) X1)
ExIxxX,XXX (p(2)) X1)]
In practice, we approximate the above expectations by / Monte Carlo samples, x(¹) Xo): E Xxxxp(Xx
p(X1
1 Σ plu x())).
(S.22)
15

---

Algorithm Training
1: Input: Dataset do:K; minibatch size K' < K; coefficients Ot., 8.,
2: repeat
3: Compute 1 and R using (S.25) for k Bx
4: Compute the empirical loss (b) in Equation (S.24)
5: Take the gradient step _,, to update b,
6: until converged
7:
return drifts b
Plug Equation (S.22) into Equation (S.21):
] E ply ()))
Σ/=1 p(y x((1) log logp(y x(o)
Vlogp(y X., Xo)
3 プ== ply X )
Σ/=1 p(y x(,))
(S.23)
wy v log x(²),
==1
where apply a softmax function to the J scalars flogp(y) x(")) to compute the sample weights wy p(y xx Ej ply x (
B Implementation Details
Training We used dataset consisting multiple simulated state trajectories æo.K to train the drift function b, (X.,, Xo) in stochastic interpolants (see Appendix B.1 for more details). We approximate the cost function in Equation (5) by the empirical loss:
1
Σ
( ((b,(i.. 2x), R.) ds,
(S.24)
K'
kEBg'
where ((::) denotes user-chosen discrepancy metric, depending on the experiment. Bx C {0: K} a subset of indices of cardinality K' < K, with
j 0g 8,00+1 + 20
(S.25)
R = á + 3 + v /0,
where N0, In) and satisfy W, d Vaz with (0, In) for all E [o, 1]. We approximate the integral over 8 in Equation (S.24) via an empirical expectation sampling from s U((0, 1)). Algorithm provides detailed description of the model training process.
Inference Our inference procedure generates trajectories conditioned on observations y using (X,, s. X,). We start by setting a specific state 2& as initial X, and iterate the learned drift model b,(X,,
over a predefined temporal grid So = 0 < S1 < < sx = 1. Within each iteration, we first compute posterior estimates { Xx] (J) J==1 using Equations (9) and (10) for J times. Then, move
one step further towards s on X en by solving Equation (7) which involves backpropagating the
2 to enforce consistency with observations yk. We iterated this process autoregressively. by setting X = =x . Empirically, we found that using a constant step size Sn across the inference process produced generally good results, although fine-tuning each step can slightly improve the performance [42, 56]. The chosen J values crossed all experiments, reported in Table S.4, were sufficient large for stable performance, with subtle variation across the dimensionality of problems. An ablation study, detailed in Appendix C.I, further confirmed that larger J offers diminishing returns. Overall, Algorithm 2 summarizes the inference process.
gradient V x,,, Σ/=1 w. wyylyy--a A(X (1)
B.1 Constructing Training Dataset
In the training stage for experiments, we require pairs of consecutive states to train the velocity model by. To generate these pairs, we proceed as follows:
16

---

Algorithm 2 Inference
1: Input: Observation yL:K. the mcasurement map A, initial state æo, model b, (x(X, Xo). noisc coefficient a, grid Sp
$1 <... < x = 1, iid. Zn ~0, Ip) for 0 N 1, step
size Sn. Monte Carlo sampling times J
2: Set L LL-1
3: Set the (As), = S - Sa. n = O
4: for k L to K do
5:
x y 6- 0s. y
6:
for = to N do
7:
X = Xs, + (X, X (As)) + Joon v (Ass)
8:
[X( 1
J j=1 + Posterior estimation (b,, Spz Xo- Xx,.)
9:
(wg))=1
Softmax (lly A(x )12)
10: X Ant1
X sn V x.,, Σj=1 Wjj l8 A(
11: end for
Øk+1 X
13:
end for
return (+
K
Figure S.5: Structure of training data. Consecutive states paired across multiple simulated trajectories construct the i, and R, defined in Equation (S.24) for training the velocity model b..
1. Trajectory simulation. We simulate T' independent trajectories, denoted as oo:k where each trajectory starts from a unique initial state vh. The state transitions within each trajectory follow the dynamics defined in Equation (1).
2. Consecutive state pairs formation. For each trajectory t, form two aligned sequences:
(a) -6: The original sequence of states with each last state z'x discarded.
(b) af.x= The sequence of states shifted by one time step with each initial state wó discarded.
The two sequences form pairs of consecutive states (xr #1+1) for k = 0,,,, 10.,,..... K
3. Concatenating trajectories. Then, concatenate T sequences of a æij and T sequences æi.x end-to-end, respectively, into two long sequences: 0o. and æi.K. where each state in the second sequence is the corresponding successor states in the first sequence.
4. Sampling training data. During training, batches of paired consecutive states are sampled. For each batch, we sample training data pairs follows:
sample states from 2 as Xo's.
(b) retrieve the counterpart states from 1:7 Xi's.
The batches of state pairs (Xo. X1) are used to construct I, and R, in Equation (S.24) for training the velocity model b, described in Algorithm
In summary, we draw many state pairs (Xo, X, and estimate the integral over in Equation (S.24) by approximated via an empirical expectation over draws (O, 1), and for every and every state pairs (Xo, X ), we independently compute Equation (S.25) with S samples of zg. so the training loss can be rewritten as:
8
C 6
b
RR/112
(S.26)
K
kEBg
s=1
17

---

MONTE CARLO SAMPLING TIMES
SAMPLING STEP SIZE <
WOLLLE-WELL
17
LORENZ
1963
21
0.0002
SR
4X
25
INCOMPRESSIBLE
SR
8x
25
NAVIER STOKES
SO
5%
25
SO
1.5625%
25
1.75
PARTICLE IMAGE VELLOCIMETRY
25
WEATHER FORECASTING
25
0.1
Table S.4: Hyperparameters for FlowDAS in the inference stage of all experiments presented in this study. SR stands for super-resolution task and SO represents sparse observation (inpainting) task.
where we randomly draws s from ( 1). Note that the training batch size equals S x K'. Figure S.5 illustrates the structure of training data, showing how consecutive states are paired across trajectories.
B.2 FlowDAS Inference Hyperparameters
We present key hyperparameters for FlowDAS in the inference stage, including the Monte Carlo sampling times J and the sampling step size & in Table S.4.
C Ablation Study
This section examines different aspects of the proposed FlowDAS, including alternative the method and the hyperparameter settings. We also provide various evaluation results. To sim- plify notation, we omit the explicit sampling distribution p(æ)) in the expectation operator, writing Be [z)) simply as E2 [x))]. The sampling distribution for æ will be specified at the end of the equation where necessary.
C.1 Monte Carlo Sampling and An Alternative
In Appendix A.3, we estimate Vlog (y X,, Xo) by
V Ex, (y) Ex, X1)
log (y X, X =
(S.27)
where we approximate the expectation term by averaging J Monte Carlo samples:
Ex. (1(x | E x Xx. Xo).
(S.28)
This Monte Carlo estimate is an unbiased estimate and the error is proportional to - When the number of Monte Carlo samples, i.e., J, sufficiently large, the estimation error converges to zero. Alternatively, one can also apply Jensen's 's inequality to estimate V log (y) X., Xo), which provides a biased estimate:
X., Xp) V Vlog / (y)Xi)) X, X dxi
V log (y)
(S.29)
V log Ex, (x) 2 VEx, (x)
The ">" arises from Jensen's 's inequality.
(V-AIX 1) 2n
Unbiased vs. biased estimation Let 7 -= p p 2ar
where Z is bounded
random variable within the finite range [0, -- As result, the logarithm function is a- Höldöl continuous with a = 1 and the gap introduced by Jensen's inequality, ic.. the Jensen gap, can be explicitly bounded [57] and given by
[log Ex, ip (x) Ex,
Mn1
(S.30)
18

---

UNBIASED
BIASED
t log p( 21)
17.29
-36.98
1 log log Ti.K)
-0.228
-1,530
W (1 01:k)
0.106
0.111
RMSE(21:K. F1:K)
0.202
0.363
Table S.5: Evaluation of unbiased vs. biased estimation. Comparison metrics between unbiased and biased estimation in the Lorenz experiments. The results demonstrate that the unbiased estimation outperforms the biased estimation.
where M is constant that satisfying log Z - log E[Z]| < M 1 - E|Z] and 7 = Additionally, because Z [0, !2ny. we have M < and 7 max 2 -
B|
V2
And for this upper-bound My when + 1, p (X, X,, Xu) will become delta distribution concentrated on X1 and Z = (i) will also have a delta distribution concentrated on p X1).
and the 7/ R 0 finally. In conclusion, this bias should be controlled by
Ex, [P X1) Ex, [logp (y X1)I
(S.31)
V
Although this bias is theoretically bounded, it still results in slight degradation in performance. Table S.5 shows the comparison for the Lorenz experiment to illustrate this point.
Hyperparameters for Monte Carlo Sampling We examine the estimation process and associated hyperparameters in Equation (S.28), where the expectation is computed using Monte Carlo method. The hyperparameter J, is referred as the number of Monte Carlo sampling iterations in Algorithm It is important to note that increasing J does not lead to an increase in neural network evaluations but only involves additional Gaussian noise simulations, which are computationally lightweight. To illustrate the effect of J, we use numerical results from the Lorenz experiment. As shown in Table 6.6, increasing J can improve performance by approximately 20%, with negligible impact on computational time.
J RMSE
3
6
12
21
30
50
0.167
0.148
0.153
0.142
0.150
0.138
Table S.6: Effect of J. The RMSE of generated state trajectories for FlowDAS is evaluated with different Monte Carlo sampling times in Equation (S.28) for the Lorenz 1963 experiment. As J increases, the RMSE initially decreases, indicating improved performance, and then stabilizes.
C.2 Posterior Estimation Methods
We also evaluate the impact of different methods for posterior estimation: as defined in Equations (9) and (10). The results presented in Table S.7, where "Istorddrr" and "2nd-order" refer to 1st- order Milstein method and 2nd-order stochastic Runge-Kutta method, respectively. "No correction" indicates forecasting purely based on the model without incorporating observation information. The results show that both Ist-order and 2nd-order estimations provide reasonable accuracy. However, the 2nd-order estimation consistently delivers better performance. This suggests that employing more accurate estimations of (X, X,, Xu) can effectively enhance model performance. Beyond the 2nd-order method, higher-order approaches like the Runge-Kuun Kutta 4th-order (RK4) method could further improve accuracy. However, these methods come with increased computational cost: 2nd- order estimation requires two neural network evaluations per step compared to one for 1st-order estimation, while RK4 requires four evaluations per step. In our experiments, we find that the 2nd-order estimation strikes a good balance between performance and efficiency, making it a practical choice. Further exploration of higher-order methods will be left for future research.
19

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IST-ORDER
2ND-ORDER
No CORRECTION
32 - 128
0.048
0.038
0.206
16a 128
0.101
0.067
0.206
Table S.7: Effect of posterior estimation. The RMSE of vorticity estimate from FlowDAS is evaluated on the incompressible Navier-Stokes task using different posterior estimation methods as defined in Equations (9) and (10), and forecasting without observations (i.e., no correction). For the super-resolution tasks 16² - 128 and 32 > 1282, both posterior estimation methods significantly outperform forecasting without observations. Among them, the 2nd-order method achieves the lowest RMSE.
C.3 Particle Collapse and Robustness o Likelihood-baat bbaed Sampling
The likelihood-baze sampling in Equation (8) can, in principle, suffer from particle collapse, which a well-known issue nigh-dimensional importance sampling where few weights dominate the normalization sum, leading to an effective sample size close to one. We provide further analysis and our mitigation strategy below.
Observed behavior. During the stochastic interpolation process, we observed that the likelihood score |yy - Ax, 11 typically decreases as the sampling proceeds, but in some cases starts to increase sharply near the end of the trajectory (typically within the final 30 to 50 steps). This abnormal increase coincides with the particle weights wy becoming heavily concentrated on single particle x indicating a collapse of the Monte Carlo estimator.
Empirically, we observed two distinct regimes:
When the likelihood term decreases as expected, the weights remain balanced across particles, ensuring stable estimation.
When the likelihood begins to rise unexpectedly, one or two particles dominate the weights (approaching 100%), leading to collapse and potential divergence.
Mitigation strategies. To prevent collapse, we employ a two-step remedy:
Resampling: When abnormally concentrated weights are detected, we immediately perform resampling to regenerate balanced particle set. This a standard correction technique in sequential importance sampling.
Alternative sampling strategy: If resampling does not stabilize the process, we switch to the alter- native sampling strategy detailed in Appendix C.I. Although this approach may yield slightly less precise estimates, it significantly enhances robustness and prevents divergence in high-dimensional settings.
C.4 Alternative Likelihood Score Approximation
In the main formulation, the likelihood score term Vogg((y Xo, X.) is estimated via Monte Carlo sampling over particles {x(") 1==1 which provides unbiased approximation at the cost of potential particle collapse high-dimensional settings. An alternative approach is to adopt DPS-type approximation [42]:
V logp(y Xo. X.) V Vlogp(y X1), where x == E X1
X X.).
This estimator replaces the Monte Carlo sampling with the conditional expectation X1, thereby avoiding weight collapse but introducing bias bounded by Jensen's 's inequality.
To assess the effect of this approximation, directly substituted the DPS-type term into Equation (8) of the main paper and re-ran the weather forecasting experiment. The quantitative comparison is summarized in Table S.8.
The DPS estimate can mitigate collapse and sometimes improve numerical stability. However, the denominator in Equation (S.34) (see derivation below) can approach zero when - O, causing the
20

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METHOD
RMSE STD
FLOWDAS
0.056 + 0.002
FLOWDAS-DPP DPS
0.050 + 0.002
Table S.8: Comparison of likelihood approximations on the weather forecasting task. While the DPS approximation slightly improves RMSE, introduces theoretical bias due to Jensen's gap and requires additional stabilization.
optimization to become unstable. To prevent this, we select a cutoff value 8, and apply the DPS-type update only for Se- In the weather forecasting experiment, we set se 0.04 via grid search. This parameter may vary across datasets and is problem dependent, with no closed-form theoretical criterion for selection. Further tuning of parameters (as,, Ba, da) could improve the method, which we leave for future work.
Overall, FlowDAS-DDP DDPS provides biased yet more stable alternative that complements main FlowDAS formulation and demonstrates the flexibility of the framework.
Short derivation of FlowDAS-DPS. To derive the DPS-type approximation, we notice that
by (x,,,, X,) á Xo + | Xo: X, + F0 Xo: Xal:
(S.32)
Besides,
X, Xo + [X,, Xo, X.) V80,Elz Xo; Xa).
(8.33)
By combining the above two equations, we obtain
(x,,(X,; Xo) d,X, (a, aa,
E X
X X,,X,]
(S.34)
3. 3,0.
C.5 Ablation on Conditioning Horizon
To study the effect of the conditioning horizon, perform an ablation by varying the number of observed past states used during inference. We evaluate performance using the Continuous Ranked Probability Score (CRPS), proper scoring rule that jointly measures accuracy and uncertainty calibration of probabilistic forecasts- the lower the CRPS, the better the performance.
NAVIER-STOKES: CRPS vs. CONDITIONING HORIZON
# CONDITIONING STATES CRPS
10
6
0.538
0.634
0.663
0.776
SEVIR: CRPS vs. CONDITIONING HORIZON
# CONDITIONING STATES CRPS
6
3
0.015
0.021
0.021
Table S.9: Effect of conditioning horizon. Longer conditioning horizons consistently improve CRPS, indicating better probabilistic forecasting performance.
The results clearly show that incorporating a longer conditioning horizon leads to consistent improve- ments in forecast accuracy and uncertainty estimation. In the Navier-Stokes task, extending the horizon from 10 steps yields approximately 30% reduction in CRPS, highlighting the benefit of temporal context in capturing complex dynamics. Similarly, in the SEVIR precipitation forecasting task, conditioning on six previous frames achieves the best predictive accuracy. These findings confirm that longer temporal conditioning enables FlowDAS to leverage sequential dependencies more effectively, which is crucial for accurate data assimilation in digh-dimmnsionll and nonlinear dynamical systems.
21

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MODEL
PARAMS (M)
TRAINING TIME (HR)
INFERENCE TIME (S/SAMPLE)
FLOWDAS
30.0
22
30,9
PDEDUFF
22.9
45
28.9
TRANSOLVER
11.2
80
12.7
FNO
1880.0
80
2.4
Table S.10: Comparison of model size, total training time, and inference time on the weather forecasting task using a single A100 GPU.
METHOD
COMPUTATION TIME (S)
RMSE STD
MAX MEMORY (GB)
1ST ORDER
17.100
0.0490 0.0026
1.8
2ND ORDER
30.871
0.0485 + 0.0035
3.9
HYBRID
205.271
0.0478 + 0.0033
70.0
Table S. S.11: Accuracy-speed trade-off for stochastic integrators. The hybrid "2nd order 400 +
Flow x 100" improves RMSE only slightly (absolute gain 0.0007) but increases computation time by ~ 7x and peak memory by ~ 18x compared with the second-order integrator.
C.6 Runtime and Parameter Comparison
We report the model sizes, total training times, and inference costs for the SEVIR weather forecasting task using a single NVIDIA A100 GPU in Table S.10. FlowDAS achieves the shortest training time among all baselines, converging roughly twice as fast as PDEDiff and nearly four times faster than Transolver, despite having a comparable model size.
The iterative correction mechanism FlowDAS introduces a modest increase in inference time compared to purely feedforward models. However, this process is essential for producing observation- consistent predictions and stabilizing long- long-horizon forecasts. Importantly, FlowDAS exhibits strong training efficiency, reaching convergence significantly faster than other baselines.
C.7 Bias-efficiency trade-offs of stochastic integrators
Our stochastic integrators in Equations (9) and (10) approximate the condiiioonee X; and therefore introduce discretization bias. We quantify the accuracy-speed trade-off on the weather forecasting task by comparing three settings: a first-order integrator, second-order integrator, and hybrid method that follows the recond-order scheme for the first 400 steps and then computes the full cechasttc-interpllant path to the endpoint X, for last 100 steps ("2nd order 400 + Flow 100"). The hybrid reduces bias but is much more costly.
We also considered full-path variant ("Flow>5000) that computes X, exactly at every step by rolling out the leftward stochastic interpolant. This requires 500 (500 + 499 + + 1) x 2.5 x 10 forward passes and retaining the entire computation graph backpropagate log p(y X1). which leads to out-of- of-memory failures even on A100 GPUs. In contrast, the first- and second-ord schemes provide stable inference with much lower cost while maintaining competitive accuracy.
These results shown in Table S.11 support our choice: conditioning on observations at each interpola- tion step delivers most of the DA benefit, whereas reducing the small integrator bias further yields only marginal accuracy gains at very high computational and memory cost. We include this analysis and the table above in the appendix for clarity.
D Additional Experiment Results
This section provides the details of our experiments, including additional experiments and results.
22

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D.1 Double-well Potential Problem
D.1.1 The Double-well Potential System
In this experiment, we investigate a ID tracking problem in dynamical system driven by the double-well potential. The system is governed by the following stochastic dynamics:
dæ (4x(x² 1) dt + Ba dE.
(S.35)
with observation model defined by A(x) = x². The observations are given by
A(x) 7 +1
(S.36)
where the stochastic force &e a standard Brownian motion with diffusion coefficient of 3x 0.2. standard Gaussian noise with standard deviation of 0.2. The v/ is the derivative of the potential (xx) = x² 2x + f. where a function independent of æ. The system describes a particle 1 and 11, with small fluctuations around these points, as
trapped in the wells located illustrated in Figure S.6a.
D.1.2 Training and Testing Data Generation
We trained the model simulated trajectories generated by numerically solving the transition equation using a temporal step size of 0.1. The training dataset consisted of 500 trajectories, each of length 100, with initial points uniformly sampled from the range [-2, In the testing stage, we introduced stronger turbulence to the system, causing the particle to occasionally switch wells (xr -)).
Double-well potential
1.0
FlowDAS
True state
Potential value
0.5
0.
State
-0.5
-1.0
-10
State
Time step
(a) Illustration of the Double-We Well Potential Prob- lem. double-well potential system, particles are typically captured at the bottoms of the wells (x = and = -1). With low probability, particles can transition between wells due the stochastic term in the transition equation.
(b) Visualization Results of Double-Well Potential Problem. We show visualization results of Flow- DAS and BPF, while the basee solver SDA fails to produce reasonable results. FlowDAS can track the dramatic change of the particles that BPF struggles to immediately react to.
Figure S.6: Illustrations and visualization results for the double-well potential problem.
D.1.3 Neural Network for Learning the Drift Velocity
In this low-dimensional double-well potential task, the drift velocity b, approximated using a fully connected neural network with 3 hidden layers, each having hidden dimension of 50. Both the input and output dimensions of the network are For the condition X, and timestep s, we empirically find that embedding Xg. similar to how is embedded, outperforms directly using X, as an additional input to the network. The intuition behind this approach that embedding X, helps the network better distinguish between the two variables. X, and X₂- The model is trained using the Adam optimizer with base learning rate 0.005, along with linear rate scheduler. Training conducted for 5000 epochs.
23

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D.1.4 Hyperparameters During Inference
Hyperparameters of the inference procedure, specified for the double well potential task, presented in Table S.4.
D.1.5 Baseline
For this task, we compare our method with SDA and the classic BPF method. For SDA, we fix the window size two and use the same training data as FlowDAS to ensure fairness. The local score network is implemented [50]. The score network consists of 3 hidden layers, each with hidden dimension of 50. The model is trained using the Adam W optimizer with base learning rate of 0.001, weight decay of 0.001, and linear learning rate scheduler. Training is conducted for 5000 epochs. For the BPF method, the particle density is set to 16384.
fully connected neural network, following the architecture proposed in
D.1.6 Results
The visual results are shown in Figure S.6b. Surprisingly, while both FlowDAS and the classic BPF method produce reasonable results, FlowDAS demonstrates superior performance tracking dramatic changes in the trajectory. In contrast, the score-based solver SDA fails to produce reasonable results. This failure arises because the starting point of SDA during optimization is purely Gaussian noise, leading to poor initial estimation of the target state. Furthermore, the cubic observation model amplifies the differences, causing the optimization gradients to explode and resulting in recovery failure.
In FlowDAS, the error is bounded by || - A(xx) )12 because the generation process begins with the previous state, which serves proximal estimate of the target state. Consequently, FlowDAS undergoes a more stable optimization process.
Compared to the classic BPF method, we find that it struggles to capture dramatic changes in the trajectory. This limitation is primarily due to the small diffusion coefficient 3a. such as B, = 0.2, which causes the predicted filtering density in BPF to concentrate around the mean value dictated by deterministic part of the transition equation. As a result, extreme cases lying the tail of the future state distribution p( p(#++1 | xx)) are often missed. This phenomenon can be explained by the truncation error arising from the finite particle space [50].
In contrast, FlowDAS is capable of immediately sampling from the true conditional distribution as the steps become sufficiently large. This allows FlowDAS to better capture the tail region of the true distribution and react to dramatic changes, even those in low-probability areas. Additionally, FlowDAS can effectively incorporate observation information, enabling balance prediction and observation even when the true underlying state occurs in a low-probability region.
D.2 Lorenz 1963
D.2.1 The Lorenz 1963 Dynamic System
To simulate this system, we use the RK4 method, which updates the solution from time in to bx+1 = bn + h using the following formulas for each variable a, b, and c:
kian h p(bn an),
kiB. h (a (p Cn) - bp)
(S.37)
kic. h. (n TCn),
k2an = p (bn + kibn - (an + Klam )
2
(an + kian (p (cx + kien
(bn + kibn
))
k2bb. =
(S.38)
2
2
kgcn = h (an + kion (bn + Kibn
T(c, + kien
2
2
24

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Location a
Location a
RoWDAS
SDA Dynamics
-2
11
-1
Time Step
Time Step
Figure S.7: Forecaseting dynamics modeling evaluation in the Lorenz 1963 task: FlowDAS
SDA. FlowDAS produces results that closely match the true states, demonstrating its ability to learn the underlying transition dynamics p(# (gg). In contrast, SDA exhibits rapid divergence from the true states. This divergence arises because SDA focus modeling the joint distribution p( 2x) rather than directly learn the transition dynamics p( mx),, which is inherently less suitable for capturing the system's underlying dynamies.
(bn + k2bnn ) - (an + k2am))
K3am = h 1
2
(12 + kiz0 (p (cn + k:zen )) (In + Kizbn
:)
kabn
(S.39)
2
2
(an + k2on [bn + k2gbn) T(cn + 1:zcn
k3cn
=
2
2
Rean h ((bn + kibb.) (an + kisaa))
kabn. h ((an + K30. (p (cn + k30. )) (bn + k3z0 )),
(S.40)
ksca = h ((an + K300 (b, + k:sb.)) - J(en + kiccn) )).
Then, the updates for an. bn. and on are:
ax+1 On + (hian 2kzan + 2k:za. kean),
1
bn+1 bn + (kio.. + 2kizbn 2kiz.n + kabn).
(S.41)
6
Cx+1 Ca + (kic. + 2kze. + 2kic.. + kACa).
6
After solving these deterministic updates, the stochastic force E5) is added to (an+1, but1, Cu+1), which leads to the Lorenz 1963 system dynamic equations described in Equa- tion (11).
D.2.2 Training and Testing Data Generation
We apply the RK4 method described in Appendix D.2.1 to generate the simulated training and testing data.
D.2.3 Hyperparameters During Inference
Hyperparameters of the inference procedure, specified for the Lorenz 1963 task, are presented in Table S.4. Noticeably, the step size Ç is smaller than in other experiments due the chaotic nature of the Lorenz 1963 system, which requires finer step size to accurately capture its dynamics.
D.2.4 Neural Network for Learning the Drift Velocity
In this task, we use fully connected neural network with 5 hidden layers, each with hidden dimension of 256, to approximate the velocity field bg. The input and output dimensions are both 3.
25

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For the condition Xo and timestep we use embeddings of dimension 4. The model is trained using the Adam optimizer with a base learning rate of 0.005, and a linear learning rate scheduler is applied. Training is conducted over 23000 epochs.
D.2.5 Baseline and Additional Results
We compare FIo with SDA and BPF. For the SDA, we use score neural network with fixed window size of 2. This local score functtin is implemented using fully connected neural network with hidden layers, each having a hidden dimension of 256. The model is trained using the AdamW optimizer with a base learning rate of 0.001 and linear scheduler over 23000 epochs. For BPF, the particle density is set to 16384.
We also implemented EnSF [58] and its performance is on par with BPF. This is expected since EnSF uses a training-free Monte Carlo estimator for the score function, which faces the same particle collapse issues, i.e., samples concentrate
high-probability regions of the transition dynamies, leading to biased score estimates. This reflects train-sampling tradeoff: EnSF saves training time but cannot learn accurate scores, especially under highly nonlinear dynamics and observation models like in our noisy Lorenz-63 settings(e.g. with collected data, avoids this issue and achieves superior performance, For the EnSF, we largely follow the official implementation and adopt the settings summarized below. The ensemble size is set to 20000 to ensure stable Monte Carlo estimation of the ensemble mean and covariance. The diffusion SDE is integrated using 500 steps with the Euler solver. Both Ea and Eb fixed to 0.001. The initial data variance is set to zero, and no covariance inflation is applied (inflation factor = 1).
atan( observation). FlowDAS, by learning the drift
METHOD
FLOWDAS
SDA
BPF
ENSF
RMSE
0.202
1.114
0.270
0.298
Table S.12: Comparison of FlowDAS with diffusion-based and ensemble-based filters. Reported RMSEs are computed on the same problem setting as described in the Section 4.1. FlowDAS achieves the lowest RMSE, demonstrating robustness under non-Gaussian dynamies and non-linear observation observations.
D.2.6 Accuracy of the Learned Dynamics
We evaluate the dynamic learning performance of FlowDAS and SDA, focusing their ability to forecast future states without using observations. For BPF, we do not evaluate its performance in this context since it explicitly incorporates the true system dynamics.
As SDA designed rely on observations and does not work in an auto-regressive manner, we adapt it for this evaluation by using the previous state as seeudo-bbeervatio. The SDA then forecasts the next state based on this input. For FlowDAS, we disable the observation-bart update step by setting the step size S. O.
Both methods are initialized with the same initial state, and we simulate 64 independent trajectories of length 25, True states are generated by solving Equation (11) using the RK4 method, starting from the same initial point. The visualization of location a across 25 time steps is shown in Figure S.7.
D.3 Incompressible Navier-Stokes Flow
D.3.1 Incompressible Navier-Stokes Flow Problem Settings
For the incompressible Navier-Stokes flow problem, we adopted the problem setting from [31] and used the provided training data. The choices of experiment parameters are as follows: v = 10
0.1, The stochastic force& & is defined as:
E(t, x. y/ (t) (6x) + W2(() (T)) + Ws(1) sin( + v)) + Wa(1) cos(8(r + v)) / + y))
+ Ws(() cos(6x) + Wa(!) (in + Wr(() cos(5(z / + v)) + sin((((((
(S.42)
For more details and insights about this problem, see [31].
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D.3.2 Hyperparameters During Inference
Table S.4 presents the hyperparameters of the inference procedure for the incompressible Navier- Stokes flow task.
D.3.3 Neural Network for Learning the Drift Velocity
We use a U-Net architecture to approximate b,, following the network proposed in [31]. The conditioning X, is implemented through channel concatenation the input. The architectural details are as follows:
Number of states conditioned on: 10.
Number of initial channels: 128.
Multiplication factor for the number of channels each stage: (1, 2, 2, 2).
Number of groups for group normalization in ResNet blocks: 8.
Dimensionality of learned sinusoidal embeddings: 32.
Dimensionality of each attention head in the self-attention mechanism: 64.
Number of attention heads in the self-attention layers: 4.
We employ the AdamW optimizer [59] with a cosine anncaling schedule to reduce the learning rate during training. The base learning rate is set to 2 10 Training is conducted with batch size of 32 for 1000 epochs.
D.3.4 Baseline
We compare FlowDAS to FNO-DA, Transolver-DA, and PDEDiff.
The model is configured with 7 stacked Fourier layers. Each layer processes data in 128-dimensional feature space and truncates the Fourier series to include 6 modes in each spatial direction. padding of applied before the Fourier operations and removed afterward. The model is provided with the previous 10 states to predict the state at the next time step. We train the model for 1000 epochs, using batch size of 64 and the AdamW optimizer with an initial learning rate of 0.0001.
FNO-DA
Transolver-DA
The model has 8 layers of transolver blocks whose hidden channels are 128 and have 32 slices, with attention heads. The model is provided with the previous 10 states to predict the state the next time step. We train the model for 1000 epochs using batch size of 64, the AdamW optimizer, and a OneCycleLR learning rate scheduler with an initial learning rate of 5e
The neural network is configured with temporal window of 11, of which the first 10 inputs are conditions and the forecast horizon is 1; an embedding layer dimensionality of 32; and base number of feature channels 256. The network depth is 5, and the activation function used is SiLU [59]. We train the model using a batch size of 64 and the AdamW optimizer, with a linear learning rate scheduler (initial learning rate 0.001) and weight decay of 0.001.
PDEDiff
During inference, the aj and a2 used is varied from task to task for both the FNO-DA and the Transolver- DA. We show the detailed parameter settings in Tables S.13 and S.14. We use AdamW optimizer for Kalman update with a learning rate of 1e -4 and maximum iteration time 2000 to ensure the loss converges. Without other indications, adopt the same optimizer settings for both FNO-DA and Transolver-DA.
D.3.5 Kinetic Energy Spectrum Analysis
We evaluate the methods from physics perspective using the kinetic energy spectrum. A better method produces results that align more closely with the kinetic energy spectrum of the true state. The kinetic energy spectrum is computed as follows:
E(kr)
Σ
E(ke, ky.)
(S.43)
P.G
kpve E bin
27

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@1
Q2
SR
4X
0.63
INCOMPRESSIBLE
1.59
NAVIER STOKES
SO
5%
0.10
SO
1.5625%
0.29
WEATHER FORECASTING
1.22
1
Table S.13, 13: Hyperparameters for FNO-DA the inference stage of all experiments presented in this study. SR stands for super-resolution task and SO represents sparse observation (inpainting) task
Q1
Q22
SR
4X
0.53
INCOMPRESSIBLE
SR
1.21
I
NAVIER STOKES
SO
5%
0.14
1
1.5625%
0.22
1
WEATHER FORECASTING
5.00
1
Table S.14: Hyperparameters for Transolver-Da in the inference stage all experiments pre- sented in this study. SR stands for super-resolution task and SO represents sparse observation (inpainting) task
where kp.q represents the wavenumbers grouped into bin E(k,, ky,) is the kinetic energy at each wavenumber and p.q are the index representing a specific discrete wavenumber along the k or ky
axis.
Since the direct outputs of both FlowDAS and PDEDiff are vorticity fields, is necessary to first convert the vorticity into velocity before computing the kinetic energy spectrum. This conversion is achieved by solving the Poisson equation Ae - to obtain v and then calculating its gradient - Vi to derive the velocity v.
We present the results of the kinetic energy spectrum analysis for the Navier-Stokes flow super- resolution and sparse observation tasks.
D.3.6 Additional Results
In this section, present additional results to Section 4.2, including the kinetic energy spectrum and visualization results for the Stikke flow super-resolution and sparse observation tasks.
16 + 1 Super-Resolution The kinetic energy spectrum is shown in Figure S.8 (a), and additional visualization results are provided Figure S.9. FlowDAS effectively reconstructs high- resolution vorticity data from low-resolution observations, while PDEDiff struggles to capture high-frequency physics.
32 + 128 Super-Resolution The kinetic energy spectrum is shown in Figure S.8 (b), and additional visualization results are provided in Figure S.10. Similar to the 162 - 1282 task, FlowDAS achieves superior alignment with the true state compared to PDEDiff.
5% Sparse Observation visualization results in Figure S.11. FlowDAS accurately recovers the vorticity field despite the highly sparse observations, significantly outperforming PDEDiff, demonstrating strong recovery of physical information.
The kinetic energy spectrum presented in Figure S.8 (d), with additional
Across all tasks, FlowDAS consistently outperforms PDEDiff, demonstrating superior alignment with the true kinetic energy spectrum and effective recovery of the vorticity field. FlowDAS not only excels in super-resolution tasks but also handles sparse observation challenges with robustness, maintaining physical coherence and accurately capturing high-frequeng, dynamics.
28

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10'
10'
Kinetic Energy E(k)
Kinetic Energy E(k)
107
10a
10-
101
GT
GT
10"*
Score-based
Score
10-3
Transliver DA
Transolver DA
(a)
10:2
(b)
Wavenumber k
Wavenumber K
10'
107
Kinetic Energy E(k)
Kinetic Energy E(k)
10a
107
10-4
10"4
HOWGAS
RNO DA Tansolver-AA
PNO DA
TransohveDD
(C)
10a
101
(d)
10a
10
Wavenumber k
Wavenumber k
Figure S.8: The kinetic energy spectrum of: (a) super-resolution task: 162 + 128²; (b) super- resolution task: 32 + 1282: (c) sparse observation task: 1.5625%; (d) sparse observation task: 5%, in the incompressible Navier-Stokes flow task. We present the kinetic energy spectrum of the true state alongside the estimations from FlowDAS and baseline methods. FlowDAS can produce results that better aligned with the true state terms of the kinetic energy spectrum, evidenced by the oscillations in the spectrum of baselines, indicating FlowDAS's superiority in recovering the physics information and effectiveness as surrogate model for stochastic dynamic systems.
t
Observation
True State
PDEDiff
FloWDAS
Navier-Stokes Flow: Super-Resolution Task (8x)
Figure S.9: Data assimilation of incompressible Navier-Stokes Sow-SSupe-Reeolllee task
(8x). Experiments were conducted at observation resolution of 16?. The goal of resortieno task is to reconstruct high-resolution vorticity data (128?) from low-resolution observations.
29

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4
Observation
True State
PDEDiff
FlowDAS
Navier-Stokes Flow: Super-Resolution Task (4x)
Figure S.10: Data assimilation of incompressible Navier-Stokes flow flow-Super- Resoluttnn task (4x). Experiments were conducted at observation resolution of 32 The goal of super-resolution task is to reconstruct high-resolution vorticity data (128?) from low-resolutie observations.
t2
Observation
True State
PDEDiff
FloWDAS
Navier-Stokes Flow: Sparse Observation Task (5%)
Figure S.11: Data assimilation of incompressible Navier-Stokes Sww-SSpare-Observaar task
(5%). Experiments conducted at observation sparsity level at 5%. The goal of sparse observation task to recover the complete vorticity field.
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D.3.7 Performance Degradation Under More Challenging Cases
In the main paper, FlowDAS evaluated the incompressible Navier-Stoker system under moderate super-resolution (SR) and sparse-observation (SO) conditions, specifically SR 4x, SR 8x, SO 5%, and SO 1. 1.5625%. To evaluate the performance under extreme conditions, we further test FlowDAS on SR 16x and SO settings with observation coverages as low as 0.75%.
These results demonstrate graceful degradation of performance as the observation coverage de- creases or the upscaling factor increases, confirming the robustness FlowDAS even under highly underdetermined measurement scenarios.
SR 16x
SO 1%
SO 0.75%
RMSE + STD 0.398 + 0.015 0.395 + 0.016 0.452 + 0.019
Table S.15: RMSE ( std) of FlowDAS Navier-Stokes under more challenging super- resolution and sparse-observation conditions. Performance decreases smoothly as the observation ratio reduced, supporting the stability and generalization of FlowDAS in regimes.
D.3.8 Trajectory-level Evaluation
To further evaluate the temporal stability of FlowDAS, we examine its performance over full generated trajectories on the incompressible Navier-Stoker system. While the main results Table focus on single-step assimilation accuracy, this analysis quantifies how reconstruction errors and their variability evolve over multiple forecasting timesteps. At each timestep, RMSE and standard deviation are computed over 50 independently sampled initial states, each propagated through 30 stochastic realizations.
RNSE 1.5625%
Sid Dev 1. 5.5022%
RMSE
RASE 16"
0.7
Sid Dav- 167
Stdd Dev- 32?
RMS5
0.4
Forecasting Timesteo
Figure S.12: Trajectory-level RMSE and standard deviation of FlowDAS on the incompressible Navier-Stokes system. The plot shows the evolution of RMSE (solid lines) and corresponding standard deviations (shaded areas) across forecasting timesteps. FlowDAS maintains stable perfor- mance over several assimilation steps, with gradual error accumulation consistent with stochastic propagation.
D.4 Particle Image Velocimetry
This section presents realistic application of our method: Particle Image Velocimetry (PIV). PIV is widely used optical technique for measuring velocity fields fluids, with many scientific applications in acrodynamics [60-62], biological flow studies [63-65], and medical research [66-68].
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Observation at f
Observation at fz
Illuminated Particles
Tripwire
Vorticity Field (True State)
Velocity Field
Tracer Particles
Figure S.13: Illustration of the real-world Particle Image Velocimetry (PIV) experiment: The flow is seeded with tracer particles illuminated by a laser sheet, and their movements are captured by camera to derive the sparse velocity field. The goal here is to recover dense vorticity field from the sparse velocity measurement.
D.4.1 Task Description
In standard PIV setup, as shown in Figure S.13, fluorescent tracer particles are seeded into a fluid flowing through a channel with transparent walls. A laser sheet illuminates the fluid, and particle movements are recorded by high-speed camera with adjustable temporal resolution. By analyzing the displacement of these tracer particles, the velocity field within the fluid can be determined at sparse locations. Unlike the task in Section 4.2, which involves recovering dense vorticity fields from sparse vorticity observations, PIV introduces slightly different DA task: recovering dense vorticity fields (ær w) from sparse velocity measurements. This observation model is defined by
y = A(u(w)) + n
(S.44)
where the A(-) sparsely samples the velocity field v and the observation noise n has standard deviation of 7 = 0.25. The relationship between the velocity v and the state (vorticity) w is given by w = V x v. To derive the velocity v from the vorticity w, we first solve the Poisson equation Ar to obtain the stream function v, and then compute the velocity v the gradient of th. This process is performed using the Fast Fourier Transform.
D.4.2 Dataset
In this experiment, use the same fluid dynamics simulation data from the NS experiments described in Section 4.2. However, we convert the vorticity data to velocity fields via Fourier transform to create synthetic PIV datasets [69]. The particle positions in our simulation are randomly initialized and then perturbed according to the simulated flow motion pattern. these synthetic images, we assume particle density of 0.03 particles per pixel. particle diameter of 3 pixels, and peak intensity of 255 for each particle in grayscale. The images are processed through a standard PIV pipeline to extract particle locations, match corresponding particles across frames, and compute sparse velocity observations. These sparse measurements are then used in DA to reconstruct the full vorticity fields.
D.4.3 Baselines
We compare our method against the SDA solver. Instead of training new scores or stochastic interpolant networks, we directly adopt the networks trained on vorticity data from the incompressible NS flow experiment evaluate FlowDAS and SDA the PIV task.
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PARAMETER
VALUE
UNIT
PARTICLE DENSITY
0.03
PARTICLE PER PIXEL
PARTICLE DIAMETER
3
PIXEL
PEAK INTENSITY
255
GRAY VALUE
Table S.16: PIV task: parameters for simulation. This table summarizes parameters for the PIV experiments in detail.
Observations and Velosity
Verticity (True State)
FlowDAS
Figure S.14: Data assimilation of Particle Image Velocimetry. The vorticity field is visualized in the same way as Section 4.2, FlowDAS outperforms SDA in of reconstruction fidelity and RMSE, recovering more detailed features even when direct observations are not available. These improvements highlight the potential of FlowDAS for real-world applications.
D.4.4 Experiment Setting
The training data, neural network (including FlowDAS and SDA) is the same as those in the incom- pressible Navier-stokes flow simulation. Figure S.13 shows the standard PIV set up. Hyperparameters of the inference procedure, specified for the PIV task, are presented in Table S.4.
D.4.5
Results
FlowDAS accurately reconstructs the underlying fluid dynamics from observed particle images, producing vorticity estimates with high precision. As shown in Figure S.14, it outperforms the baseline SDA in terms reconstruction fidelity. Quantitatively, FlowDAS achieves a lower average RMSE of 0.118, compared to 0.154 for SDA. This performance gap highlights the robustness of FlowDAS and its potential utility in fluid dynamics applications.
D.5
Weather Forecasting
D.5.1
Baselines
We compared FlowDAS FNO-DA, Transolver-DA, and PDEDiff.
The model is configured with stacked Fourier layers. Each layer processes data in 128-dimensional feature space and truncates the Fourier series to include 64 modes in each spatial direction. padding of applied before the Fourier operations and removed afterward. The model is provided with the previous 6 states to predict the state at the next time step. We train the model for 1000 epochs, using batch size of 200 and the Adam W optimizer with an initial learning rate of 0.0001.
FNO-DA
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BACKBONE OF FLOWDAS
RMSE 4
CSI(720) (0.3)
CSI(710) (0.5) 1
FNO
0.053
0.718=0.015
0.568=0.022
TRANSOLVER
0.056=0.002
0.702=0.018
0.540 0.0288
UNET
0.056=1.002
0.703=0.017
0.540=0.021
Table S.17: Comparison of backbone network architectures of the drift model on the weather forecasting task. All metrics are averaged over multiple runs. Subtle improvements observed when using a FNO network as the backbone of the drift model.
Transolver-DA The model has 8 layers of transolver blocks whose hidden channels are 128 and have 32 slices, with & attention heads. The model is provided with the previous 6 states to predict the state at the next time step. We train the model for 1000 epochs using a batch size of 200, the AdamW optimizer, and OneCycleL.R learning rate scheduler with an initial learning rate of 5e-4.
The score neural network configured with a temporal window of 7, of which the first inputs conditions, and the forecast horizon is 1; embedding layer dimensionality of 32; and a base number of feature channels 256. The network depth is 5, and the activation function used is SiLU [59]. We train the model using batch size of 200 and the Adam optimizer, with linear learning rate scheduler (initial learning rate 0.001) and a weight decay of 0.001.
PDEDiff
During inference, the 0j and a2 used is varied from task to task for both the FNO-DA and the Transolver-DA. We show the detailed parameter settings in Tables S.13 and S.14. We use an AdamW W optimizer for Kalman update with learning rate of le and maximum iteration time of 2000 to ensure the loss finally becomes stable. Without other indications, we adopt the same optimizer settings for both FNO-DA and Transolver-DA.
D.5.2 Training Details
We experiment with three backbone architectures U-Net, FNO, and Transolver to learn the drift term and compare their performance and provide numerical results in Table S.17. The architectural details are as follows:
UNet: shown in Appendix D.3.3, the only change is the number of states conditioned on. Here we condition on previous 6 states.
FNO: this model is configured with 7 stacked Fourier layers. Each layer processes data a 128-dimensional feature space and truncates the Fourier series to include 64 modes in each spatial direction. A padding of is applied before the Fourier operations and removed afterward. The model is provided with the previous 6 states, the X, and the time additional channel concatenated to the previous states and the X, to predict the drift term.
Transolver: this model has 5 layers of transolver blocks whose hidden channels are 128 and have 32 slices, with 8 attention heads. The sinusoidal timestep embeddings are applied. The inputs the previous states, the X, and the time and the output is the drift term.
D.5.3 Results
We present additional visualizations to demonstrate FlowDAS's ability of capturing unknown system dynamics on this weather forecasting task in Figures S.15 to S.17. FlowDAS successfully generates accurate estimates of the weather in the future frames in an autoregressive manner. Table 3 provides numerical results of FlowDAS as well as other baseline methods. FlowDAS consistently outperformed DA baselines both in terms RMSE and CSI metries.
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Gondition
Observation
True State
FlowDAS
c+ 10
+ 20
30
2++ 40
È C+
+ 60
Figure S.15: Additional result Data assimilation of weather forecasting Sparss-OOserrattion task). The underlying PDE of the dynamical system is unknown. This experiment was conducted at an observation sparsity level of 10%, showing the capacity of gongtrraecctor forecasting.
Condition
Observation
True State
FloWDAS
++
++
2+330 30
t+ 40
++
60
Figure S.16: Additional result 2: Data assimilation of weather forecasting (Sparse- Observvtinn task). The underlying PDE of the dynamical system is unknown. This experiment was conducted at an observation sparsity level of 10%, showing the capacity of long-trajectory forecasting.
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Condition
Observation
True State
FlowDAS
++
Figure S.17: Additional result 3: Data assimilation of weather forecasting (Sparse-Observalee task). The underlying PDE of the dynamical system is unknown. This experiment was conducted at an observation sparsity level of 10%, showing the capacity of long-trajectory forecasting.
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Reviews:
[{"review_id": "zHxHshpIgf", "reviewer": "Reviewer_Qtzs", "summary": "The paper presents a data assimilation framework based on a stochastic interpolator introduced in reference [30], used to model probabilistic transitions between state vectors across time steps. The method considers short-step conditional transitions, rather than learning a global noise-to-data map. Observations are incorporated at each interpolation step through a drift correction term. Without observations it reduces to autoregressive probabilistic forecasting. It is evaluated on benchmarks Lorenz-63, SEVIR, Navier–Stokes, and a weather forecasting task.", "questions": "The paper addresses an important problem and presents an interesting application of stochastic interpolators to data assimilation. However, the methodological contribution appears incremental. The framework is similar to existing approaches, as it extends a data assimilation approach that employs score-based diffusion models of reference [28] and stochastic interpolators in reference [31]. The main distinction seems to be the use of stochastic interpolators instead of posterior sampling diffusion models for modeling short-step conditional transitions, but the contribution appears somewhat incremental.\n\nThe overall presentation of the method could be more streamlined. The proposed method in Section 3 is presented based on modifications of components of the stochastic interpolator in [31] to incorporate observation conditioning. However, the presentation focuses heavily on these adaptations, rather than a self-contained method. \n\nThe implementation details in Section 3.2 are short, with details only in Appendix B. This makes the inference steps difficult to understand from the main paper.\n\nThe inconsistent use of uppercase and lowercase symbols in the notation makes the derivations more difficult to follow.\n\nThe paper would benefit from a more thorough discussion of computational efficiency. Runtime, scalability, and comparisons to baseline methods are not sufficiently addressed.", "limitations": "There are no significant potential negative societal impact.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- The paper addresses an interesting and timely problem, with applications to high-impact domains such as weather forecasting.\n\nWeaknesses:\n- The contribution appears somewhat incremental, as it builds on similar existing work on stochastic interpolators and diffusion models applied to data assimilation.\n- The inference procedure is not presented very clearly, and some parts are presented as an extension of stochastic interpolators.", "was_revised": true, "final_justification": "After reading the reviews and the responses, I appreciate the additional comparisons in terms of running time, the comparisons with traditional algorithms such as EnKF, and the clarifications. However, the contribution still appears somewhat incremental compared to works such as SDA and PDEDiff. I am raising my score accordingly to 4.", "rebuttal": "First of all, we sincerely thank the reviewer for their valuable feedback and constructive suggestions, which have helped us improve the clarity, accuracy, and completeness of the paper! \n\n&nbsp;\n\n**A. “The contribution appears somewhat incremental.”**\n\nResponse: \n\nWe would like to highlight the novelty and significance of our contributions. FlowDAS is **not a simple application of existing stochastic interpolants or diffusion models**, but a principled and technically novel framework for solving the **scientifically important problem of data assimilation (DA)**—which differs fundamentally from pure forecasting.\n\n- **Data assimilation is harder and more scientifically meaningful than pure forecasting.** Unlike forecasting, which predicts future states given the present, data assimilation requires reconstructing a latent trajectory given *partial and noisy observations over time*. This is critical in real-world systems such as weather prediction and oceanography, etc. It demands algorithms that are *measurement-consistent*, temporally coherent, and robust under sparse observations, that FlowDAS directly addresses.\n\n- **Beyond existing stochastic interpolant (SI) methods.** Prior SI methods (e.g., [31]) focus exclusively on probabilistic forecasting and cannot incorporate observations. FlowDAS makes the first extension of SI to the DA setting by introducing:  \n  (1) A **likelihood-based drift correction** (Eq. 7) that enables measurement conditioning during rollout;  \n  (2) An **unbiased estimator for the gradient of the observation likelihood** (Eq. 8); and  \n  (3) A **Monte Carlo sampling accelerator using stochastic integrators** (Eqs. 9–10) for efficient inference.\n  (4) A **flexible backbone architecture**, where using neural operators (e.g., FNO) instead of UNet yields even better results (see Table S.11, to be moved into main text).\n\n  These advances together enable **observation-conditioned generation**, which has not been addressed in prior SI literature.\n\n\n- **Advantages over diffusion-based approaches.** FlowDAS offers key improvements over diffusion-based methods like SDA and PDEDiff [1]. **SDA** models the dynamics via a joint score over entire trajectories, which limits temporal resolution and interpretability. **PDEDiff** improves on this by using conditional scores for state transitions, making it closer in spirit to FlowDAS. However, both methods require **sampling from Gaussian noise**, which leads to instability (see our toy double-well experiment in Appendix D.3), reduced physical interpretability, and degraded performance as shown in all experiments. In contrast, FlowDAS performs direct **state-to-state transitions** with an explicit transition density $p(x\\_{k+1} \\mid x\\_k, y\\_{k+1})$, enabling more stable, efficient, and physically grounded inference.\n\nTogether, these contributions establish FlowDAS as the **first generative data assimilation framework based on stochastic interpolants**. It represents a substantial and necessary step forward for building physically grounded, observation-aware generative models for scientific applications.\n\n\n&nbsp;\n\n\n**B. “The inference procedure is not presented very clearly.”**\n\nThank you for this valuable feedback. We agree that a more detailed explanation of the inference procedure would enhance the clarity of the main paper. While we had placed a full, detailed breakdown in Algorithm 2 (Appendix B) due to initial space limitations, we will incorporate it directly and insert the following description into Section 3.2 in the revision:\n\n  **Inference** Given a trained drift model $\\hat{b}_s$, FlowDAS performs inference in an autoregressive manner to estimate the latent state trajectory $ \\{ \\hat{x}\\_{1:K} \\}$. At each time step $k$, we begin from the known state $\\hat{x}_k$ and predict the next state $\\hat{x}\\_{k+1}$ using a discretized stochastic interpolant over a time grid $s_0 = 0 < s_1 < \\cdots < s_N = 1$. We set the interpolant state $X\\_{s\\_0} \\gets \\hat{x}\\_k $ and retrieve the next observation $ y\\_{k+1} $. For each interpolation step $ s\\_n$, we simulate a forward transition using the learned drift, given in Eq.7, and noise. To enforce observation consistency, we generate $J$ posterior endpoint samples $ \\hat{X}\\_1^{(j)}$, and compute their likelihoods under the measurement model, i.e., $ || y - \\mathcal{A}(\\hat{X}\\_1^{(j)}) ||_2^2 $. A softmax over these values gives importance weights $ \\{ w\\_{1:J} \\}$. We then apply a correction to the interpolant using a weighted gradient $- \\zeta\\_n \\nabla\\_{X\\_{s\\_n}} \\sum \\_{j=1}^J w\\_j || y - \\mathcal{A}(\\hat{X}\\_1^{(j)}) ||_2^2$, where $\\zeta\\_n$ denotes the step size. After reaching $s=1$, we set $ \\hat{x}\\_{k+1} \\gets X\\_{s\\_N} $ and proceed to the next time step. Repeating this procedure over all $ k $ yields the full trajectory. \n\n&nbsp;\n\n**C. “The presentation focuses heavily on these adaptations, rather than a self-contained method.”**\n\nWe would like to hightlight that **data assimilation naturally consists of two essential components**: a probabilistic forecasting model and an observation-conditioned correction mechanism. Prior work such as [31] addresses only the forecasting part. To clearly situate our contribution, we chose to introduce that work early on, but this does not mean FlowDAS is not self-contained.\n\nFlowDAS is a complete generative DA method. In the final version, we will revise Section 3 to present FlowDAS more coherently as a standalone algorithm, and move contextual references to prior work into a separate discussion in the Section 2 to improve readability.\n\n&nbsp;\n\n**D. “The inconsistent use of uppercase and lowercase symbols in the notation makes the derivations more difficult to follow.”**\n\nOur notation intentionally distinguishes between random variables and their realizations:  \n- Uppercase letters (e.g., $X, I, R$) denote random variables or stochastic processes.  \n- Lowercase letters (e.g., $x, y$) represent specific realizations or observations.\n\nFor example, $X\\_0, X\\_1 \\sim p(X\\_0, X\\_1)$ are random variables, while $x\\_0, x\\_1$ are concrete samples. This convention is common in probabilistic modeling and used consistently throughout. In the final version, we will add an explicit note at the start of Section 2 to make this distinction clear to readers.\n\n\n&nbsp;\n\n**E. \"The paper would benefit from a more thorough discussion of computational efficiency. Runtime, scalability, and comparisons to baseline methods are not sufficiently addressed.\"**\n\nWe agree with the reviewer on the importance of computational efficiency and have added a detailed performance comparison in the appendix. The results, using a single A100 GPU on the SEVIR dataset, highlight FlowDAS’s practical viability:\n\n| Model       | Params (M) | Total Training Time (hr) | Inference Time per Sample (s) |\n|:-----------:|:----------:|:-------------------------:|:------------------------------:|\n| FlowDAS     | 30.0       | 22                        | 30.9                           |\n| SDA         | 22.9       | 45                        | 28.9                           |\n| Transolver  | 11.2       | 80                        | 12.7                           |\n| FNO         | 1880.0     | 80                        | 2.4                            |\n\nThe slightly longer inference time for FlowDAS stems from its iterative correction process, which is essential for achieving accurate and observation-consistent predictions. While this leads to higher per-sample inference time, it is balanced by significant **training efficiency**, as FlowDAS reaches convergence much faster than all competing baselines. We believe this reflects a favorable trade-off in practical deployment scenarios. A more detailed discussion will be included in the final version.\n\nTo further improve inference speed, one promising direction is to apply **post-hoc distillation**, where a faster surrogate model is trained to approximate the FlowDAS sampler. Another avenue is to explore hybrid formulations that **integrate variational inference** [2] (as suggested by Reviewer 3gVf) techniques into the framework, potentially offering both computational gains and improved flexibility.\n\n&nbsp;\n\n## Reference\n[1] Shysheya, A., Diaconu, C., Bergamin, F., Perdikaris, P., Hernández-Lobato, J. M., Turner, R., & Mathieu, E. (2024). On conditional diffusion models for PDE simulations. Advances in Neural Information Processing Systems, 37, 23246-23300.\n\n[2] Morteza Mardani, Jiaming Song, Jan Kautz, and Arash Vahdat. A variational perspective on solving inverse problems with diffusion models. arXiv preprint arXiv:2305.04391, 2023."}, {"review_id": "z4RoMu3zBI", "reviewer": "Reviewer_3gVf", "summary": "FlowDAS stochastic interpolants to improve upon the Score-based Data Assimilation Model by Rozet. Stochastic interpolants: mapping from noise$ + x_0 \\rightarrow x_1$. One progresses the dynamics from $x_t$ to $x_{t+1}$ by learning the score for $P(x_{t+1} | x_t)$. At inference time the drift is augmented by a Bayes-term so every Euler–Maruyama step is nudged toward the incoming observation, giving trajectories that are “analysis-consistent” as they are generated .", "questions": "Could variational inference over the interpolant path yield accurate density spread equivalent to the EnKF, or perhaps a better density estimation procedure?\n\nHow does performance degrade as the observations become very sparse (for example, lower than 1%)? Would localization need to be incorporated into the method at any point?", "limitations": "Yes.", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths\n\nThe data assimilation step is already integrated into the prediction step. As a result, there is no need to separate out the prediction and analysis step as in standard data assimilation. However, this comes with the drawback that it makes this model harder to integrate into existing predictive architectures for real-world problems, since this model is an all-in-one data assimilation solution. Particularly since current predictive weather models are done at a very fine resolution, then FlowDAS would need to scale well in order for the predictive model to also compete.\n\nTraining is stable and interpretable, in contrast to global noise-to-data diffusion chains by taking advantage of stochastic interpolants. The progression of the stochastic interpolants makese it so that the model starts off at the first point with no noise, introduces stochasticity in the middle, and then results in the target.\n\nFlowDAS offers significantly improved results over SDA, which has a similar framework but only aims to generate a state trajectory conditioned on the observations. In addition, this opens up FlowDAS to be well integrated with the autoregressive data assimilation framework compared to the full trajectory generation in SDA.\n\nThe use of stochastic interpolants in data assimilation has not been explored before, making this paper interesting as it takes the first steps towards what seems to be a promising avenue of exploration.\n\nWeaknesses\n\nStochastic integrators introduce bias; accuracy–speed trade-offs under very stiff dynamics remain to be quantified.\n\nWhile each step is stochastic, the framework still produces a single trajectory unless multiple MC traces are run which makes uncertainty quantification difficult.\n\nSummary\n\nGiven its multitude of strengths, I am leaning towards a strong accept for this paper. I feel like this is a strong technical paper with good results whose approach would be interesting to the NeurIPS community.", "was_revised": true, "final_justification": "After the rebuttal process, I decided to maintain my score as a 6 (Strong accept). The paper is theoretically interesting by bringing in stochastic interpolants into the data assimilation process, and should open up further avenues of discussion for score-based approaches for data assimilation.", "rebuttal": "First of all, we sincerely thank the reviewer for the positive evaluation and encouraging comments, as well as for the insightful suggestions that helped us further improve the paper! :)\n\n## Weakness\n**A. “Stochastic integrators introduce bias; accuracy-speed trade-offs under very stiff dynamics ...”**\n\n1). **Accuracy-speed trade-offs** Indeed, the stochastic integrators in Equations (9) and (10) trade exactness for efficiency and introduce numerical bias. To evaluate this trade-off, we compared three settings: first-order integrator, second-order integrator, and a hybrid method that computes the full stochastic interpolant path to the endpoint $X\\_1$ for the final 100 steps (“2nd order × 400 + Flow × 100”). As shown in the table (now included in Appendix D), **the full-path method increases computation time by 7× and peak memory usage by 18× compared to the second-order integrator, while improving RMSE by only ~0.1%.**\n\nWe initially considered the full Flow*500 method, i.e., computing $X\\_1$ exactly at every step by fully rolling out the leftward stochastic interpolant, but this would require about 500+(500+499+...+1) $\\approx$ 250,000 forward passes and retaining the entire computation graph at each step. This would cause out-of-memory failures even on A100 GPUs. This substantial cost arises because computing the full-path endpoint requires storing the entire interpolation path to backpropagate $\\log p(y|X\\_1)$. In contrast, the 1st- and 2nd-order methods yield stable, efficient inference while maintaining competitive accuracy.\n\nWe believe this experiment validates our design choice: the benefit of observation conditioning significantly outweighs the small bias introduced by the integrator. We will clarify this point in the final version and include the trade-off analysis and table in the appendix.\n\n\n|             Method              | Computation Time |      RMSE       | Max Memory Used |\n|:------------------------------:|:----------------:|:---------------:|:----------------:|\n|         1st order              |     17.100 s     | 0.0490 ± 0.0026   |     1.8 G     |\n|         2nd order              |     30.871 s     | 0.0485 ± 0.0035   |     3.9 G     |\n| 2nd order × 400 + Flow × 100   |    205.271 s     | 0.0478 ± 0.0033   |    70.0 G     |\n\n\n2). **More stiff dynamics** In our experiments on the Navier–Stokes system, known for its stiffness and chaotic behavior, we have observed stable rollout behavior and robust error control. While we do not explicitly tailor our integrators for stiffness (e.g., via implicit solvers or adaptive step size control), the **short-step nature of the interpolant and the sequential correction mechanism naturally mitigate stiffness-related instabilities**.\n \nIn addition, we are currently extending FlowDAS to a real-world ocean dynamics application, where the governing equations follow the **Quasi-Geostrophic (QG) model**. These systems exhibit more complex behavior, such as inverse energy cascades and long-range spatial correlations, making them even more challenging than standard Navier–Stokes in certain regimes. Our preliminary results indicate that FlowDAS still performs well under this setting, though further gains are likely possible through domain-specific strategies tailored to QG-type dynamics. We view this as a promising direction for advancing the applicability of FlowDAS to operational-scale geophysical systems.\n\n**B. “Difficult uncertainty quantification...”**\n\nWhile FlowDAS generates a single trajectory per sampling run, its stochastic formulation naturally supports uncertainty quantification via ensemble sampling. In our weather forecasting task, we already report ensemble standard deviation as a measure of uncertainty (see Table S.10). For the Navier–Stokes experiments, we have similarly conducted Monte Carlo sampling to quantify uncertainty and will include the ensemble standard deviation in the updated version of the paper. This is done by running multiple inference trajectories under the same observation setting and computing the spread of predictions. As shown in the table below, the uncertainty increases as the observation becomes sparser or lower in resolution, which aligns with expectations.\n\n|       RMSE ± Std        |   SR 4x     |  SR 8x     |  SO 5%     | SO 1.5625%  |\n|:-----------------------:|:-----------:|:----------:|:----------:|:-----------:|\n|                         | 0.038 ± 0.001 | 0.067 ± 0.050 | 0.071 ± 0.005 | 0.123 ± 0.010 |\n\nThis ensemble-based strategy provides a practical and interpretable way to assess uncertainty within the FlowDAS framework.\n\n&nbsp;\n\n## Questions\n**A. “Could variational inference over the interpolant path yield accurate density spread...”**\n\nThank you for this insightful question. Since FlowDAS is based on MCMC-style sampling, it is natural to ask whether variational inference (VI), a widely used alternative in inverse problems and posterior approximation, could provide comparable or even improved density estimates. We address this in three parts:\n\n- **Can VI be integrated into the FlowDAS framework?**  \n  **Yes.** Following RED-Diff [1], we can construct a RED-FlowDAS variant. All theoretical components generalize naturally to the stochastic interpolant setting (see our brief derivation).\n\n- **How does VI compare to EnKF and FlowDAS?**  \n  * Compared to **EnKF**, VI is more flexible: it accommodates general likelihoods (not limited to Gaussian noise) and models complex state transitions via KL divergence minimization, which goes beyond EnKF’s linear assumptions.  \n  * Compared to **FlowDAS**, VI provides an explicit density estimate, while FlowDAS samples from the posterior and thus models the posterior distribution indirectly. However, MCMC may better explore complex multi-modal posteriors, whereas VI can be mode-seeking due to the nature of KL divergence.\n\n- **Additional benefits of VI:**  \n  VI avoids the need for conditional score estimation, can be faster than MCMC, and is amenable to stochastic and distributed optimization. These advantages make it a promising direction for extending FlowDAS.\n  \n\n**B. “Performance degradation under challenging cases & Localization?”**\n\n1). **Performance degradation under extremely challenging cases** We have conducted additional experiments using super-resolution (SR) and sparse-observation (SO) settings with as little as **0.75%** coverage. The results, shown below, indicate a **graceful degradation** of performance as the amount of observed information decreases. This supports the robustness of FlowDAS even under highly limited measurement scenarios.\n\n|     RMSE ± Std     |   SR 16x     |   SO 1%     |  SO 0.75%   |\n|:------------------:|:------------:|:-----------:|:-----------:|\n|                    | 0.108 ± 0.010 | 0.148 ± 0.014 | 0.177 ± 0.019 |\n\nThese results will be included in the updated appendix.\n\n2). **Localization** Unlike ensemble-based filters such as EnKF, which rely on covariance estimation and therefore require localization to suppress spurious long-range correlations, **FlowDAS does not estimate or use a background covariance matrix**. Instead, it performs observation-conditioned updates directly through a likelihood-based correction at each step, guided by the measurement operator A. This naturally confines the influence of observations to their spatial support, eliminating the need for explicit localization.\n\n&nbsp;\n\n## (**Optional**) Brief Derivation for VI-FlowDAS\nFor completeness, we include a brief derivation showing how variational inference can be integrated into FlowDAS.\n\nWe could format the data assimilation question in a VI settings and that is we want to fit a family of densities $q({X}_1 \\mid {X}_0,{Y})$ to a intractable posterior distribution $P({X}_1 \\mid {X}_0,{Y})$, where the ${X}_1$ is the state variable we want to predict, and ${X}_0$ is the current state and ${Y}$ is the observation on ${X}_1$. And the variation method is based on  KL minimization\n\n$\n    \\min_q KL\\big(q({X}_1 \\mid {X}_0,{Y}) || P({X}_1 \\mid {X}_0,{Y}))    \n$\n\nwhere $q := \\mathcal{N}(\\mu, \\sigma^2 I)$ is a variational distribution. And this objective can be expand:\n\n$\nKL\\big(q({X}\\_1 \\mid {X}\\_0,{Y}) \\| P({X}\\_1 \\mid {X}\\_0,{Y})\\big) = -{E}\\_{q({X}\\_1 \\mid {X}\\_0,{Y})}\\big[\\log P({Y} \\mid {X}\\_1,{X}\\_0)\\big]\\+KL\\big(q({X}\\_1 \\mid {X}\\_0,{Y})\\| p({X}\\_1 \\mid {X}\\_0)\\big) + \\log p({Y},{X}\\_0) \n$\n\nThe last term is the observation likelihood that is **constant** w.r.t.~$q$. Thus, to minimize the KL divergence, it suffices to minimize the variational bound (omit the last term). This brings us to the next claim.\n\n${Proposition 1}$. The KL minimization is equivalent to minimizing the variational bound, that itself obeys the **velocity matching loss** in the stochastic interpolants framework: \n\n$\n\\min\\_{\\{\\mu, \\sigma\\}}-{E}\\_{q({X}\\_1 \\mid {X}\\_0,{Y})}\\left[\\log P({Y} \\mid {X}\\_1,{X}\\_0)\\right] +\\int_{0}^{1} \\frac{1}{2\\sigma\\_s^{2}} {E}\\_{q({X}\\_s \\mid {X}\\_0,{Y})} \\Big[\\big\\|\\hat{b}^{q}\\_s({X}\\_s, {X}\\_0) - \\hat{b}^{p}\\_s({X}\\_s, {X}\\_0) \\big\\|^2_2 \\Big] ds\n$\n\n where $q({X}_s | {X}_0,{Y}) = \\mathcal{N}( \\alpha_s {X}_0 + \\beta_s \\mu, (\\beta_s^2 \\sigma^2 + \\sigma_s^2 s)I)$ and $\\hat{b}^{q}_s({X}_s, {X}_0)$ can be explicitly computed:\n\n$\n\\hat{b}^{q}_s({X}_s, {X}_0) = \\dot{\\alpha}_s{X}_0 + \\dot{\\beta}_s{X}_1^{q}+\\dot{\\sigma}_s{W}_s \n$\n\nwhere ${X}_1^{q}$ is produced by sampling from $q({X}_1 \\mid {X}_0,{Y}) := \\mathcal{N}(\\mu, \\sigma^2 I)$. \n\nFor the $\\hat{b}^{p}_s({X}_s, {X}_0)$, this is exactly the stochastic model we trained.\n\n&nbsp;\n\n## Reference\n\n[1]. Morteza Mardani, Jiaming Song, Jan Kautz, and Arash Vahdat. A variational perspective on solving inverse problems with diffusion models. arXiv preprint arXiv:2305.04391, 2023."}, {"review_id": "bh5iGjNqAn", "reviewer": "Reviewer_Ztob", "summary": "This paper proposes FlowDAS, a method to perform data assimilation using the stochastic interpolants generative modelling framework. More specifically, the model learns the state transition dynamics and, at generation time, incorporates a “correction” term that guides the generated states towards the observations, and is approximated through a Monte Carlo sampling procedure. Experiments are performed on a range of dynamical systems ranging from Lorenz-63 to Navier-Stokes, and a real-life precipitation dataset, where FlowDAS shows promising performance.", "questions": "1. What do the authors mean that FlowDAS “provides clearer physical insight into one-step dynamics” / “training … provides clearer physical insight into one-step dynamics”? Concretely, what becomes clearer in the stochastic interpolant framework versus, for example, diffusion models?\n2. Could the authors clarify how expensive the method is at sampling time? Could they also compare the training costs of different architectures?\n3. For the SDA baseline, the authors of the paper mention that employing one or a few steps of Langevin sampling (after the predictor step) are very important for good performance (section 3.3: \" In practice, we find that few LMC steps are necessary.\"). Is this something the authors have also employed in the experiments?\n4. Could the authors comment more on the limitations of their approach?\n5. Could the authors include a discussion of the related work of [4]?", "limitations": "It is unclear how costly the framework is at sampling time. I would imagine it should be less costly than diffusion-based formulations, but a clarification would be helpful.\n\nDue to the inclusion of several previous time steps (10 previous states in the Navier-Stokes experiment), the method might have large memory requirements. These might not be prohibitive in the settings considered in the paper, but might become too restrictive to be used in, for example, medium-range weather modelling.\n\nThe conditioning approach might also become costly if the likelihood $p(\\textbf{y} | \\mathbf{X}_1^{(j)})$ is costly to evaluate (not the case in here).", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n1. **Extensive and varied experiments** -  The experimental section is comprehensive, covering a wide range of dynamical systems, with increasing complexity. The paper also provides additional experiments in the appendix.\n2. **Clear, detailed Methodology** - The authors are comprehensive and clear in their description of the methodology.\n3. **Thorough Experimental Design Documentation** - Both the main text and appendix specify all implementation details (network architectures, conditioning scenarios, data-generation procedures, etc.), ensuring a good level of transparency.\n4. **Ability to handle non-linear observation operators + non-Gaussian noise** - This is something that is tricky to achieve for several posterior estimation techniques in the diffusion model literature, whereas, as far as I understand, this should not be a problem in FlowDAS.\n\n**Weaknesses**\n1. **Inappropriate SDA baseline configuration for tasks involving forecasting**\n   - **SDA’s original scope**: The chosen baseline (Score‐based Data Assimilation, SDA) was designed for all-at-once conditional generation given observations throughout a trajectory. It is not optimised for forecasting, where future observations are unavailable.\n   - **Follow-up adaptation to forecasting**: Indeed, this observation lies at the basis of follow-up work by Shysheya et al. [1], where the authors show that SDA's joint score formulation fails when the task under consideration has a forecasting component, and proposes three autoregressive strategies to deal with this:\n     - **Guidance conditioning** - condition on previously-generated states through the same mechanism as the one used for conditioning upon observations (guidance), which does not require any retraining\n      - **Amortised approach** - a new **conditional** score network is trained to predict the next state(s) conditioned on a number of previous states\n      - **Universal amortised approach** - similar to b), but the authors vary the conditioning scenario during training to allow for more flexibility at sampling time. \n   - **Unclear SDA implementation**: The manuscript does not specify which strategy it uses to incorporate previous states into SDA (and I am assuming it is similar to one of the choices presented in Shysheya et al. [1]]): \n     - If via guidance, then the literature of diffusion models clearly shows that conditioning through the architecture is much more effective than through guidance, so this baseline would not be entirely fair.\n     - If via amortisation, that’s a better setup, but in the light of [1] a stronger baseline would be represented by the universal amortised approach. This is because as proven in [1], for the plain amortised approach longer history (e.g., in the NS experiment a history of 10 states) tends to hurt performance. This was also observed in [2].\n   - **Recommendation**: A more appropriate baseline for experiments including a forecasting component (e.g. Lorenz-63, weather forecasting on SEVIR) would be, in my opinion, the universal amortised version of [1], not the original all-at-once SDA formulation.\n    - **Joint/Conditional score** - Related to this, the authors claim throughout the manuscript that SDA models the joint score. This is true in the original formulation, but in their experimental setup, either by feeding in the previous states as conditioning information, or by conditioning through guidance, the SDA score also becomes a conditional one.\n2. **Lack of computational cost analysis**\n   - For sampling, I could not find the value used for $N$, the length of the discretised interpolation interval.\n   - The network architectures for the baselines seem to be different, so it would help to understand how that affect the computational cost during training.\n3. **Ablation on conditioning horizon** - The authors claim that they observed “markedly improved performance on weather-forecasting tasks” by conditioning on multiple previous states. This is an interesting insight, also discussed to some extent in [1], and performed in SOTA weather forecasting models such as GenCast [3], where they condition on the previous two states. However, the authors do not compare to conditioning on fewer steps / one step to clearly show that the performance is \"markedly improved\".\n4. **Missing limitations discussion** - The authors mention in the paper checklist that the limitations are discussed in the appendix, but I could not find them clearly stated.\n5. **Incomplete coverage of a few related methods**\n   - Given that some tasks have a strong forecasting component, I would have expected better coverage of the related paper [4]. The experimental setup from NS is taken from there, and the two frameworks are clearly very related (with FlowDAS also incorporating the DA component).\n   - The claim that “diffusion models’ non-autoregressive nature makes it hard to deal with long sequence rolling-out problems in high-dimension complex dynamics” overlooks successful high-dimensional autoregressive diffusion methods (e.g. GenCast [3]) and should be revised.\n6. **Typographical and notational errors**\n   - Eq S.20 - Should denote the conditional score $\\nabla \\text{log} p (\\mathbf{X}_s|\\mathbf{X}_0)$, not the joint score $\\nabla \\text{log} p (\\mathbf{X}_s, \\mathbf{X}_0)$;\n   - Abstract: Phrase “A good modelling”;\n   - L40: Missing verb - “Existing DA methods **are** split into two categories.”\n\n**References**\n\n[1] Shysheya, A., Diaconu, C., Bergamin, F., Perdikaris, P., Hernández-Lobato, J. M., Turner, R., & Mathieu, E. (2024). On conditional diffusion models for PDE simulations. Advances in Neural Information Processing Systems, 37, 23246-23300.\n\n[2] Lippe, P., Veeling, B., Perdikaris, P., Turner, R., & Brandstetter, J. (2023). Pde-refiner: Achieving accurate long rollouts with neural pde solvers. Advances in Neural Information Processing Systems, 36, 67398-67433.\n\n[3] Price, I., Sanchez-Gonzalez, A., Alet, F., Andersson, T. R., El-Kadi, A., Masters, D., ... & Willson, M. (2023). Gencast: Diffusion-based ensemble forecasting for medium-range weather. arXiv preprint arXiv:2312.15796.\n\n[4] Chen, Y., Goldstein, M., Hua, M., Albergo, M. S., Boffi, N. M., & Vanden-Eijnden, E. (2024). Probabilistic Forecasting with Stochastic Interpolants and F\\\" ollmer Processes. arXiv preprint arXiv:2403.13724.", "was_revised": true, "final_justification": "After the rebuttal, I decided to maintain my rating of 4 (Borderline Accept). Several concerns were addressed, including clarifications on baseline implementations, additional details on computational cost, ablations for the conditioning scenario, and a commitment to expand the discussion of limitations. However, I did not increase my score because some contributions remain incremental relative to the baselines. Furthermore, the proposed framework appears to require substantial computational resources, both in terms of memory and inference time, due to the large number of steps needed to preserve sample diversity. While the authors suggest that operating in latent space could mitigate these costs, learning a sufficiently compressed latent representation that still retains the necessary information is itself a non-trivial challenge.", "rebuttal": "We thank the reviewer for their detailed, balanced and constructive feedback, and we are encouraged by their recognition of the potential of our work.\n\n## Weaknesses:\n**A. “Inappropriate SDA baseline configuration for tasks involving forecasting.”**\n\nThe SDA results reported in our paper for Navier-Stokes and SEVIR are already based on the universal amortised SDA variant proposed in [1]. The model is conditioned on previous states (P=10 for NS, P=6 for SEVIR) with varying conditioning scenarios during training. We will rename this baseline to **PDEDiff** for clarity.\n\nFor Lorenz-63, we originally used the naive joint-score SDA. We have now added the PDEDiff (P=1, C=1 as FlowDAS) to this setting as well.\n\nWe will revise the manuscript to clarify the SDA configurations used and correct any references to joint vs conditional score formulations. Notably, FlowDAS achieves the best performance across most metrics, even compared to BPF, which has access to the exact forward model. While PDEDiff improves upon SDA by modeling conditional scores, it still falls a bit short of FlowDAS.\n\n| Metric                          | FlowDAS | SDA     | PDEDiff | BPF    |\n|:-------------------------------|:-------:|:-------:|:-------:|:------:|\n| $\\log p(x\\_{2:K} \\| x\\_1)$           | 17.29   | -332.7  | 8.156   | 17.88  |\n| $\\log p(y \\| x\\_{1:K})$             | -0.228  | -6.112  | -1.977  | -1.572 |\n| W1 (Trajectory)                | 0.106   | 0.528   | 0.516   | 0.812  |\n| RMSE                           | 0.202   | 1.114   | 0.871   | 0.270  |\n\n**B. “Lack of computational cost analysis”**\nResponse:\n1). Regarding the Discretization steps, our stochastic interpolant uses N = 500 time steps during sampling. We will add this to the implementation details section.\n2). We have now added quantitative runtime and parameter comparisons (for the weather forecasting task using a single A100 GPU) to the appendix. \n\n| Model       | Params (M) | Total Training Time (hr) | Inference Time per Sample (s) |\n|:-----------:|:----------:|:-------------------------:|:------------------------------:|\n| FlowDAS     | 30.0       | 22                        | 30.9                           |\n| SDA         | 22.9       | 45                        | 28.9                           |\n| Transolver  | 11.2       | 80                        | 12.7                           |\n| FNO         | 1880.0     | 80                        | 2.4                            |\n\nThe slightly longer inference time for FlowDAS stems from its iterative correction process, which is essential for achieving accurate and observation-consistent predictions. While this leads to higher per-sample inference time, it is balanced by significant **training efficiency**, as FlowDAS reaches convergence much faster than all competing baselines. We believe this reflects a favorable trade-off in practical deployment scenarios. A more detailed discussion will be included in the final version.\n\nTo further improve inference speed, one promising direction is to apply **post-hoc distillation**, where a faster surrogate model is trained to approximate the FlowDAS sampler. Another avenue is to explore hybrid formulations that **integrate variational inference** [5] techniques into the framework, potentially offering both computational gains and improved flexibility.\n\n**C. “Ablation on conditioning horizon”**\n\nTo assess the impact of the conditioning horizon, we conducted a controlled ablation using the CRPS metric on both Navier–Stokes (NS) and SEVIR experiments by varying the number of observed previous steps used during inference. The results are summarized below:\n\n* **Navier–Stokes CRPS vs. Conditioning Horizon**\n\n| #Conditioning States | 10    | 6     | 3     | 1     |\n|:----------------:|:-----:|:-----:|:-----:|:-----:|\n| CRPS             | 0.538 | 0.634 | 0.663 | 0.776 |\n\n* **SEVIR CRPS vs. Conditioning Horizon**\n\n| #Conditioning States | 6     | 3     | 1     |\n|:----------------:|:-----:|:-----:|:-----:|\n| CRPS             | 0.015 | 0.021 | 0.021 |\n\nThese results clearly show that incorporating a longer observation history leads to consistent and significant improvements in forecasting performance. In particular, we observe a **~30% CRPS reduction in NS** when increasing the conditioning length from 1 to 10 steps. Similarly, in SEVIR, although the absolute values are small due to the scale of the task, conditioning on 6 previous frames led to the best predictive accuracy. This supports our claim that incorporating temporal context is important for accurate data assimilation, especially in high-dimensional forecasting tasks.\n\n\n**D. “Missing limitations discussion”**\n\nTo improve clarity, we will include an explicit **Limitations and Future Work** section in the revised manuscript.\n\n> **Limitations and Future Work**\n\n> FlowDAS has so far been validated on controlled dynamical systems (e.g., Lorenz-63, Navier–Stokes), and its generalizability to more complex, real-world settings such as numerical weather prediction remains to be explored. In particular, we have not yet studied its performance under Sim2Real conditions or applied it to operational-scale, high-dimensional systems like those addressed by GEN_BE. \n\n> Although FlowDAS supports probabilistic inference via sampling, the current sampling speed remains a limitation. One possible solution is to apply post-hoc distillation, where a faster surrogate model is trained to mimic the FlowDAS sampler. Exploring hybrid formulations that incorporate variational inference techniques could be another promising direction to improve efficiency. \n\n**E. Incomplete coverage of a few related methods**\n\n1. (**Better coverage of [4]**) The reviewer is correct that our Navier-Stokes experiment follows the setup established in [4], and we will certainly add this citation in the relevant section.\nTo further clarify the context of our work, we will also expand our discussion in Related Work section to explicitly describe how FlowDAS builds upon and extends such autoregressive forecasting approaches by incorporating an observation-driven data assimilation component. To reflect this, we plan to revise the paragraph talking about ‘Diffusion model-based methods’ in Section 2 into ‘Generative model-based methods’ and **insert the following paragraph** discussing [4]:\n\n> A closely related line of work is the stochastic interpolant-based forecasting framework recently proposed in [4], which formulates the prediction task as conditional sampling of future system states given the current state, using a fictitious, non-physical stochastic process governed by a learned SDE. While this approach enables probabilistic forecasting, it does not incorporate observation-driven corrections during inference. In contrast, FlowDAS augments the stochastic interpolant dynamics with a measurement-consistent correction term at each step, enabling integration of observational data and thus unifying generative modeling with data assimilation.\n\n2. (**non-autoregressive vs. autoregressive diffusion models**) We will revise the sentence in the related work section to reflect this more balanced view:\n\n> Additionally, non-autoregressive diffusion models may struggle with long sequence forecasting in high-dimensional systems, though autoregressive diffusion approaches, like GenCast [3], have shown promising results in such settings.\n\n**F. Typographical and notational errors**\n\nWe corrected the notation errors. These edits will be included in the final version.\n\n&nbsp;\n\n## Questions\n**A. “FlowDAS provides clearer physical insight into one-step dynamics?”**\n\nWe agree that the phrase 'provides clearer physical insight' could imply an unintended comparison. Our intention was simply to state that our method offers insights that are physically clear. To better reflect this, we will revise the phrasing to '**provides clear physical insight**' and ensure this meaning is unambiguous.\n\n**B. “For the SDA baseline, … a few steps of Langevin sampling …, is this something the authors have also employed in the experiments?”**\n\n**Yes**, we apply this predictor-corrector (PC) sampling settings proposed in the SDA baseline. Specifically, the predictor step is 512 and the corrector step is 3. As we empirically find, these settings show a good balance between the good performance and computation burden. The settings are kept across the Lorenz, NS, SEVIR experiments.\n\n&nbsp;\n\n## Limitations:\n**A. “Inclusion of several previous time steps -> large memory requirements”**\n\nWe agree that memory usage can grow with the number of input states, especially in high-dimensional settings such as medium-range weather forecasting. In fact, in our ongoing collaboration with an industry partner, we are applying FlowDAS to real-world meteorological data with shape (71, 181, 360), where 71 is the number of channels. This indeed is significantly higher than in our current Navier–Stokes and SEVIR experiments.\n\nTo address this, a promising direction we are actively exploring is lifting FlowDAS into a **latent space**, inspired by works such as SLAMS [6], which can substantially reduce computational and memory overhead while preserving dynamics. We will add this as a concrete point in the discussion of future work.\n\n**B. “Costly conditioning if the likelihood is costly to evaluate”**\n\nWe agree with the reviewer. In our current experiments, the likelihood evaluation is relatively cheap, so **this is not a bottleneck**. For settings where the likelihood is expensive to compute, developing approximated likelihood evaluations would be a promising direction for future work.\n\n&nbsp;\n\n## Reference\n[5] Morteza Mardani, Jiaming Song, Jan Kautz, and Arash Vahdat. A variational perspective on solving inverse problems with diffusion models. arXiv preprint arXiv:2305.04391, 2023.\n\n[6] ​​Qu, Yongquan, et al. \"Deep generative data assimilation in multimodal setting.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024."}, {"review_id": "FkKEihDte6", "reviewer": "Reviewer_CZwM", "summary": "The paper presents _FlowDAS_, a technique that leverages dynamics modeled by stochastic interpolants to perform data assimilation. Given observations, the method proceeds by augmenting the original drift vector field (learned by training a stochastic interpolant) with the likelihood score, approximated by Monte Carlo. Experiments on several data assimilation benchmarks demonstrate superior performance of FlowDAS to other ML-based data assimilation methods (such as score-based data assimilation) on several metrics evaluating accuracy of state estimation and uncertainty quantification.", "questions": "- What do the acronyms for FlowDAS stand for? I don't believe this is introduced.\n- In data assimilation, typically the objective is to compute the filtering distribution $p(x\\_{k+1} | y\\_1, \\ldots, y\\_{k+1})$. Instead, one step of FlowDAS targets sampling from the distribution $p(x\\_{k+1} | x\\_{k-l:k}, y\\_{k+1})$. This leads to a few questions:\n    - Can you please provide more details on how to roll out FlowDAS to assimilate data sequentially? Assuming Markovian dynamics for simplicity, given data $y_{k+1}$, FlowDAS essentially modifies the distribution $p(x\\_{k+1} | x\\_{k})$ to get $p(x\\_{k+1} | x\\_{k}, y\\_{k+1})$. Now suppose we get  samples $x\\_{k+1}^{(n)} \\sim p(x\\_{k+1} | x\\_{k}, y\\_{k+1})$ and a new observation $y_{k+2}$. Then, do we take samples from $p(x\\_{k+2} | x\\_{k+1}^{(n)}, y\\_{k+2})$ for all $n$? Thus effectively sampling from $\\int p(x_{k+2} | x_{k+1}, y_{k+2}) p(x\\_{k+1} | x\\_{k}, y\\_{k+1}) dx_{k+1}$?\n    - If so, it is still unclear to me how this distribution should be interpreted. If it is meant to be $p(x_{k+2} | x_{k}, y_{k+1}, y_{k+2}) = \\int p(x_{k+2} | x_{k+1}, y_{k+2}) p(x\\_{k+1} | x\\_{k}, y\\_{k+1}, y\\_{k+2}) dx_{k+1}$, then at the first step, we should be sampling from $p(x\\_{k+1} | x\\_{k}, y\\_{k+1}, y\\_{k+2}) $ instead of $p(x\\_{k+1} | x\\_{k}, y\\_{k+1})$. I wonder if this causes any problems.\n- I wonder why FlowDAS does not suffer from particle collapse when computing the likelihood score (8), which is a well-known issue with importance sampling in high dimensions (i.e. one of the weights $w\\_j$ become much more dominant compared to others, which leads to \"collapsing\" all the contribution to particle $\\hat{X}\\_1^{(j)}$).\n- I am confused by the use of the Wasserstein distance to evaluate the deviation of $\\hat{x}\\_{1:K}$ from the ground truth $x\\_{1:K}$. Is this measuring the distance between two delta measures, one centered at $\\hat{x}\\_{1:K}$ and the other at $x\\_{1:K}$? If so, what does this mean? It seems more natural to me to consider other uncertainty estimation metrics such as the log-likelihood $\\log p(x\\_{1:K} | \\hat{x}\\_{1:K})$ or CRPS.\n- I also don't understand the use of the metric $\\log p(\\hat{x}\\_{2:K} | \\hat{x}\\_1)$. This does not make a comparison with either the ground truth $x_{1:K}$ nor the observations $y$ so in what sense does it evaluate the quality of the predictions?", "limitations": "Limitations are not explicitly addressed in the work as far as I can see.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "__Strengths:__\n- The proposed method is simple and elegant; It is almost plug-and-play once the stochastic interpolant is trained, and there is no need for additional training to assimilate observations.\n- The results come naturally equipped with uncertainty quantification.\n- Experiments are extensive, especially in the supplementary materials, and show clear promise of the methodology.\n\n__Weaknesses:__\n- The writing feels rushed in places. There are several misprints peppered throughout the paper. For example:\n    - Line 147: \"they does not\" -> \"they do not\"\n    - Line 168: \"an Multi-Layer Perceptron\" -> \"a Multi-Layer Perceptron\".\n    - Caption of Table 2: \"Naiver-Stokes\" -> \"Navier-Stokes\".\n    - The sentence in Lines 69-70 \"struggling in discontinuity situations...\" feels rushed and can be improved. For example, what do the authors mean by \"discontinuity situations\".\n- To my understanding, FlowDAS does not sample from the filtering distribution $p(x_k | y_1, \\ldots, y_k)$, which is usually the target distribution of data assimilation. This raises several questions. See __Questions__ below for more details.\n- I do not understand the reasoning behind the choice of some of the UQ metrics. See __Questions__ below.\n- Comparisons with classical filtering baselines are largely missing. In particular, the benchmark problems considered are settings where traditional data assimilation methods such as the ensemble Kalman filter (EnKF) or its variants, such as LETKF, can be applied without problems. Hence, without comparisons to these baselines, it is unclear what merit the proposed method brings.\n- There are related methodologies proposed in the literature, such as EnSF (see [1]) and DiffDA (cited in work), that also consider \"step-by-step\" generation similar to FlowDAS, but are not compared. Again, it is unclear what advantages FlowDAS brings over these methodologies.\n\n[1] Bao, Feng, Zezhong Zhang, and Guannan Zhang. \"An ensemble score filter for tracking high-dimensional nonlinear dynamical systems.\" Computer Methods in Applied Mechanics and Engineering 432 (2024): 117447.", "was_revised": true, "final_justification": "The proposed methodology provides an elegant, new approach to data assimilation that nicely makes use of the properties of stochastic interpolants and how they can condition on observations. The experiments also show promising results, both quantitatively and qualitatively. However, I still have several concerns that became more apparent during the discussion. First, the method is not theoretically sound, as it does not sample from the filtering distribution. This leads to concerns regarding the robustness and reliability of the method. Second, it became apparent that the uncertainty quantification was not properly evaluated. The new results on CRPS show that FlowDAS actually struggles compared to the baseline models, although further investigation is necessary, as also mentioned by the authors. For these reasons, I am willing to give an acceptance, but I consider it borderline.", "rebuttal": "We thank the reviewer for their balanced and constructive feedback, and we are encouraged by their recognition of the potential of our work.\n\n## Weaknesses:\n**A. “The writing feels rushed in places”**\n\nWe thank the reviewer for the careful reading and for pointing out these writing issues. We corrected all the noted misprints. Specifically, we now clarify that “discontinuity situations” refer to assimilation tasks involving abrupt state changes or non-smooth dynamics that often challenge traditional filtering methods. \n\n**B. “Comparisons with classical filtering baselines are largely missing.”**\n\nWe have added EnKF as classical filtering baselines in the Lorenz 63 experiment. The updated RMSE comparison is shown below:\n\n|         | FlowDAS | SDA   | PDEDiff | BPF   | EnSF  | EnKF  |\n|---------|---------|-------|---------|-------|-------|-------|\n| RMSE    | 0.202   | 1.114 | 0.871   | 0.270 | 0.298 | 0.318 |\n\nThe highly nonlinear system dynamics induce non-Gaussian ensembles, which breaks the assumption of EnKF. Our nonlinear observation model in this case further deteriorates its performance. In contrast, FlowDAS is data-driven and does not rely on linearity or Gaussianity assumptions. As a result, it achieves the lowest RMSE across all baselines, including classical filters BPF and EnKF, and diffusion-based methods, e.g., SDA and PDEDiff. This supports FlowDAS’s suitability for nonlinear and non-Gaussian data assimilation tasks.\n\n\n**C. “It’s unclear what advantages FlowDAS brings over EnSF and DiffDA, etc.”**\n\nWe have added **EnSF** to the Lorenz-63 experiment (details below). Its performance is on par with BPF. This is expected since EnSF uses a training-free Monte Carlo estimator for the score function, which faces the same **particle collapse issues**, i.e., samples concentrate in high-probability regions of the transition dynamics, leading to biased score estimates. This reflects a **train-sampling tradeoff**: EnSF saves training time but cannot learn accurate scores, especially under highly nonlinear dynamics and observation models like in our noisy Lorenz-63 settings(e.g., our $\\text{atan}(x\\_1)$ observation). FlowDAS, by learning the drift with collected data, avoids this issue and achieves superior performance.\n\nWe do not include a direct comparison to **DiffDA** for the following reasons:\n\n1). **Mismatch in data and task setup**: DiffDA is tailored for **ERA5 reanalysis**, which involves many atmospheric variables. In contrast, SEVIR is a simplified dataset with only vertically integrated liquid fields. Adapting DiffDA to SEVIR is non-trivial and out of scope for this work, but we are happy to include it in future studies.\n\n2). **Architectural differences**: DiffDA uses an external numerical weather model as its forecasting component and applies a diffusion model purely as a denoiser. FlowDAS handles both forecasting and assimilation jointly with a single learned stochastic model, which is more modular and general-purpose.\n\nWe will clarify this distinction in the final version and emphasize that FlowDAS targets **general, model-agnostic data assimilation** without relying on external solvers.\n\n&nbsp;\n\n## Questions:\n**1. “What do the acronyms for FlowDAS stand for?”**\n\nWe apologize for the omission. FlowDAS stands for Flow-based Data Assimilation with Stochastic Interpolants. Here, “Flow” refers to the use of stochastic interpolant flows to model the posterior distribution, and “DAS” is short for Data Assimilation. We will revise the manuscript to introduce and define this acronym explicitly at first mention.\n\n**2. “More details on how to roll out FlowDAS”**\n\n* To improve clarity in inference, we revised the Inference paragraph in the main text to explicitly explain how FlowDAS proceeds in sequential data assimilation (see below). \n* To directly address your questions on the target distribution that FlowDAS samples from: In our current setup, we sample $x\\_{k+1} \\sim p(x\\_{k+1} | x\\_k, y\\_{k+1})$ with $n=1$, and proceed sequentially. That is, we assimilate one observation at a time and do not marginalize over multiple future samples. FlowDAS targets a local posterior conditioned on the current state and the next observation only. We agree that extending FlowDAS to sample from $p(x\\_{k+1} | x\\_{k}, y\\_{k+1:k+P})$ would provide a more expressive and generalized model and can be better suitable to approximate the full filtering distribution. We mention this as a promising direction for future work.\n\n**Inference** Given a trained drift model $\\hat{b}_s$, FlowDAS performs inference in an autoregressive manner to estimate the latent state trajectory $ \\{ \\hat{x}\\_{1:K} \\}$. At each time step $k$, we begin from the known state $\\hat{x}_k$ and predict the next state $\\hat{x}\\_{k+1}$ using a discretized stochastic interpolant over a time grid $s_0 = 0 < s_1 < \\cdots < s_N = 1$. We set the interpolant state $X\\_{s\\_0} \\gets \\hat{x}\\_k $ and retrieve the next observation $ y\\_{k+1} $. For each interpolation step $ s\\_n$, we simulate a forward transition using the learned drift, given in Eq.7, and noise. To enforce observation consistency, we generate $J$ posterior endpoint samples $ \\hat{X}\\_1^{(j)}$, and compute their likelihoods under the measurement model, i.e., $ || y - \\mathcal{A}(\\hat{X}\\_1^{(j)}) ||_2^2 $. A softmax over these values gives importance weights $ \\{ w\\_{1:J} \\}$. We then apply a correction to the interpolant using a weighted gradient $- \\zeta\\_n \\nabla\\_{X\\_{s\\_n}} \\sum \\_{j=1}^J w\\_j || y - \\mathcal{A}(\\hat{X}\\_1^{(j)}) ||_2^2$, where $\\zeta\\_n$ denotes the step size. After reaching $s=1$, we set $ \\hat{x}\\_{k+1} \\gets X\\_{s\\_N} $ and proceed to the next time step. Repeating this procedure over all $ k $ yields the full trajectory. \n\n**3. Particle collapse issue for FlowDAS**\n\nThis is a very insightful comment! This is indeed a well-known issue in high-dimensional importance sampling and a valuable observation for our method.\n\nWe observed that the likelihood score can first decrease and then unexpectedly increase near the end of the sampling trajectory. When this occurs, we switch to the alternative sampling strategy described in the supplementary material. While it may occasionally produce slightly suboptimal results, it is more robust compared to the importance sampling approach in the main text. We will explicitly mention this issue and our workaround in the revised paper, and plan to investigate more principled solutions in future work.\n\nMore specifically, we found that particle collapse tends to occur during the final 30–50 steps of the sampling process. In these cases, the likelihood term $||y - Ax||_2^2$, which is expected to decrease steadily, begins to rise unexpectedly—eventually diverging and causing sampling failures. Upon examining the importance weights, we observed a clear pattern:\n* When the likelihood is decreasing as expected, the weights are distributed relatively evenly across particles.\n* When the likelihood increases abnormally, the weights become highly concentrated on a single particle (close to 100%), indicating a collapse of the Monte Carlo estimator.\n\nTo address this, we use the following two-step strategy:\n* **Resampling**: When the weights are abnormally concentrated, we immediately resample a new set of particles. This is a standard remedy for particle collapse in importance sampling.\n* **Alternative sampling strategy**: If resampling does not resolve the issue, we switch to the alternative sampling method described in the supplementary. Although it is not always as precise, it significantly improves robustness and prevents divergence.\n\n**4 & 5. Metrics used when evaluating FlowDAS**\n\nWe clarify that these metrics were only used in the Lorenz experiments, where our goal was to benchmark FlowDAS against SDA. Since SDA used these evaluation metrics, we adopted the same for direct comparability. Below, we explain the meaning of each metric and we will clarify them in the revised text: \n* Expected log-prior $\\mathbb E\\_{q(x\\_{1:K} | y)} [\\log p(x\\_{2:K} | x\\_1)] $: This measures whether the generated trajectories conform to the known prior dynamics. Specifically, given a sampled trajectory $x\\_{1:K} \\sim q$, we evaluate how likely it is under the true transition model $p(x\\_{k+1} \\mid x\\_k)$ for $k = 1, \\ldots, K-1$. A higher value indicates better physical consistency with the governing equations. It penalizes trajectories that deviate from the expected evolution under the dynamics, even if they match observations.\n* Expected log-likelihood $\\mathbb E\\_{q(x\\_{1:K} | y)} [\\log p(y | x\\_{1:K})] $: This evaluates how well the generated trajectories explain the observed data. For each sampled trajectory, we compute the likelihood of the full observation sequence $y$, and average this across samples. This score reflects data fidelity, i.e., higher values indicate that the sampled trajectories are more consistent with the observations under the assumed noisy and nonlinear observation model.\n* Wasserstein distance $W\\_1(p, q)$: This is computed between the estimated and ground-truth trajectory distributions. In our case, it becomes a trajectory-wise distance between delta measures centered at $\\hat{x}\\_{1:K}$ and $x\\_{1:K}$. While simple, it provides an interpretable measure of deviation in trajectory space.\n\n&nbsp;\n\n## Appendix\n**Implementation of EnSF** \n| Setting                          | Value         |\n|----------------------------------|---------------|\n| Ensemble size                    | 20,000|\n| Diffusion SDE steps              | 500           |\n| SDE solver                       | Euler  |\n| $\\varepsilon_a$, $\\varepsilon_b$ [1] | 0.001, 0.001   |\n| Initial data variance            | 0             |\n| Inflation factor                 | 1             |\n\n[1] Bao, Feng, Zezhong Zhang, and Guannan Zhang. \"An ensemble score filter for tracking high-dimensional nonlinear dynamical systems.\" Computer Methods in Applied Mechanics and Engineering 432 (2024): 117447."}, {"review_id": "iqazTecs2V", "reviewer": "Reviewer_RCK2", "summary": "This paper propose FlowDAS, a stochastic interpolant-based framework for data assimilation in high-dimensional, partially observed dynamical systems. It unifies probabilistic forecasting and observation assimilation by modeling local state transitions and incorporating measurement consistency. The paper is well-structured and clearly written, with a concise and transparent presentation of the method. The approach is applied to a variety of tasks, including Lorenz-63, Navier–Stokes flows, and real-world weather forecasting.", "questions": "1. **On the implementation of 3D-Var in high-dimensional settings (Line 261):**\n    The variational assimilation baseline approximates the forecast error term using an L2 norm: $\\alpha_1 \\|x - x^\\prime_{k+1}\\|^2_2$. While this may be reasonable for low-dimensional systems, it ignores the covariance structure of forecast errors in high-dimensional cases such as Navier–Stokes or weather forecasting. This simplification may degrade assimilation quality, as the actual forecast errors often exhibit complex cross-correlations. More refined covariance approximations such as those used in GEN_BE [1,2,3] could provide a stronger baseline. Since both diffusion- and interpolant-based AI methods aim to model these dependencies, it would be more meaningful to compare them against improved variational baselines.\n\n2. **On computational cost comparisons:**\n\n   Traditional variational methods (e.g., 3D-Var) are often computationally efficient. It would be helpful to provide a comparison of computational complexity or runtime between FlowDAS, SDA, and classical variational methods—even if FlowDAS is more expensive—so that readers can better understand the cost-accuracy trade-offs.\n\n3. **On ensemble sampling and uncertainty modeling in inference:**\n\n   I may have misunderstood, but although the method claims to support sampling-based probabilistic inference, it appears that only a single analysis sample $\\hat{x}_k$ is used at each step during inference (as shown in Algorithm 2 in the Appendix). This raises the question: Does FlowDAS support generating an ensemble of analysis samples at each assimilation step, and then propagating these samples to produce the next-step forecast /analysis ensemble. Such a capability would be analogous to ensemble filtering methods like the EnKF, where uncertainty is explicitly carried forward through the flow. Clarifying whether FlowDAS supports this kind of flow-dependent uncertainty evolution would help contextualize its probabilistic modeling power.\n\n[1] Descombes, Gael, T. Auligné, Francois Vandenberghe, D. M. Barker, and Jerome Barre. \"Generalized background error covariance matrix model (GEN_BE v2. 0).\" *Geoscientific Model Development* 8, no. 3 (2015): 669-696.\n\n[2] Bannister, Ross N. \"A review of forecast error covariance statistics in atmospheric variational data assimilation. I: Characteristics and measurements of forecast error covariances.\" *Quarterly Journal of the Royal Meteorological Society: A journal of the atmospheric sciences, applied meteorology and physical oceanography* 134, no. 637 (2008): 1951-1970.\n\n[3] Bannister, Ross N. \"A review of forecast error covariance statistics in atmospheric variational data assimilation. II: Modelling the forecast error covariance statistics.\" *Quarterly Journal of the Royal Meteorological Society: A journal of the atmospheric sciences, applied meteorology and physical oceanography* 134, no. 637 (2008): 1971-1996.", "limitations": "Yes, the authors have discussed the limitations.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strength:\n\n1. The paper addresses an important and timely problem in scientific machine learning and data assimilation.\n2. The proposed framework is conceptually novel, combining stochastic interpolants with observation-conditioned inference.\n3. FlowDAS enables unified probabilistic forecasting and assimilation, and avoids some limitations of score-based diffusion models (e.g., global sampling, lack of interpretability).\n\nWeaknesses:\n\n1. The implementation of baseline methods, particularly the 3D-Var setup, appears suboptimal and underexplored. Stronger or more representative model-driven baselines could better contextualize the performance of FlowDAS.\n2. The paper lacks a detailed comparison of computational cost with other baselines.\n3. Although the method is designed for probabilistic forecasting, the rolling forecast experiments use only a single sample per step, without evaluating or visualizing uncertainty propagation. This limits the demonstrated benefit of using stochastic interpolants.", "was_revised": true, "final_justification": "The author didn't compare their method with 3DVar, which is a crucial baseline for numerical weather prediction experiments. Without such comparisons, it becomes challenging to fully assess the effectiveness of the proposed approach. Putting aside the experimental rigor, I do acknowledge that the proposed algorithm is innovative from a methodological perspective, and it will likely be of interest and value to the AI research community. Therefore, I will maintain my evaluation as borderline accept.", "rebuttal": "We thank the reviewer for their balanced and constructive feedback, and we are encouraged by their recognition of the potential of our work.\n\n**A. On the implementation of 3D-Var in high-dimensional settings**\n\nWe agree that the background error covariance matrix is a critical component of modern data assimilation systems used in numerical weather prediction (NWP). Like in our SEVIR settings, the B matrix measures how an error of the VIL at one location is related to errors at other locations and is quite critical for modern variational data assimilation (VAR) systems [1]. However, it is always hard and complicated to compute this B matrix, especially in high-dimensional systems like the SEVIR and NS. Taking the SEVIR as an example, we only take VIL into consideration and its size is $1\\times 128\\times 128$, and its B-matrix would have a shape of $16384 \\times 16384=268,435,456$. Even though the B matrix is a positive semi-definite, symmetric matrix, it would still be very hard to compute, and we would need to resort to highly specialized scientific software like GEN_BE v2.0 [2]. \n\nGiven this, and the fact that FlowDAS is intended as a general-purpose, model-agnostic framework, looking into this specific question setup would be an important but orthogonal direction to our current contributions. We would be happy to discuss this point further and add it in the discussion section if the paper is accepted.\n\nBesides, we have added EnKF as a baseline in our Lorenz 63 experiments. In EnKF, the forecast error covariance $B$ is approximated using the sample covariance of the forecast ensemble, which avoids explicit construction of $B$ and instead operates directly on the ensemble. We observe that FlowDAS still outperforms the EnKF in terms of RMSE. Besides, both BPF and EnKF achieve similar RMSE, which is expected as they both have the knowledge of the forward dynamic model. \n\n|         | FlowDAS | SDA    | PDEDiff | BPF    | EnKF   |\n|---------|---------|--------|---------|--------|--------|\n| **RMSE**| 0.202   | 1.114  | 0.871   | 0.270  | 0.281  |\n\n> Note: As suggested by Reviewer Ztob, we also include **PDEDiff**, the universal amortised extension of SDA proposed in [4], as an additional baseline for completeness. PDEDiff trains a conditional score network to autoregressively predict the next state(s) given a number of past states, enabling more flexible and accurate sequential generation compared to the original all-at-once SDA. This helps address the forecasting limitations of SDA’s joint score formulation.\n\nFinally, as we transition FlowDAS to real-world weather forecasting applications in collaboration with industry, the design and approximation of the background error covariance will definitely be an important consideration for future work.\n\n&nbsp;\n\n**B. On computational cost comparisons:**\n\nWe agree with the reviewer on the importance of computational efficiency and have added a detailed performance comparison in the appendix. The results, using a single A100 GPU on the SEVIR dataset, highlight FlowDAS’s practical viability:\n| Model       | Params (M) | Total Training Time (hr) | Inference Time per Sample (s) |\n|:-----------:|:----------:|:-------------------------:|:------------------------------:|\n| FlowDAS     | 30.0       | 22                        | 30.9                           |\n| SDA         | 22.9       | 45                        | 28.9                           |\n| Transolver  | 11.2       | 80                        | 12.7                           |\n| FNO         | 1880.0     | 80                        | 2.4                            |\n\nThe slightly longer inference time for FlowDAS stems from its iterative correction process, which is essential for achieving accurate and observation-consistent predictions. While this leads to higher per-sample inference time, it is balanced by significant **training efficiency**, as FlowDAS reaches convergence much faster than all competing baselines. We believe this reflects a favorable trade-off in practical deployment scenarios. A more detailed discussion will be included in the final version.\n\nTo further improve inference speed, one promising direction is to apply **post-hoc distillation**, where a faster surrogate model is trained to approximate the FlowDAS sampler. Another avenue is to explore hybrid formulations that **integrate variational inference** techniques into the framework, potentially offering both computational gains and improved flexibility.\n\n&nbsp;\n\n**C. On ensemble sampling and uncertainty modeling in inference:**\n\nFor uncertainty modeling, yes, **FlowDAS does support ensemble-style sampling**. At each assimilation step, we can generate multiple samples from the posterior $p( x\\_{k+1} | x\\_k, y\\_{k+1}) $, allowing us to form an ensemble of analysis trajectories. These samples can then be propagated forward through the system dynamics, enabling flow-dependent uncertainty evolution similar to EnKF.\nIn our experiments, we used this capability to assess uncertainty. For instance, in the Navier–Stokes (NS) experiments, we conducted inference 25 times independently and used the resulting ensembles to compute standard deviations of RMSE across different settings. We apologize for not making this explicit in the main text. Below we report the mean and standard deviation (std) of RMSE over 25 sampled trajectories (with some additional experiments on more challenging assimilation scenarios suggested by Reviewer 3gVf):\n\n|          RMSE           |   SR 4x   | SR  8x   | SR 16x   |  SO 5%   | SO 1.5625% |  SO 1%   | SO 0.75%  |\n|:-----------------------:|:------:|:------:|:------:|:------:|:-------:|:------:|:------:|\n| Mean                    | 0.038  | 0.067  | 0.108  | 0.071  | 0.123   | 0.148  | 0.177  |\n| Std                     | 0.001  | 0.050  | 0.010  | 0.005  | 0.010   | 0.014  | 0.019  |\n\nWe will clarify this ensemble sampling capability and its usage in uncertainty quantification in the revised paper.\n\n&nbsp;\n\n## Reference\n[4] Shysheya, A., Diaconu, C., Bergamin, F., Perdikaris, P., Hernández-Lobato, J. M., Turner, R., & Mathieu, E. (2024). On conditional diffusion models for PDE simulations. Advances in Neural Information Processing Systems, 37, 23246-23300."}]

Author rebuttal:

Meta-review: The average rating is 4.4, with all reviewers recommending acceptance (4, 4, 6, 4, 4). Reviewers all agree on the technical correctness of the paper and on the soundness of the approach, even if some of them judge the contribution to be incremental. The discussion has been constructive and improvements have been made to the paper, including clarifications and additional results.

Recommendation: acceptance.
Decision: Accept (poster)