LD-EnSF: Synergizing Latent Dynamics with Ensemble Score Filters for Fast Data Assimilation with Sparse Observations
Authors: Pengpeng Xiao
Phillip Si
Peng Chen
Venue: ICLR 2026 Poster (2026)
Keywords: Data Assimilation
Latent Dynamics
Physical Models
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We develop LD-EnSF, a score-based data assimilation method that learns latent dynamics and operates fully in latent space, removing the need for expensive forward simulations.

Abstract: Data assimilation techniques are crucial for accurately tracking complex dynamical systems by integrating observational data with numerical forecasts. Recently, score-based data assimilation methods emerged as powerful tools for high-dimensional and nonlinear data assimilation. However, these methods still incur substantial computational costs due to the need for expensive forward simulations. In this work, we propose LD-EnSF, a novel score-based data assimilation method that fully eliminates the need for full-space simulations by evolving dynamics directly in a compact latent space. Our method incorporates improved Latent Dynamics Networks (LDNets) to learn accurate surrogate dynamics and introduces a history-aware LSTM encoder to effectively process sparse and irregular observations. By operating entirely in the latent space, LD-EnSF achieves speedups orders of magnitude over existing methods while maintaining high accuracy and robustness. We demonstrate the effectiveness of LD-EnSF on several challenging high-dimensional benchmarks with highly sparse (in both space and time) and noisy observations.

Published as a conference paper at ICLR 2026
LD-ENSF: SYNERGIZING LATENT DYNAMICS WITH ENSEMBLE SCORE FILTERS FOR FAST DATA ASSIMILA- TION WITH SPARSE OBSERVATIONS
Pengpeng Xiao!.2
Phillip Si Peng Chen
1 GGorrga Institute of Technology
Yale University
p. p.xiaodya le. e.eu,,, ipsi6, 6, pchen 02 ) @ga @gatech.edu
ABSTRACT
Data assimilation techniques are crucial for accurately tracking complex dynamical systems by integrating observational data with numerical forecasts. Recently, score- based data assimilation methods emerged as powerful tools for high-dimensional and nonlinear data assimilation. However, these methods still incur substantial com- putational costs due to the need for expensive forward simulations. In this work, we propose LD-EnSF, a novel score-based data assimilation method that eliminates the need for full-space simulations by evolving dynamics directly in compact latent space. Our method incorporates improved Latent Dynamics Networks (LDNett) to learn accurate surrogate dynamics and introduces history-aware LSTM encoder to effectively process sparse and irregular observations. By operating entirely in the latent space, LD-EnSF achieves speedups orders of magnitude over existing methods while maintaining high accuracy and robustness. We demonstrate the effectiveness of LD-EnSF on several challenging high-dimensional benchmarks with highly sparse (in both space and time) and noisy observations.
INTRODUCTION
Data assimilation (DA) (Sanz-Alonso et al.) 2023) plays a central role in improving simulation accuracy for complex physical systems by integrating observational data into numerical models. is extensively applied in real-world domains such as weather forecasting (Schneider et al., (022). computational fluid dynamics (Carlberg et al., 2019), and sea ice modeling (Zuo et al. (021). where systems exhibit intricate interactions and uncertainties. Classical Bayesian filtering methods, such as the Kalman Filter (Kalman 1960). Ensemble Kalman Filter (EnKF) (Evensen) 1994) and particle filters (Künsch 2013), are widely used because of their computational efficiency. Variants such the Local Ensemble Transform Kalman Filter (LETKF) (Hunt et al. (007) and adaptive sampling meth- ods (Bishop al. (001) further extend the applicability of EnKF. However, standard EnKFs struggle to handle high-dimensional and nonlinear systems because of quadratic complexity in dimensionality and the linearized posterior assumption. Variational methods like 4D-Var Var (Rabier & Liu 2003) offer
improved accuracy but require complex optimization with repeated forward simulation Ianulations, leading to high computational cost. Ensemble methods such as 4DEn Var (Desroziers et al., 2014)) learn linear tangent models to accelerate the optimization process, but the approximation gap can be significant. Though data-driven methods such as Tensor- Var (Yang et al., (025) aim to accelerate the 4DVar through kernelized representations, they still require solving complex optimization problems and differentiating through multi-step dynamics, making direct application to high-dimensional systems nontrivial.
To overcome these limitations, the Ensemble Score Filter (EnSF) (Bao et al. 2024b) has been developed for sigh-iimensional and nonlinear data assimilation, offering linear complexity and more accurate posterior approximation. Unlike the EnKF, EnSF encodes probability densities via score functions and generates samples by solving reverse-time stochastic differential equations (SDEs). However, it performs poorly under sparse observations, where the likelihood gradient is zero in the unobserved regions. Latent-EnSF (Si & Chen 2025) addresses this issue by using Variational
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Autoencoders (VAEs) (Kingma & Welling 2014) to project both states and observations into shared latent space, where score-based filtering can be effectively applied. The latent representations enable more informative gradients. thus mitigating the limitations of sparsity. After assimilation, the system dynamics are evolved in the original full space using existing simulation methods. While this approach is flexible and model-agnostic, it remains computationally demanding because of the complexity of propagating full-space dynamics, limiting its applications to real-time data assimilation and resource--onstraineed settings.
Surrogate models, especially neural network- -basdd approaches, offer solution to this problem (Rudy
al.
2017 Champion et al. 2019 Lu et al.
2021 [i et 2020 Floryan Graham 2022
Vlachas
et al. 2022. Bonev et al.) 2023] Chen et al. 22024; Yue et al.) 2025a b QQu et al.
2025 These
models aim to learn reduced representations of the system and directly evolve the dynamics in
low-dimensional latent space, thereby avoiding repeated calls to the original numerical simulator.
Among these approaches, Latent Dynamics Networks (LDNeets) (Regazzoni et al.) 2024) outperform traditional surrogate models by achieving higher accuracy while using significantly fewer trainable parameters across several complex systems. It jointly learns a smooth latent representation and its temporal evolution without requiring separate encoder. Crucially, LDNets evolve latent states con- ditioned on system parameters (e.g.. initial conditions, system coefficients). This strong dependency means that inaccuracies in the parameters lead to significant prediction errors. Yet it also provides an opportunity: by integrating data assimilation, these parameters can be estimated and corrected during inference, which makes them particularly suitable for data assimilation.
Offline learning
Online deployment
Phase 1
0
Phase 2
Yus
X1.
(x-1
Time
H
Dyn net
Obs Encoder
Dyn net
f
Yı
8 Rg
(5c 14
E1
EnSF
Rec net
Obs Encoder
62
4e+a)
[9c]
x(t.
Magnitude
S D
net
&1
Time
The framework of the LD-EnSF method. Offline learning: In phase 1, the LDNet is trained based on the dataset to capture the latent dynamics. In phase 2, an LSTM encoder trained to encode the observation history V1. matching the latent variable and parameter ut of the trained I.Net at time f. Online deployment: at each assimilation time step, the LD-FEnSF assimilates an ensemble of prior latent pairs {st, We with LSTM encoded latent pairs (&t, úz). The posterior latent states are then used to reconstruct the full states at arbitrary time and space points.
Figure 1:
In this work, we accelerate the assimilation process by avoiding the costly forward propagation of full dynamics while addressing the limitations of EnSF regarding sparse observations. We introduce the Latent Dynamics Ensemble Score Filter (LD-EnnF),, see Fig. for its general framework. Our approach leverages L.NNtts model and preserve system dynamics within very low-dimensional latent space where EnSF is applied. Furthermore, we incorporate a Long Short-Term Memory (LSTM) (Hochreiter) 1997) encoder for mapping observations the latent space, enabling the efficient use of past observations, especially in scenarios with high observation sparsity. The code for data generation, model training, and inference can be found https: / github......... com/ pengpeng can.ppengxxia//0--- aaa.aoolddeesst Our main contributions are summarized as follows:
We propose LD-EnSF, an enhanced data assimilation model developed based the Latent- EnSF for Bayesian filtering. We replace the disconnected VAE and forward propagation model with more cohesive framework, using I.Netts as an improved surrogate model
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perform the assimilation process in low-dimensional latent space. This significantly reduces computational costs and enables real-time data assimilation with large ensembles.
We advance LDNets with novel initialization scheme, a two-stage training strategy, and im- proved network architectures, enabling high accuracy and wow-dimensional representations of complex dynamics.
We propose a new sparse observation encoder based on LSTM, designed to align both latent states and system parameters from LDNets. This encoder effectively captures historical and iregularly spaced observations, enabling accurate and robust joint data assimilation of states and parameters.
We demonstrate the superior accuracy, efficiency, and robustness of LD-EnSF through high- dimensional data assimilation examples of increasing complexity, including Kolmogorov flow, tsunami modeling, and atmospheric modeling, with extreme spatial and temporal sparsity.
2
BACKGROUND
In this section, we introduce the basic concepts and problem setup for data assimilation in the context of filtering, and present the ensemble score filter (EnSF) and its latent space variant, Latent-EnSF.
2.1 PROBLEM SETUP
We denote 23 Rd as dz-dimensional state variable of dynamical system at (discrete) time t E Zt, with initial state ro- Given the state It-1 at time / 1, with / 1,2,.... the evolution of the state from to is modeled as
It (x)-1,
(1)
where F; Rd d.. + Id is non-linear forward map, and u E Rd represents a du.-dimensional uncertain parameter. By E Rdy we denote d, dyddiennsional noisy observation data, given as
Yt = H (3x)) + Tb
where Rd + Rd is the observation map, and "t represents the observation noise.
Due to the model inadequacies and input parameter uncertainties, the model of the dynamical system in Eq. [ may produce inaccurate predictions the ground truth. The goal data assimilation to find the best estimate, denoted by åt, of the ground truth, given the observation data yı (yn, 1/2, ". (3) up to time f. This requires us to compute the conditional probability density function of the state, denoted as P(x:(Via), which is often non- non-Gassiian. In the Bayesian filtering framework (Appendix I). data assimilation is formulated as two-step process of prediction and update, where accurately approximating the prior and posterior densities remains a challenge. This also extends to inferring the uncertain input parameter u,, which remains a challenging task under sparse or noisy observations.
2.2
ENSF AND LATENT-ENSF
Built on the recent advances in score-based sampling methods (Song et al. 2021 EnSF (Bao et al. 2024b) draws samples from the posterior P( using its score V log P(ralVi:t through Monte Carlo integration for the prior score in the prediction step. and damped likelihood score in the update step; see details of the method and algorithm in Appendix A.2 EnSF utilizes the explicit likelihood function and diffusion process to generate samples without assuming system linearity, proving effective for nonlinear, high-dimensional systems like the Lorenz 96 model (Bao al.) 2024b) and guasi-geostrophin dynamics (Bao et al.) 2024a). However, its performance is significantly hindered in sparse observation scenarios, where the score of the likelihood function vanishes in unobserved state components (Si & Chen 2025)..
To address the limitations of EnSF with sparse observations, Latent-EnSF (Si & Chen 2025) performs data assimilation in a shared latent space, where both states I, and observations Y/: are encoded using coupled variational autoencoder (VAR). The VAE, comprising state encoder Estate observation
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encoder Eobs- and decoder D, is trained minimize loss function balancing state-observation consistency, reconstruction errors, and latent distribution regularization. After training, Latent-EnSy applies EnSF in the latent space using encoded states Easte (2,) and observations Eahx (14). Assimilated latent samples are decoded into the full state space and propagated via the dynamical model in Eq. [ This method mitigates the vanishing score issue faced by EnSF for sparse observations and achieves high accuracy even under extreme sparsity, where the vanilla EnSF fails, such as using only 0.44% of state components for shallow water wave propagation problem.
While Latent-EnSF improves sampling efficiency over the vanilla EnSF, its forward evolution still requires numerical propagation of the full dynamical system, which is computationally prohibitive for large-scale and real-time applications. Moreover, Satent-EnSF's latent states exhibit oscillatory and non-smooth behavior, making it challenging to construct stable dynamical model the latent space approximating complex physical dynamics. These challenges motivate the method we develop in this work.
METHODOLOGGY
In this section, we present LD-EnSF, a fast, robust, and accurate data assimilation method that avoids using full-model dynamics during assimilation. It accomplishes this by learning latent representation of the system state via LDNets and effectively incorporating sparse observations using an LSTM encoder, which extends the VAE encoder in Latent-EnSF by accommodating not only historical observations but also irregular spatial sparsity.
3.1 LATENT DYNAMICS NETWORK
Latent Dynamics Network (LDNet) (Regazzoni et al., 2024) has been demonstrated as one of the most capable methods in capturing low-dimensional representations of many complex dynamics. The architecture consists of dynamics network Fe., which evolves the latent state st, and a reconstruction network Rey, which maps latent states back to the full state space at any spatial point.
To extend and improve LDNet for learning more complex dynamics with varying initial conditions in the context of data assimilation, we propose three new variants: (1) shifting initial latent state, (2) two-stage training and fine-tuning, and (3) new architecture of the reconstruction net.
The dynamics network Fe,, as shown Fig. ! offline learning phase 1, takes the latent state 8 E IR and parameter as input and outputs the time derivative of s:-1:
Š Fo, (st- Mz),
t= 0,1,,,.....
The latent state 8; is updated from the one-step forward Euler scheme as follows:
$t = 8:-1 + At Št-1, t t= 0,0,,,,....
where At the time step, set as At T/n for n steps. Unlike Regazzoni et al. 2024). we initialize the latent state as s instead of so to accommodate varying initial conditions.
The reconstruction network Ray, also shown Fig. [ reconstructs from the latent state 8, an approximate full state E(,,E) at any spatial query point Ω in domain Ω as:
a(t, () Ro, (st, E). = 0,
To train the I.Nett, we propose a two-stage training strategy. First, we jointly train both the dynamics network and the reconstruction network by minimizing the loss:
N ΣΣΣ j=1
C()2.0z) = NM
(5)
where {2, 1 are N trajectories, and M is the number of spatial points & in each trajectory. In the second stage, we fine-tune the reconstruction network R₂, with fixed latent representations from the dynamics network, effectively reducing reconstruction errors. This novel training strategy ensures accurate and efficient modeling of both the latent dynamics and the full state reconstruction.
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Moreover, to enhance the reconstruction power for more complex dynamics, we propose to employ a SraNet-based architecture (He et al., (0166) and integrate Fourier encoding (Tancik et al. 2020
Qiu et al. 2024 Salvador & Marsden 2024) to better capture high-frequency spatial components, defined as Ç + (cos 18, B3], where E Rmxd is a trainable parameter matrix, and m is the hyperparameter controlling the dimensionality of the encoding.
3.2
OBSERVATION ENCODING
To map the observations into the learned latent space, we propose training a separate observation encoder employing an LSTTM (Hochreiter) T997) to address the challenge of encoding sparse and noisy observations. In contrast to the VAE encoder used in EanS, which constructs latent representations of the sparse observations step f, the LSTM encoder leverages temporal correlations in sequential observations y1.t, effectively learning a nonlinear time-delay embedding (Noakes) 1991, Takens 2006). In addition, while the VAE in Latent- Latent-EnSF can only handle observations on a regular grid, the LSTM encoder is capable of effectively handling observations random/irregular locations.
As shown Fig. 1 the LSTM encoder, denoted as Eay RRd, xt - IRd+dd, parameterized by trainable parameters 03, with historical observations yı.t up to time / as input. The output of the LSTM network is pair consisting of the approximate latent state & E TRd,, and parameter üe E RRdu;
(&e, ú.) Egg (V1.)),
(6)
where y = (2xe) is the noiseless sparse observation at time t. This encoder facilitates the assimila- tion of not only the state as in Latent-Ens EnSF but also the parameter ue.
To train the LSTM encoder, we minimize the following loss:
((0g) - Nn
Σ Σ j=1
(1
-
(1)) 111a0 -u - Ir).
where the N trajectories are sampled from the LDNet training data, with s, provided by the trained LDNet. This loss enforces the alignment between the encoded observations and the corresponding latent states and parameters, enabling additional parameter estimation during data assimilation. While proper weighting between these two terms be explored, we found that common normalization or standardization of the states and parameters to similar ranges works well without weighting.
3.3 LD-ENSF: LATENT DYNAMICS ENSEMBLE SCORE FILTER
After training the LDNet and LSTM offline, we integrate EnSF to perform data assimilation in the latent space, as illustrated in the online deployment phase in Fig. Let K₁ = (st, u.) denote the augmented latent state. The corresponding latent observation encoded by the LSTM network in Eq. is given by o, ($z, iz). The latent observation data can be approximately modeled as
(8) where we take the identity map Hattmn (Fee) = and estimate latent observation noise f via LSTM encoding of the true observation noise "/. from Eq. [ as detailed in Appendix E..] To this end, the Bayesian filtering problem in latent space is formulated in two steps. In the prediction step, we have
6: Haaten + %/..
P(melona-1)) / P
with the transition probability P(se (5)-1) derived from the latent dynamics in Eqs. 3 and 4 where we draw samples of u, from the empirical posterior of u;-1- the update step, we have
P(eo(O1.) P(é, N.) ((c()).)))))
(10)
with the likelihood function P(d, (K)) defined through the latent observation model (Eq. g We
can then employ EnSF to solve this Bayesian filtering problem in the latent space, with the latent dynamics (LD) evolved as a surrogate the full dynamics. We present one step of the LD-EnSF in Algorithm
As shown in Fig. [ the assimilation process runs entirely in the latent space, removing the need to map back to the full state space during assimilation. Once latent states are obtained, the reconstruction network can be used to recover the full system state any desired spatial location. Moreover, the smoothness of the latent trajectories, as we will show in our experiments Section 4.1 enables accurate interpolation in time, allowing the full state to be evaluated at any continuous time point.
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Algorithm 1 One step of LD-EnSF
Input: Ensemble of the latent states and parameters { from distribution P(Rt-1(011-1) and the observations V1:e. Dynamics network Fa,, observation encoder EØg.
Output: Ensemble of the latent states and parameters {ree from posterior distribution P(
1: Simulate the latent dynamics in Equations 3 and 4 from the ensemble of samples { } to
obtain samples {Kr} following ((Kr/@1:-11) where we draw samples of {ue} based on (ue-1).
2: Encode the historical observations by the LSTM network as 9r = Eg, (11:+).
3: Draw an ensemble of random samples { from the standard normal distribution NO, I).
4: Run the reverse-time SDE of EnSF from each fit,1 to obtain the posterior sample Ki = Rt,0-
EXPERIMENTS
We consider three experimental examples: (1) chaotic system modeled by Kolmogorov flow with uncertain viscosity, (2) shallow water wave propagation in tsunami modeling with uncertain initial conditions (for earthquake locations), and (3) forced hyperviscous rotation on sphere with earth-like topography in atmospheric modeling with an uncertain forcing term. Below, we briefly present these problems, with detailed equations and the setup for data generation provided in Appendix B
Kolmogorov flow. Kolmogorov flow is canonical turbulent system driven by a sinusoidal body force, parameterized by an uncertain Reynolds number Re (Kochkov et al.] (02T). The spatial resolution is set as 150 x 150. We burn in for 100 time steps, then simulate 200 time steps with step size of 8t 0.04. Observations are provided every steps on a regular 10 10 grid (0.44% of the domain) and on 100 randomly placed points for evaluating LD-EnSF, resulting in 40 assimilation
steps.
Tsunami modeling. We use simplified shallow water equations for tsunami modeling, where the initial condition is specified as a Gaussian bump randomly placed in the spatial domain, following the setup in Si & Chen (2025). For the simulation,
discretize the spatial domain into uniform grid of 150 x 150. The simulation is carried out over 2, 000 time steps with a time step size of öt zs 21 seconds. Observation data are provided
regular 10 x 10 grid (and 100 random/irregular points to test LD-EnSF) for every 40 simulation steps, leading to 50 assimilation steps.
Atmospheric modeling. This setup is adapted from the planetswe example in the Well (Ohana et al. 2024). The spatial resolution is set as 512 x 256. We simulate for half a year (21 days as we set 42 days for a year), totaling 504 hours for approximately 30,240 time steps with an adaptive time step size of about öt xe 60 seconds, where the initial condition is derived from the 500 hPa pressure level of the reanalysis dataset ERAS (Hersbach et al.) 2020). Observations are provided on a regular
16 x 8 grid for every 8 hours of simulation, leading to 63 assimilation steps, featuring extreme spatial
(~~0.1 1 1%) and temporal (~00.2% sparsity.
4.1 OFFLINE LEARNING OF LDNETS
LDNets are trained via hyperparameter search (Biewald 2020) with latent states discretized at obser- vation times. The hyperparameter selection and normalization method are detailed in Appendix C.I In our current experimental settings, we use a static utt). LDNet can also accommodate slowly varying u(l) without modification. If u, changes more drastically, an explicit dynamical model W4-1 = (uz)) would need to be learned in addition to the latent state dynamics. The test error of LDNet as a surrogate model reported in Table compared with the VAE used in Latent-Ent EnSF (Si & Chen 2025) with a Latent Diffusson Model (Rombach et al. (022) architecture. For a fair comparison, we also construct a latent dynamical model by integrating the VAE with an LSTM (referred as VAE-dyn) jointly trained to predict the latent state as &saee (2++1) LSTM(Estme (Ia), .... Estate (xr )). This setup allows the LSTM to propagate the initial latent state. As shown in Fig. 3 while the VAE effectively learns latent representations of the full dynamics and achieves low reconstruction error (with VAE-dyn exhibiting similar state reconstruction error), the LSTM in VAE-dyn fails to maintain stable long-term term latent predictions, leading rapid accumulation of reconstruction errors over time. In contrast, our LDNet achieves the lowest approximation errors in all three examples. Note that the original LDNet fails to capture the varying initial conditions in the tsunami example (resulting

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in large errors), while our improved LDNet accommodates this. Furthermore, the original LDNet exhibits much larger errors than ours in capturing
dynamics for the Kolmogorov flow.
Table 1: Relative RMSE (averaged in time) for the approximation of full dynamics by LDNet (original ours), VAE reconstruction, and VAE-dyn (with LSTM for latent dynamics).
LDNet
WAE
0.5
Value
Example
VAB
VAE-dya
LDNet (original)
LDNet (ours)
Simulation Time (houms)
Simulation Time (hours)
Kelmegorov
0.0131
0.964
0.0349
0.0123
Tsunami
0.0309
0.1837
0.0168
Figure 2: Smoother latent states of LDNet (left)
Atmospheric
0.0856
0.483
0.1042
0.0656
than those of VAE (right) for tsunami modeling.
Kolmogorov Flow
Tsunami Modeling
Atmospheric Modeling
100
Net (ours)
100
1x-10-1
Relative RMSE
Relative RMSE
LDNet (ours)
Relative RMSE
LDNit (ours)
VAE
VAE
VAE
10-1
dyn
WiE-dyn
VAl-dyn
Net (original)
10-1
LDNet (original)
(original)
1x10--1
10
10-2
Sx 10
6.0 8.0
0.0
2.9
8.8 11.7
5.0 10.0 15,0 20.0
Simulation Time (seconds)
Simulation Time (hours)
Simulation Time (days)
Figure 3: Relative RMSE for the full-state approximation by LDNet (original and ours), VAE reconstruction, and VAE-dyn (with LSTM trained to predict the VAE latent dynamics).
Compared to the latent states of VAE, the latent states of LDNet are noticeably smoother, as shown in Fig. 2 for the tsunami example, and in Fig. G Appendix for other examples. The smoother latent states make it much easier for the latent observations (mapped from the sparse observations by an observation encoder) to match the predicted latent states. This can help to enhance the data assimilation accuracy of the LDNet-basee LD-EnSF compared to the VAE-based Latent- EnS.. supported by results Section 4.3 Moreover, the smooth latent states of LDNet facilitate accurate temporal interpolation, allowing for the reconstruction of full states arbitrary times besides those observation times (Fig. in Appendix
4.2 OFFLINE LEARNING OF OBSERVATION ENCODER
After training LDNets, we generate sequence the latent states 8; in Eq. 4fr for each input parameter sequence ne by running the latent dynamical model (Eq. 3 The observation operator H is set sparse sub-sampling matrix that selects the state values from grid points. The observational training data this stage do not include observation noise. Using single-layer LSTMs, we are able to achieve an average test error of 0.07% for Kolmogorov flow, 0.5% for tsunami modeling, and 2.53% for atmospheric modeling. Detailed training configurations are provided in Appendix C.I
Irregular observations. For fair comparison with the VAE encoder in Latent-EnSEE we use sparse observations on equidistant grid points the following experiments. However, our LSTM encoder can also effectively handle arbitrarily located observations, which is crucial in practical applications. In Appendix [F.4] we provide robust and accurate assimilation results using the LSTM encoder with 100 irregularly sampled points for Kolmogorov and tsunami, and 128 for the atmospheric case.
4.3 ONLINE DEPLOYMENT LD-ENSF
To deploy the trained LDNets and LSTM encoder for online data assimilation, we initialize an ensemble 20 samples (s-1, ). each comprising the latent state s 1 = 0 and parameter uo for the assimilation. For the Kolmogorov flow, uo represents the Reynolds number Re and is randomly drawn from a uniform distribution 4(5500, 1500]). For the tsunami modeling, uh E L4(0, 0.5]. [0, 0.5)) represents the coordinates of the local Gaussian bump for initial surface elevation, which is randomly sampled in the lpper-left quarter of the domain. In the atmospheric modeling experiment, we define the time-dependent forcing using un 2((11,)))) [1, 4)), where the two components represent
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the amplitude h and latitudinal spread of seasonally and diumally shifting hotspot the geopotential field. The assimilation process for the complex variation of the dynamics is visualized in Appendix H
Kolmogorey Flow
Tsunami Modeling
Atmaspheric Modeling
2005
1037
8::5
Relatta ROMSE
0000
Ers
Relattive RONSE
Ens
Laver Shii--dy3
6095
LD-EXS"
E
Latork-8:55 E:5F-------- dyn LETIF
Helative
5
1095
126
10
20
50
10
20
40
60
Assimilation Stop
Assimilation Step
Assimilation Stop
Figure 4: Relative RMSE with uncertainty estimate of the assimilation results for the three examples.
LETKF is not shown in the atmospheric modeling case because of divergence.
Comparison of assimilation accuracy. We compare the assimilation accuracy of EnSF, Latent-Eni EnSF (by VAE). Latent-EnSPP Eis---nFFFF---ee dyn (by VAE-dyn), LETKF (a state-of-te of-the-art art variant of the EnKF (Hunt al. (207)). 4DEn Var (for tsunami modeling). and LD-EnSF with 10% observation noise for all examples, shown in Fig. 4 We can observe that the full-space methods LETKK and especially EnSF fail to assimilate highly sparse and noisy observations in the high-dimensional setting. In particular, the EnSF fails because of noninformative gradients, which resolved by the modifications made in the latent space methods. Latent- -anntt-ESFF-yyn, which uses VAE-dyn as surrogate forward model, has decreased performance compared with Latent- EnSF because of limited forecast accuracy. LD-EnSF achieves the smallest assimilation errors, as also visualized in Fig. for the atmospheric example the final assimilation step. For the extremely sparse observational data in this example, 0.1% % data in space and 0.2% in time, our LD-EnSF still preserves high assimilation accuracy with around 5% relative RMSE for 10% observation noise, while LETKF gives rise to numerical instability issues (to satisfy the CFL condition) that stopped the assimilation early (not reported in the figure).
Ground Truth
LD-EnSF
Latent-EnSf
EnSF
Zonal wind (mms)
20
Observation
LD-EnSF Enro
Latent-EnSF Error
EnSF Error
10
-10
20
Figure 5: Zonal wind velocity field at the 500 hPa pressure level from an atmospheric simulation with a given forcing field. Top-left: ground truth; bottom-left: sparse observations. Remaining columns: assimilation results (top) and corresponding errors (bottom) at the final time step.
Comparison of computational cost. We evaluate the computational efficiency of LD-EnSF compared other methods. EnSF, Latent-EnSI, and LETKF all require simulating the full dynamics during data assimilation, whereas LD-EnSF evolves only surrogate latent dynamics. As shown in Table 2 compared to the full dynamics used by other methods, evolving latent dynamics by LD-EnSF achieves speedups of 2 x 105, 4 x 1 and 5 x 10t times (T,) for the three examples. As LD-EnSF performs all data assimilation steps in the latent space, it does not require transforming assimilated latent states back to full states at each assimilation step. In contrast, Latent-EnSF reconstructs or decodes the latent states to full states at every step, incurring additional reconstruction time (T,). Additionally, LDNet learns much lower-dimensional latent representation (10, 12, and 52 dimensions compared to 400, 400, and 512 dimensions Latent- Latent-EnSF), which further reduces the assimilation time (Ty). Meanwhile, LETKF incurs significantly higher computational costs because of its assimilation process. The high efficiency of LD-EnSF not only enables real-time data assimilation and the use of larger ensembles to capture extreme events, but also provides increasing advantages in online cost as
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the number of assimilation cycles grows. A detailed comparison of online gains versus offline cost is given in Appendix C.2
Table 2: Comparison in wall-clock runtime in seconds. We denote the time for evolution of the dynamics as Ta. the time for data assimilation as 7,. and the time for reconstructing the full state from the latent state as Tr (only needed at the last time step for L.DEEnFF). The assimilation dimension is denoted as Da. All results are for the full trajectories of ensemble of 100 samples. The device is AMD 7543 CPU by default (64 processors for parallel simulation in atmospheric modeling), unless GPU (a single NVIDIA RTX A6000 GPU) is specified.
Esample
Kolmogono Flow
Tsunsmi Modeling
Atmosphericc Modeling
Metric
LETKF
EaSF
Latent-EoSF
LD-EaSF
LETKF
EnSF
Latent-EaSF
LD-BnSF
BoSF
Latent-EoSF
LD-EaSF
To (8)
10,820
10,829
10, 929
0.049
211.30
211,30
211.30
0.050
35, 603
35, 803
35, 603
0.070
Ty
12,720
40.13
0.71
0.35
10,440
83.86
0.68
0.37
69,620
27.36
2.42
1.42
(GPU))
6.86
0.0018
5.11
0.0014
35,08
0.017
dimension D. 45, 000 45,000
400
67, 67,500
500
400
303,216
203
512
Robustness and sensitivity of LD-EnSF. Note that we construct the LSTM encoder without observa- tion noise. To assess the robustness of LD-EnSF to observation noise, we perform data assimilation with varying noise levels (no noise, 5%, 10%, and 20%) for all examples. The relative RMSE of assimilated full states, parameters, and latent states is shown in Appendix E.5 While the assimilation errors increase with noise, the increase remains modest, and the errors drop significantly in the initial phase. This demonstrates the robustness of LD-EnSF, achieving high accuracy for both the states and the parameters (averaged over 20 experiments) despite highly sparse and noisy observations. In addition, we examine the effect of non- Gaussian and heteroskedastic noise in Appendix [E.2, and evaluate
generalization in Appendix F.3 Sensitivity and stability analyses with respect to latent dimension, ensemble size, and observation sparsity are presented in Appendix [F]
We present additional ablations in Appendix G systematically eval- uating the architectural choices the LDNet surrogate and the observation encoder, their training strategies, and the impact of their integrated latent-space assimilation. These results isolate each component's 's contribution and provide concrete support for the design choices in LD-EnSF.
Additional ablation studies.
RELATED WORK
Score-based data assimilation: Score-based methods have emerged as promising tools for nonlinear data assimilation. Bao et al. (2024b) introduced a filter that integrates diffusion models into Bayesian filtering for state estimation, while [ao et al.) (2024a) proposed training-free ensemble score estimation method that has been successfully applied to geophysical systems. Rozet & Louppe (2023) used conditional score-bases based generative models to reconstruct entire trajectories; however, these smoothing approaches assume access to future observations and thus differ from the real-time filtering required in practical scenarios, where only past and present data are available. [Cen et al. (2025) extended Rozet & Louppe (2023) by incorporating stochastic interpolants and adapting the framework for filtering. However, their surrogate model relies solely on score-based predictions, and their dynamics are propagated in the full-state space, resulting in significantly higher computational costs compared to latent-space approaches. Huang et al. (2024) proposed a conditional generation method, but its performance degrades after several assimilation steps, with simple interpolation baseline outperforming it, indicating limitations in handling states that deviate substantially from the training distribution. They also rely on expensive score-function training and full-state generative modeling, contrasting with the training-free EnSF that we utilize. More recent work includes the state- observatbb augmented diffusion (SOAD) model (Li et al., (024c) and sequential Langevin sampling (Ding et al. 2024), both of which require expensive training of the score function and differ from the training-free free EnSF that we build upon in this work.
Latent-space methods have been shown to improve both the accuracy and efficiency of data assimilation (Peyron et al. 2021 Bachlechner et al. 2021 Penny et al. 2022 Pawar & San 2022 Cheng et al. 2023 Mücke et al. 2024). For instance, Chen al. (2023) integrated Feedforward Neural Networks (FNN) with the EnKF to model latent dynamics, while Li et al. (2024b) proposed using Spherical Implicit Neural Representations (SINR) and Neural ODEs (Chen et al.) 2018) with the EnKF. These approaches mainly focus on Kalman-basee filtering, while our work takes different direction. We extend score-based EnSF to handle sparse observations by introducing
Latent space assimilation:

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Published as a conference paper at ICLR 2026
new LSTM observation encoder, while maintaining high computational efficiency through latent- space assimilation with LDNets. Deep Bayesian filtering (DBF) Tarumi et al. (2024) also performs filtering in learned latent space, but unlike LD-EnSF it trains end-to-end generative model with linear latent transitions and a Gaussian inverse observation operator, rather than pairing a learned surrogate with training-free EnSF update. Meanwhile, |Frion al. (2024) performs DA with learned Koopman prior, and Singh et al. (2024b) links a Koopman-style latent evolution with an online DA loop. Similar LD-EnSF, these method also operate in a compressed latent space.
Deep learning and data assimilation: A growing line of work integrates deep learning with classical variational and ensemble-basee data assimilation. Examples include VAE-enhanced variational methods (Xiao et al. 2025b c) and efffiittr-eeficient hybrid Varar systems (Xiao et al., 2025a). Gottwald Reich (2021) couple doody-coo surrogate modeling with the EnKE. [Singh et al. (2024a) introduce recursive operator framework for semilinear PDEs that supports both forecasting and DA. In parallel, several learned 4D- Var systems extend classical variational DA. Examples include LEVDA (Si & Chen 2026), Tensor-Var Var (Yang al. 2025), 4DVarNets (Fablet al. 2023 Beauchamp et al. 2023b), their ensemble and uncertainty-aware variants (Beauchamp et al. 2023a). fast baseed DA surrogates (Wang et al. 2024), and large-scale hybrids (Li al. 2024a).
6 CONCLUSION AND FUTURE WORK
In this work, we developed LD-EnSF, a robust, efficient, and accurate method for high-dimensional Bayesian data assimilation large-scale dynamical systems with sparse and noisy observations. By integrating LDNets with improved initialization, training, and architectures, alongside LSTM-based historical observation encoding, L.DEESF achieves a smooth, low-dimensional latent representation while enabling fast latent-space dynamics evolution and joint assimilation of state variables and system parameters. Numerical experiments on three challenging examples demonstrated its superior accuracy and efficiency compared to LETKF, EnSF, and Latent-EnSF.
For more complex uncertain system parameters, such as high- high-dimensional stochastic processes and spatially varying random fields, future work should explore effective strategies for encoding these into -ww-dimensional latent representations. For more complex dynamical systems that are difficult to capture with latent dynamics over long time horizons, our framework could be extended by iteratively applying LD-EnSF over shorter intervals, with the latent model adaptively retrained at each stage.
ACKNOWLEDGMENTS
This work was supported in part by the National Science Foundation under Grant No. 2325631, 2245111, as well as 2025 IDEaS + Cloud Hub at Georgia Tech, with support from Microsoft.
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A ADDITIONAL BACKGROUND
A.1 BAYESIAN FILTER FRAMEWORK
In the Bayesian filter framework (Dore al. 2009). the data assimilation problem becomes evolving P( 11) to P(ra/ (/r.)) from time t- to This includes two steps: prediction step followed by an update step. In the prediction step, we predict the density of 2₁, denoted P(ra/Mitt-1) from P(z1-1l1:=11) and the forward evolution of the dynamical model in Eq. 1 as
P(21881.-11) = P(ax/ler-
(11)
where (())) represents transition probability. In the update step, given the new observa- tion data yt, the prior density P(r+(yn=-11) from the prediction is updated the posterior density P(rdlyi) by Bayes' rule as
1 P(ue P(zx 11:+-1)
P(zelvna)
(12)
Here, P(ye (are) is the likelihood function of the observation data Y/ determined by the obser- vation model in Eq. 2 and Z is the model evidence or the normalization constant given as = P(y ((r
which is often intractable to compute.
A.2 ENSEMBLE SCORE FILTER
A comparison between previous score-based assimilation methods and our LD-EnSF is shown in Table 3
Table 3: Comparison of EnSF, Latent-EnSF, and LD-EnSE.
Methods
Sparse Observations Dynamics
EnSF
X
X
Latent-EnSF
V
X
LD-EnSF
In EnSF, given physical time I, we define a pseudo-diffusion time 7 T =0,, at which progress
dæ,, = f T)dT + g(T)dW,
(13)
driven by d dimeennnsao Wiener process W. Here, we use It,r to indicate the state at physical time and diffusion time The drift term f( 7) and the diffusion term g(7) are chosen as
f f r) = dlog O+,
d dlog ar g2
dr 21,7 (2 dr
(14)
dr
with a, = 1 - e.) and 82 = T, where &a a small positive parameter to avoid log a /dr being undefined (e.g., Ca 0.01 in experiments). This choice leads the conditional Gaussian distribution
21,, NNa, (21),
(15)
which gradually transforms the data distribution taken as F1,0 = Da ~ P(xx(Vi:) at T = close to a standard normal distribution at T = This transformation process can be reversed by integrating an SDE from to 7 = 0 as
dr.,- [x....)) (r) log P(.,,)
9(r)dW,
(16)
where Wi is another Wiener process independent of W. and V, log (a is the score of the density P(x,,r (v1:)) with the gradient V, taken with respect to Xt,r- By this formulation, ,t,r follows the same distribution with density P(Zt,rlyidd in the forward and reverse-time SDEs.
To compute the score V, log ((rr,)1)1))) in Eq. 16 the EnSF (Bo et al. 2024b) uses
v log P(x (2x)
V log (11--1)
>h(T) log P(ye
(17)
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where the damping function h(r) -- is chosen to monotonically decrease in [0, 11, with h(1) 0 and h()) = 1. The likelihood function ((3)(221=>) in the second term can be explicitly derived from the observation map in Eq. 2 The first term be approximated using Monte Carlo approximation, with samples drawn from the distribution P(
With the score function V log (lxte) evaluated in Eq. 17 the samples from the target distribution ( (/ can be generated by first drawing samples from N(0, I) and then solving the reverse-time SDE in Eq. [6] using, e.g., the Euler-Maruyama scheme. The workflow for a single-step data assimilation using the EnSF is summarized in Algorithm Zn in Appendix A.2
The algorithm of EnSF is presented in Algorithm 2 where state samples evolved forward according to the physical model and then refined via sore-based sampling in the reverse-time process.
Algorithm/ 2 One step of EnSF
Input: Ensemble of the states (12-1) } from distribution P(24-1l(1a-1) and new observation yt- Output: Ensemble of the states { from distribution P(æ+ (1:a).
1: Simulate the forward model in Equation from ( to obtain samples (Ir,o} following
2: Generate random samples ( from standard normal distribution (O, I).
3: Solve the reverse-time SDE in Equation 16 starting from samples { using the score in
Equation 17 to obtain }
B DETAILS OF EXPERIMENTS SETTINGS
B.1 TSUNAMI MODELING
We consider the shallow water equations for tsunami modeling, which are widely used to model the propagation of shallow water where the vertical depth of the water is much smaller than the horizontal scale. These equations are frequently applied in oceanographic and atmospheric fluid dynamics. In this study, we adopt simplified form of tsunami modeling:
dv
dt fk gVn.
(18)
dn
- ((1 + H)v),
dt
where is the two-dimensional velocity field, and 17 represents the surface elevation. Both v and constitute the system states to be assimilated. Here, H = 100m denotes the mean depth of the fluid, f is the full latitude varying Coriolis parameter, and g is the constant representing gravitational acceleration. k refers to the unit vector in the vertical direction. We define two-dimensional domain of size L, where = 10 in each direction. The initial condition is a displacement of the surface elevation modeled by a Gaussian bump with its center randomly and uniformly distributed in the lower-left quarter the domain. The boundary conditions are set such that v = Over time, the wave dynamics become increasingly complex due to reflections at the boundaries. The spatial domain is discretized into a uniform grid of 150 x 150, following the setup of (Si & Chen (025). The simulation is carried out over 2000 time steps using an upwind scheme with time step size of öt xs 21 seconds. Including the initial condition, the dataset comprises 2001 time steps. We generate 200 trajectories, dividing them into training (60%), validation (20%), and evaluation (20%) sets.
B.2 KOLMOGOROV FLOW
In the second example, we consider the Kolmogorov flow with uncertain Reynolds number, a para- metric family of statistically stationary turbulent flows driven by a body force. This incompressible fluid is governed by the Navier-Stokes equations (Kochkov al. 202T):
dv dt
-- V
V'v Vp+f. Re
p
(19)
V vv
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The external forcing term f is defined as f sin 0.1v, where & (E1, (2) is the spatial coordinate, E (1,0),
the velocity field, p is the pressure field, and p 1 denotes the fluid density. The fluid velocity v is the state variable to be assimilated. The spatial domain is defined as [0, 2x|² with periodic boundary conditions and fixed initial condition. The flow complexity is controlled by the Reynolds number Re. The spatial resolution is set to 150 x 150. We simulate the flow over 300 time steps with step size of to 300. A total of 200 trajectories are generated, with the Reynolds number Re randomly sampled from the range [500, 1 1500]. These trajectories are divided into training (60%), validation (20%), and evaluation (20%) sets.
0.04 and take the data from time steps 100
B.3 ATMOSPHERIC MODELING
In the third example, model the atmosphere by adopting the PlanetSWE formulation from Ohana ct al. (2024). This system serves as simplified approximation of the primitive equations used in atmospheric modeling at single pressure level.
Ov
Vv gVh ~V. 20 x X
-
Øt
Oh
= HV v (hv) - (
Ot
where v is the two-dimensional velocity field, ht denotes the deviation of the pressure surface height from the mean H, and Z represents the Coriolis parameter. The term 4 denotes hyperviscosity. The initial conditions are derived from the 500 hPa pressure level in the ERAS reanalysis dataset. F is an external foreing term that introduces both daily and annual seasonality.
t (c(() 2T day'
(t(t) = sin
2x year
0max
F(6, 0, ny cos(Ø - do(()) exp (- (_(0- 02()))
o2
where á and 0 denote longitude and latitude, respectively; t is the simulation time; ht is the forcing amplitude; and (1) and (l) represent the seasonally shifting longitude and latitude centers of the forcing. 0max = 0.4 is the maximum latitudinal declination; a = T/2 controls the latitudinal spread of the forcing; day and year are time scale constants corresponding daily and annual cycles. This dataset defines a 42-day year to increase simulation complexity. In our setting, we simulate half year period (21 days), with snapshots selected every hours, resulting total of 63 time steps, with a spatial resolution of 512 x 256. A total of 200 trajectories are generated by sampling the forcing amplitude h. 20.01, 30) and the latitudinal spread parameter a U[1, 4]. These trajectories are split into training (60%), validation (20%), and evaluation (20%) sets.
C TRAINING DETAILS
C.I HYPERPARAMETER CHOICE
Normalization. Prior training, normalize the data following the approach of (Regazzoni et al., 2024). For bounded data, we scale variables, including the parameter u, the space coordinate ç, and the state variable x, to the range [-1, 1). For unbounded data, standardize the variables mean and unit variance. The time step At in Eq. 4 is treated as a hyperparameter during training.
To determine the optimal hyperparameter choices for LDNets in examples, automate the hyperparameter search using Bayesian optimization (Biewald, 2020). The range of hyperparameters considered is listed in Table 4 The term "downsample time steps" refers to the number of time steps
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sampled from the original dataset. Meanwhile, "At normalize" is constant used to scale the time step At during latent state evolution. It serves as a practical tool to ensure numerical stability and adjust the effective temporal scale of the latent dynamics, rather than representing physical time unit. The details of further training and optimized hyperparameters are shown in Table 5 When fine-tuning the reconstruction network, we the number of epochs 1000 and the learning 10 We also present the training parameters of the LSTM encoder in Table
Table Hyperparameter search range for LDNets training
Datasct
Tsunami modeling
Kolmogorov flow
Atmospheric modeling
Downsampled time steps
20
20 50
25 63
At
0.03 0.05
0.04
0.04 0.05
Num of latent states
8 20
- 15
10 100 depth
Dynamics
depth
depth
10
2 15
width
width
width
10 - 200
20 200
100 200
Reconstruction net
depth
depth 2 15
depth
5 12
width
width
width
100 400
20 700
300 500
Fourier embedding dim
0 - 50 0.1 - 0.8
- 50
10 100
StepLR (gamma)
0.1 0.9
0.1 0.7
Batch size
16
16
4
Table 5: Training details for LDNets
Dataset
Tsunami modeling
Kolmogorov flow
Atmospheric modeling
Downsample time step
50
63
At
0.036
0.04
0.04
Space points
5000
5000
10000
Dynamics net
MLP
MLP
MLP
8 hidden layers
hidden layers
8 hidden layers
50 hidden dim
200 hidden dim
200 hidden dim
ReLU
ReLU
ReLU
Reconstruction net
MLP
MLP
MLP
10 hidden layers
14 hidden layers
15 hidden layers
300 hidden dim
500 hidden dim
500 hidden dim
ReI.U
ReLU
ReI.U
Fourier embedding dim
10
50
Initialization
Glorot normal
Glorot normal
Glorot normal
Adam (Ir)
10-3
10-3
10-3
StepLR R (gamma, step size)
(0.6, 200)
(0.7,200)
(0.6, 200)
Batch size
2
Epoch
2000
2000
2000
Loss
MSE
MSE
MSE
C.2 OFFLINE COST vs. ONLINE GAIN
Training the L.Netts takes 12.08 hours for the Kolmogorov flow, 10.76 hours for tsunami modeling, and 21.86 hours for atmospheric modeling process can be further accelerated by utilizing multiple GPUs, and further code optimization may improve efficiency and reduce computational costs.
single NVIDIA RTX A6000 GPU. The training
Although LD-EnSF incurs a higher initial training cost, this cost quickly offset by its significantly lower per-cycle assimilation cost. Based practical cost breakdown, LD-EnSF becomes more computationally efficient than Latent-EnSS when the ensemble size exceeds 100 in single assimila- tion cycle. To quantify this, we use the following total cost formulation for assimilation cycles.
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Table 6: Training details for LSTM encoder
Dataset
Tsunami modeling
Kolmogorov flow
Atmospheri modeling
Hidden layers Hidden dim
1
256
128
512
Initialization
Glorot normal
Glorot normal
Glorot normal
Adam (Ir)
10
0.002
10
CosineAnnealingL. (Tmax, eta min)
(5000, 10 10"4)
(5000, 10 10"4)
Epoch
20000
75000
75000
Dropout
0.1
0
(Ty, Tr, and Ta Table 2 are computed based on 100 trajectories):
Twtsl = Tdata + Turaining + 100
(Ty + + + TT)
Take atmospheric modeling as an example. Combining the data in Table 2 and given that Taata = 298 seconds, it is easy to that when >> 100, LD-EnSF becomes more efficient than other methods. Furthermore, there are many scenarios where real-time assimilation is crucial. For example, in tsunami modeling, occurs, we need predictions as quickly as possible for effective forecasting. In such cases, LD-EnSF about 10y times faster than Latent-EnSF and other methods during assimilation, making highly suitable for time-sensitive applications.
tsunami spend time offline training beforehand; however,
D SMOOTHNESS OF LATENT STATES
comparison of the VAE and LDNet latent spaces for the Kolmogorov flow and atmospheric modeling shown Fig.
Kolmogorov
Kolmagoray
Atmospheric
ospherlc
5.0
Vohu
2.5
0.0
0.0
0.0000
-1
2.5
0.5
-2
-0.0005
(seconds)
Simuletion Time
Simulation Idays)
Simulation (days)
Figure 6: Latent states of Kolmogorov flow and atmospheric modeling.
Due the smoothness of the LDNet latent space, the output can be evaluated at arbitrary continuous time steps by interpolation. In the tsunami modeling, Fig. repprrts the reconstruction eттor based on these interpolated latent states, where the model trained on 50 time steps sampled from total of 2000 physical time steps. During testing, the latent trajectories are interpolated to 400 time steps for reconstructing the full physical states.
ROBUSTNESS TO NOISE AND DISTRIBUTION SHIFT
E.1 ESTIMATING THE UNCERTAINTY OF LATENT STATES
To estimate latent observation noise fx. we first train the LSTM encoder using noise-free observations. We then encode noisy observations and compare the resulting latent states to the reference latent states obtained from trained LDNet. t is then estimated the standard deviation of the difference between the noisy and reference latent states. Fig. shows the estimated ft in which we assume a uniform noise level across latent dimensions.
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Average error: 0.0222
Relative RMSE
0.06
0.04
0.02
0.0
2.9
5.8
8.8
11.7
Simulation Time (hours)
Figure 7: Test error of reconstructing full physical states from interpolated latent states in tsunami modeling.
Tunnami Modeling
Kolmogorav Flow
Atmospheric Madeling
Latent state uncertainty
0.06
No nolse
Latent uncertainty
0.10
No noise
0.12
0.04
20% naise
10% noise
0.06
0.08
5%
10%
20% naise
Letent State uncertaiin
0.10
BX noise
2015 noise.
10% noise
0.06
0.04
0.02
0.04
0.02
0.02
0.00
0.00
0.00
20
40
0
10
20
30
40
0
20
40
60
Assimilation
Assimilation step
Assimilation step
Figure Estimated uncertainty in the latent space, measured as the standard deviation of the difference between the encoded latent states (from noisy observations) and the true latent states obtained from the trained I.NNet.
E.2 NON-GAUSSIAN HETEROSKEDASTIC NOISE
To evaluate the robustness of LD-EnSF under more realistic observation conditions, we conduct experiments using several non- Gaussian and heteroskedastic noise models in the tsunami assimilation task. All noise distributions are scaled to achieve mean noise-to-signal ratio of approximately 10%.
Table repports the assimilation eтrоr across these settings.
Table 7: Assimilation error under different observation noise models.
Noise Type
Gaussian
Nonstationary Gaussian
Sinusoidal Het- eroskedastic
Beta
Multivariate Gaussian
Assim. Error
0.053
0.086
0.061
0.072
0.064
The nonstationary Gaussian setting introduces time-dependent mean that varies assimilation steps but is spatially uniform. The sinusoidal heteroskedastic setting applies temporally varying Gaussian noise whose variance oscillates between 0% and 20% of the signal amplitude, following sine waveform. The Beta noise is sampled from scaled Beta distribution (e.g., Beta(2, 5)), inducing asymmetric, bounded perturbations. Finally, the multivariate Gaussian model introduces correlated noise with covariance matrix = AAA., where A contains i.i.d. standard normal entries.
Despite the increased variability and structural complexity of these noise models, LD-EnSF con- sistently achieves low assimilation error, demonstrating robustness beyond the standard isotropic Gaussian setting.
E.3
GENERALIZATION
While fitt-f--iistibution (OOD) generalization is not the main focus of this work, it remains important consideration for real-world deployment. We evaluate the robustness LDNet under distribution shift by testing it trajectories initialized with Gaussian bumps displaced outside the
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training region. We evaluate the OOD generalization of LDNet by testing it on trajectories whose initial Gaussian bump lie outside the training region. Specifically, while training conditions are sampled from [0, 0.5L] x [0, 0.5L), test on locations such as [0.25L. 0.32L], [0.25L, 0.54L], etc. (Table 8
Table 8: LDNet test enror under increasing displacement from the training region. The horizontal axis indicates the center shift L. of the initial Gaussian bump.
Distance to Train Region
0.02L
0.04L
0.06L
0.08L
0.10L
LDNet Error
0.0553
0.0720
0.0889
0.104
0.122
As expected, the error increases with distance from the training region, indicating a consistent degradation under distribution shift. However, the extent of generalization is problem dependent. For instance, the original LDNet paper (Regazzoni et al. 2024) demonstrates strong extrapolation in time on unsteady Navier-Stoker flows.
E.4 UNSTRUCTURED OBSERVATIONS ASSIMILATION RESULTS UNDER NOISE
By leveraging the LSTM encoder, LD-EnSF accommodate unstructured observations arbitrary locations. However, training a new LSTM is required for different sets of observation points. In the following results, 100 observation points (0.44%) are randomly selected the observation set. Training the LSTM observation encoder to the observations with latent states using the hyperparameter setting in Table 6 achieves a test relative RMSE of 0.094% for the Kolmogorov flow, 0.83% for the tsunami modeling and 2.68% for atmospheric modeling. The visualizations are in Appendix H
Under random observation settings, present the assimilation results in Fig. We conduct 20 experiments, each with a different ground-truti trajectory, and report the mean and uncertainty of our method. The figure also includes results under different noise levels, demonstrating the robustness of LD-EnSF. For the Kolmogorov flow and atmospheric modeling, since the data is normalized to have a mean of and standard deviation of 1, Gaussian noise applied with a standard deviation proportional to the specified noise level, e.g., standard deviation of 0.1 for 10% noise. For the tsunami modeling, due to the long-tailed nature of the data distribution, Gaussian noise added adaptively based the value each observation point.
Note that Fig. 9 (bottom middle) shows the results of parameter estimation, where the purple line corresponds the no-assimilation baseline and exhibits high variance. It corresponds to the no- assimilation baseline and reflects the prior distribution of the parameters before any data assimilation applied. The reported variance is computed across multiple test trajectories, and the wide spread of the purple curve reflects the variability in true parameter values across these cases. This highlights the inherent difficulty of parameter estimation in the absence of assimilation, particularly when the parameter is not directly observed and affects the system only indirectly. contrast, the assimilation results (depicted by the remaining curves) show substantially reduced variance and lower mean errors. For instance, under 20% observation noise, the mean estimation error is approximately 10%, with a standard deviation of around 15%. This performance is comparable across most benchmarks, though atmospheric modeling remains more challenging because of the weak sensitivity of the observed field the external forcing parameter. This makes the inverse problem more ill ill-posed, especially under sparse observations. In terms of convergence behavior, the estimated parameters generally stabilize within approximately 10 assimilation steps, which is consistent with other benchmark settings.
with different ground-truth
E.5 STRUCTURED OBSERVATIONS ASSIMILATION RESULTS UNDER NOISE
Fig. 10 shows the assimilation performance of LD-EnSF under varying levels of observation noise. Results are averaged over 20 runs with different ground-truth trajectories, and both the mean and uncertainty estimates are reported. Note that the inferred parameter exhibits higher uncertainty, which is expected, as recovering the system parameters from limited and noisy data is inherently ill-posed.
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Full states
Parameter u
Latent
10195
57X
noise
50%
0775
58% noise
Reletive RMSSE
138
375 notse
Relattiv RASE
55, naise
10% notse
.6% notse
10%
10%
L085
notse
Original
5%
2012
Original
Original
0.2%
10
20
30
40
10
20
30
40
10
20
50
40
Assimilaaion Stop
Assimilation Step
Assimilation Stop
20175
Full states
20250
Parameter u
satent states
1305
No noise
noise
$090
58% noise
10% noise
Relattve RMSE
10%
15% notse
20% notse
20% noise
MFC noise
Reletve
notse
Original
Relatbu
Original
Original
175
0.295
0.2%
25
50
12
25
50
25
37
50
Assimilation Step
Assimilation Step
Assimilation Step
Full states
Parameter u
Latent Latentsstates states
No noise
noise
Reletive AMSE
30% noise
11% noise
Rebative RMSE
57%
355 natse
20% noise
Rebative RMSE
-99
10% noise
55, noise
to96
Original
Original
Original
208
0.85%
100
20
30
40
50
60
100
20
30
40
50
60
10
20
30
40
50
60
Assimilation Stop
Assimilation Step
Assimilation Step
Figure 9: Assimilation results for unstructured observations in the Kolmogorov flow(top), tsunami modeling (middle), and atmospheric modeling (bottom). The left panel shows the relative RMSE of full states, while the middle and right panels display the of the assimilated parameters and latent states, respectively, compared to the latent states at the true parameters. For reference, errors of the original (unassimilated) quantities are also included.
F SENSITIVITY AND STABILITY ANALYSIS
Empirically, we conducted additional experiments that vary the latent dimension, ensemble size, and observation sparsity for the tsunami modeling problem. To facilitate comparison, To facilitate comparison, the assimilation error shown in this section is defined as the relative RMSE at the final time step, while the surrogate model error refers the relative RMSE averaged time.
F.1 LATENT DIMENSION SENSITIVITY
In our tsunami modeling experiments, we investigate how varying the latent dimension influences both the surrogate model accuracy and the assimilation performance, while keeping other hyperparameters unchanged. As shown in Table clear failure mode emerges when the latent space is restricted to 1 or 2 dimensions. The best performance, in terms of relative RMSE, is achieved with moderate latent dimension between 5 and 10. While high-dimensional latent space (e.g., 500 dimensions) does not lead to complete failure, it results in significantly higher errors compared to these more balanced settings.
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Full states
Parameter u
Latent states
101/
508
noise
50%
6095
58% noise
10% notse
noise
Reletive RMSSE
17%
375 notse
.6% notse
noise
Relative RMSE
10%
10% noise
55, naise
1995
25% notse
Original
2012
Original
0,35
Original
0.19
0.25
10
20
30
40
10
20
30
40
10
20
50
40
Assimilation Stop
Assimilation Stop
Assimilation Stop
20125
Full states
Parameter u
staent states
130%
No noise
noise
nolse
305
0275
5%% noise
Relatbv AMSE
55% nolie
Relatbw RMSE
MSC noise
10% noise
10% noise
20% noise
Reletve
25% notse
20% notse
Original
1275
Original
Original
15p
20
0.2%
255
25
50
0.195
25
37
50
25
37
50
Assimilation Step
Assimilation Step
Assimilation Step
Full states
Parameter u
Lttent states
No noise
noise
Reletive AMSE
30% noise
10% noise
Rebitive RMSE
355 natse
20% noise
Rebative RMSE
-99
10% noise
55, noise
loss
Original
Original
Original
200
455
0.85
10
20
30
40
50
60
1.0
20
30
40
50
60
10
20
30
40
50
60
Assimilation Step
Assimilation Step
Assimilation Step
Figure 10: Assimilation results for grid-based observations in the Kolmogorov flow(top), tsunami modeling (middle), and atmospheric modeling (bottom). The left panel shows the relative RMSE of full states, while the middle and right panels display the of the assimilated parameters and latent states, respectively, compared to the latent states at the true parameters. For reference, errors of the original (unassimilated) quantities are also included.
Table 9: Effect of latent dimension on the LDNet surrogate error (in relative RMSE) and assimilation performance for tsunami modeling.
Latent Dim
3
50
100
200
500
LDNet Error
0.861
0.459
0.0225
0.0216
0.0213
0.0239
0.0271
0.0230
0.0473
Assim. Error
0.869
0.387
0,0430
0.0364
0.0371
0.0504
0.0528
0.0474
0.0993
F.2 ENSEMBLE SIZE SENSITIVITY
EnSF consists of two components: the prior score estimation (done through Monte-Carlo sampling) and the likelihood score estimation. The latent observation in the likelihood is defined over the entire latent space, whose contribution dominates the construction of the posterior score, point explored in Latent-EnSF (Section 4.1.1). With one ensemble member, the algorithm effectively reduces a MAP estimate and yields similar accuracy. For LDDEnSF,, increasing the ensemble size has minimal impact on the average error, as shown Figure II which reports both the root-meat -quare error (RMSE) and the Continuous Ranked Probability Score (CRPS) Matheson & Winkler (1976). While the mean errors remain stable, the variance across 20 different trajectories increases with ensemble size, reflecting the uncertainty introduced by sparse and noisy observations. Note that small ensemble size was shown to be sufficient in the latent assimilation, see Figure in (Si Chen 2025). while a large ensemble size has to be used for the full/original space assimilation increase its accuracy.
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CRPS Values vs. Ensemble Size
RMSE Values vs. Ensemble Size
160%
Ensemble Size 2
Ensemble Size 2
Ensemble Size
Ensemble Size
Ensemble Size
Reletive RMSE
Ensemble
Ensemble Size 20
Ensemble Size 20
CRPS
Ensemble Size 40
Ensemble Size 40
10-3
1%
0.2%
12
25
37
50
0
12
25
37
50
Assimilation Step
Assimilation Step
Figure 11: CRPS and RMSE values for varying ensemble size in tsunami modeling
F.3 SURROGATE MODEL SENSITIVITY
The accuracy of the surrogate models (LDNet and LSTM encoder) plays critical role in the overall assimilation performance. In our setting, the LSTM is trained on latent trajectories generated by LDNet, poor LDNet accuracy also affects the quality of the observation encoder. Therefore, is essential to ensure that LDNet reaches reasonable level of fidelity. In Table I0 the assimilation егror decreases as LDNet error decreases during training. Notably, even when the LSTM achieves similar test errors across different L.NNet models, the assimilation accuracy still depends strongly on the quality of L.NNtt.
This gap between surrogate and assimilation error is expected. While the LDNet is evaluated using full and accurate latent input, the assimilation process relies on partial and noisy observations, which are encoded by separately trained LSTM. The LSTM itself introduces approximation error when reconstructing latent trajectories, and during testing we additionally add 10% noise to the observations. Therefore, it is reasonable, and theoretically consistent, that assimilation results exhibit higher error, e.g., 0.0371, than the surrogate model error, c.g., 0.0213 at epoch 1999.
Table 10: Effect of LDNet training duration surrogate and assimilation accuracy in tsunami modeling.
Epoch
99
299
399
499
799
999
1999
LDNet Error
0.144
0.0794
0.0483
0.0359
0.0309
0.0256
0.0213
LSTTM Error
0.0041
0.0033
0.0033
0.0030
0.0029
0,0032
0.0028
Assim. Frror
0.252
0.0975
0.0693
0.0499
0.0443
0.0411
0.0371
F.4 OBSERVATION DENSITY SENSITIVITY
We also study the impact of observation density on assimilation accuracy. As expected, the assimila- tion error increases as fewer observations are available, due to the reduced information available for inference. The results are summarized in Table
Table 11: Assimilation error under varying observation density in tsunami modeling.
Observation Density
0.08%
0.10%
0.16%
0.25%
0.44%
1.00%
Assimilation Error
0.129
0.0891
0.0476
0.0504
0.0371
0.0353
F.5 7 SENSITIVITY
To test the sensitivity LD-EnSF to misspecified 4, we varied the scalar value used in latent space. As shown in Fig. 12 using by smaller than the estimated value leads mild increase in error
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20275
Full states
Latent states
190%
Parameter U
2.0% 2.0
1538
2,0% t 1pppee
2.0% 2.
1.5 Spriinae
629
6255
Ralabtiv RMSE
30%
Reletive
0.811
0.811
Dellltiu
Tc
0.0%
0.2%
275
25
50
12
25
50
25
37
50
Assimilation Step
Assimilation Stop
Assimilation Stop
Figure 12: Assimilation results for misspecified 9/. in Eq.
during assimilation. When f is larger than the estimated value, the initial error is higher, but the assimilation still converges to low error. Overall, LD-EnSF remains stable across wide range f values, indicating that the method is not highly sensitive to moderate misspecification of the latent noise level.
G ABLATION STUDY ON KEY COMPONENTS
We evaluate how each component of LDNet, the observation encoder, the training pipeline, and integration for data assimilation contributes overall performance.
G.1 LDNET ARCHITECTURE AND TRAINING STRATEGY
Several improvements have been made to LDNet, including the incorporation of Fourier encoding, the use of ResNet blocks, and finettuning strategy. In Table [2] we use the Kolmogorov flow example to isolate and evaluate the individual contributions of these architectural enhancements. We observe that the combination ResNet and Fourier embedding largely improves the performance of LDNet.
Table 12: LDNet performance on Kolmogorov Flow with different architectural ablations.
Kolmogorov Flow
Fourier & ResNet
w/ ResNet
w/ Fourier
ResNeu/Fourier
LDNet Error
0.0223
0.0237
0.0268
0.0340
As shown in Table 13 fine-tuning of the reconstruction network improves accuracy beyond initial training of our proposed LDNet, and both stages significantly outperform the original LDNet.
Table 13: Relative RMSE for different training strategies in tsunami modeling.
Training
Fine-tuning
Original LDNet
Tsunami
0.0213
0.0168 0.0123
0.1837 0.0349
Kolmogorov
0.0223
While some works (Tancik et al.) 2020) keep the Fourier feature matrix B fixed by sampling from Gaussian distribution, in this paper we follow Salvador & Marsden (2024) and treat as trainable parameter. As shown in Fig. [13] whether B is fixed or trainable does not affect the training behavior. This observation is consistent with Appendix A.3 of TTnncik et al. (2020), where optimizing the Fourier feature parameters yields nearly identical results. In addition, using trainable removes the need to tune the sampling distribution of the fixed random matrix.
G.2 OBSERVATION ENCODER ARCHITECTURE
We compare our LSTM observation encoder with a CNN-based encoder composed of residual blocks and intermediate self-attention layer, which is also used by SSi & Chen (2025). For tsunami
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10a
Fixed Trainable
Rixed B Trinable
Training Locs
Voileation Loss
10"4
Setting
Test Error
10-2
Fixed B
0.0221
Trainable B
0.0220
10
0
500
1000
1500
2000
500
1600
1500
2000
Step
Step
Figure 13: Training loss, validation loss, and test error in Kolmogorov with either fixed or trainable B in the Fourier encoding.
modeling, we evaluate four configurations and compare the assimilation error: (1) a CNN encoder with direct full-state reconstruction from encoded observations; (2) LD-EnSF with CNN encoder; (3) an LSTM encoder alone; and (4) LD-EnSF with an LSTM encoder. The LSTM encoder, which incorporates temporal context, achieves significantly lower latent representation error (0.5% vs. 0.98%) than the CNN encoder. As shown in Fig. this also translates to significantly improved assimilation accuracy.
200556
CNN encoder only
CKN encoder LD-ErSF
LSTM envoder only
Relative RMSE
0090
enudder
10%%
6%
12
255 37
50
Assimilation Step
Figure 14: Comparing observation encoders with and without filtering. The colored regions indicate estimated uncertainties for observation noise at 10% level.
G.3 TWO PHASE TRAINING OF LDNET AND OBSERVATION ENCODER
Although LDNet can, in principle, be trained jointly with the LSTM encoder simplify the workflow, we find that balancing the different loss terms is nontrivial and does not lead to improved performance. To evaluate this approach, experimented with combined training objective:
Listal Liatent + L1.STM + Lohs:
where the first terms defined in Eqs. lqs.55lnde and 7 and Luts measures the reconstruction from LSTM-predicted latents. This joint training strategy results in a surrogate model error of 0.0241 and assimilation error of 0.0693 for tsunami modeling, which are higher than the results obtained using separate training in Table [
G.4
WITH AND WITHOUT LATENT-SPACE ASSIMILATION
In Fig. [5| we compare the reconstruction accuracy of two approaches for tsunami modeling under different noise levels: (i) directly reconstructing the full state from latent observations encoded by the LSTM encoder (denoted as LSTM-only, without any filtering), and (ii) LD-EnSI, which performs ensemble-based filtering in the latent space. The LSTM-only baseline performs well when there is no observation noise. However, as noise increases (e.g.. at 5% and 10%). its performance degrades significantly. In contrast, LD-EnSF maintains a lower reconstruction error by effectively incorporating both prediction uncertainty and observation uncertainty through latent-space filtering. Moreover, LD-EnSF is able to estimate output uncertainty using the ensemble, while directly reconstructing the
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full state using the LSTM encoder and reconstruction network cannot. Reconstruction from arbitrarily located sparse sensors has been studied independently by Williams et al. (2024): Tong & Chen (2026). This highlights the benefit of applying the EnSF on top of the LSTM encoder, especially under noisy conditions.
0% noise
5% noise
10% noise
Letent states
states
Latent states
LSTM EnSF
LSTN LD-EnGF
Roletive AMSE
Rolattve HIMSE
18
0.50
12
50
25
50
Full states
Full states
Full states
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Figure 15: Comparison between the LSTM-only baseline and LD-EnSF assimilation results on tsunami modeling with 0%, 5%, and 10% observation noise. The colored regions indicate uncertainty estimated by the ensemble.
H ASSIMILATION RESULTS VISUALIZATION
This section visualizes the assimilation process, including observation points, reconstructed states, and assimilation errors. Figs. 17 and 16 show results for tsunami modeling and the Kolmogorov flow, respectively. comparing structured (10 x 10 grid) and randomly sampled (100 points) observations. In Fig. [7] although the trajectory starts from a misspecified initial condition, assimilation corrects it over time. In Fig. 16, the trajectory with an incorrect Re deteriorates rapidly, while assimilation with LD-FnSF effectively reduces the error. The assimilated dynamics for atmospheric modeling are shown in Fig. [8] A comparison of different data assimilation methods at the final time step, in terms of meridional wind and geopotential height, is presented in Fig. 19
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Figure 16: Visualization of the vorticity field V of the ground truth of the Kolmogorov flow in at a Reynolds number of Re = 674.37 (Ist row) and the perturbed dynamics at He 1469.5 (2nd row), with their difference shown in the 3rd row. Sparse observations (10 x 10 from 150 150 grid) (4th row) are assimilated into an ensemble of 20 LDNet trajectories via LD-EnSF. The 5th row shows one trajectory from the ensemble, starting from a deviated Re 1469.5. The 6th row presents assimilation errors. The last three rows correspond to 100 randomly sampled observation points and their respective assimilation results, where "r" denotes random.
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Figure 17: Visualization of the surface elevation in tsunami dynamics: (1st row) ground truth, (2nd row) LDNet predictions from known initial conditions, and (3rd row) prediction errors. Sparse observations (10 10 from 150 x 150 grid) (4th row) are assimilated into an ensemble of 20 LDNet trajectories via LD-EnSF. The 5th row shows one trajectory from the ensemble, starting from a deviated initial condition. The last three points and their respective assimilation results, where 'r' denotes random.
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Figure 19: Comparison of LD-EnSE Latent-EnSh and EnSF in terms of meridional wind and geopotential height at the final assimilation step.
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Reviews:
[{"review_id": "OgbA194Zw9", "reviewer": "Reviewer_ogtz", "summary": "This paper proposes LD-EnSF, a score-based data assimilation framework that combines Latent Dynamics Networks (LDNets) and an LSTM-based observation encoder to perform Bayesian filtering directly in a low-dimensional latent space.\n\nTraditional data assimilation methods such as EnKF or EnSF are computationally expensive because they operate in the full physical space and require repeated forward model simulations. LD-EnSF negates this by learning surrogate dynamics in latent space and performing all filtering steps there, using an ensemble score filter (EnSF) to update the posterior distribution. The LSTM encoder processes sparse and irregular observations and aligns them with the latent states and parameters.\n\nEmpirical validation is conducted on three different systems (Kolmogorov flow, tsunami modeling, and atmospheric dynamics) and under severe spatial and temporal sparsity. The method demonstrates strong improvements in accuracy and speed, with orders of magnitude reductions in runtime while maintaining or improving assimilation accuracy.", "questions": "1. Theoretical Guarantees:\nCan the authors comment on whether any convergence guarantees exist for the latent-space assimilation process, particularly as a function of the surrogate model error?\n\n2. Dynamic Re-training:\nHow feasible is LD-EnSF in systems where governing dynamics change over time? Could partial retraining or online fine-tuning mitigate the offline cost?\n\n3. Uncertainty Quantification:\nHow reliable are uncertainty estimates when mapped back to physical space?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "Clear Motivation and Relevance\n- The paper tackles a key limitation of recent score-based filters, their high computational cost and poor performance with sparse observations. \n\nSolid Technical Design\n- The integration of latent surrogate dynamics (LDNet) and score-based Bayesian filtering (EnSF) is smart.\n- The introduction of a history-aware LSTM observation encoder effectively extends the latent assimilation framework to handle irregular and sparse data.\n\nComprehensive Experiments\n- The authors test across multiple physical systems of increasing complexity.\n- Results include both structured and unstructured observation setups, multiple levels and differing types of noise, and sensitivity analyses.\n\nExtensive Ablation and Robustness Studies\n- Appendices systematically evaluate noise robustness, out-of-distribution generalization, latent dimension sensitivity, ensemble size, and architectural design choices.", "weaknesses": "Limited Theoretical Novelty\n- The proposed method primarily combines existing techniques (LDNet, EnSF, LSTM encoding). While the combination is well-executed and impactful the theoretical advancement is modest. The novelty lies more in the integration and empirical rigor.\n\nBenchmark Coverage and Positioning\n- Comparisons are limited to EnSF, Latent-EnSF, and LETKF. While these are strong and relevant baselines, the paper could benefit from a clearer discussion of recent efficient variational and diffusion-based data assimilation approaches, such as Tensor-Var: Efficient Four-Dimensional Variational Data Assimilation (Yang et al., 2025) and DiffDA: A Diffusion Model for Weather-Scale Data Assimilation (Huang et al., 2024). Although implementing these methods in the current setup may not be straightforward, a more explicit positioning in the Related Work section would help situate LD-EnSF within the broader landscape.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9P3TnX2kTN", "reviewer": "Reviewer_cjhs", "summary": "The paper proposes LD-EnSF, a score-based Bayesian filtering method that performs all assimilation steps in a low-dimensional latent space learned by an improved Latent Dynamics Network (LDNet) and coupled to a history-aware LSTM observation encoder. The method extends EnSF (ensemble score filtering) to handle severe spatiotemporal sparsity without resorting to costly full-state simulations by: (i) learning smooth latent dynamics that can be time-stepped (Euler) with parameter inputs, plus a stronger reconstruction network; (ii) mapping sparse, possibly irregular observations $𝑦_{1:𝑡}$  to latent pairs $(𝑠^𝑡, 𝑢^𝑡)$ with an LSTM; and (iii) running EnSF entirely in latent space then decoding to physical space only when needed. Overall this presents a principled way of approaching the problem. Experiments on Kolmogorov flow, tsunami (shallow water), and a forced hyperviscous rotating atmosphere show lower assimilation RMSE than EnSF, Latent-EnSF (VAE), and LETKF, with very large runtime speedups because full-space propagation is replaced by latent dynamics", "questions": "Fourier Neural Operators have shown great success recently in weather and PDE modelling. Why have no discussion been provided in that regards?\n\nLatent noise modeling: In Eq. (8) the identity $H_{latent}$ and a scalar 𝛾^_{𝑡}  are assumed via empirical estimation. How sensitive is EnSF’s update to misspecifying 𝛾^_{𝑡} across latent dimensions? Could you learn a diagonal or low-rank covariance in latent space cost-effectively? \n\nAre proposals for $u_{t}$ purely resampling from the previous posterior, or is there diffusion/jitter? What happens when u varies slowly or remains static? \n\nPhysical time vs latent  Δt: Since Δt is tuned, how does the method behave if the observation sampling rate changes (e.g., sparser/faster streams) without re-training? \n\nWhen LDNet struggles (very long horizons, regime changes), could LD-EnSF seamlessly “fallback” to occasional full-model nudging (a la hybrid LD-EnSF/4D-Var) while keeping most steps latent?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "Decoder-free assimilation loop: All filtering happens in latent space (state + params), avoiding per-step decoding and cutting both compute and error accumulation.\n\nEfficiency at scale: Small latent dimension + ensemble updates → orders-of-magnitude cheaper than full-state DA; design is hardware-friendly and parallelizable.\n\nRobustness features: Reverse-SDE damping and simple latent noise modeling make the update stable under severe sparsity/noise.\n\nClear training recipe: Two-stage LDNet training with well-specified schedules/hparams improves reproducibility and stability.\n\nExtensive Evaluation : Demonstrated on Kolmogorov flow, tsunami (shallow water), and a rotating atmosphere covering increasing complexity and different observation settings.\n\nThorough ablations: Sensitivity to latent rank, ensemble size, observation density/cadence, noise level, and architecture choices; includes OOD initial-condition tests.\n\n\nReproducibility: Detailed setups (observation models, training schedules, metrics (see Appendix) ) and systematic reporting make results believable and repeatable.", "weaknesses": "Overall the paper presents a clear contribution with thorough experimentation. Following are my major concerns : \n\nRelated Work Missing : \n\nThe paper under-cites several very relevant 2024–2025 works in  that would strengthen positioning:\n\nNeural Operators for DA and Semilinear PDEs : Fourier Neural Operator and SFNO have presented great result in PDE/wether modeling but no discussion have been provided in regard to them. Additionally Semilinear Neural Operator (ICLR 2024) that proposes a recursive neural-operator framework that explicitly addresses prediction and data assimilation for semilinear PDEs have also not been cited ; \nNeural Koopman priors and Koopman-based DA : Frion et al., “Neural Koopman Prior for Data Assimilation” formulates DA with a neural Koopman prior (now a TSP 2024 article), and KODA (arXiv 2024) integrates forecasting with an online data-assimilation loop using Koopman-guided components. Both connect directly to low-dimensional latent linearizable dynamics for DA, much like LD-EnSF’s latent evolution;  Modern 4D-Var and deep 4D-Var variants : The paper cites classical 3D/4D-Var (e.g., Rabier & Liu 2003) but misses recent learned or hybrid 4D-Var systems that tackle cost and sparsity with neural parameterizations—e.g., 4DVarNet (end-to-end DA backboned on variational objectives), En4DVarNet for uncertainty, 4DVarFormer (attention-based 4D-Var surrogate with rapid multivariate analyses), and operational-scale hybrids like FuXi-En4DVar. These are important baselines or at least conceptual references for the atmosphere case and the efficiency narrative.\n\nAction item : by adding a paragraph in Related Work discussing (SNO/NO-DA, ClimODE, Koopman-DA, deep/hybrid 4D-Var), explaining what LD-EnSF gains with a training-free score component and learned latent dynamics, and why that’s preferable under extreme sparsity.\n\nTechnical Weaknesses :\n \nLatent observation model = identity. The filter assumes $𝐻_{latent}(𝜅_𝑡)=𝜅_𝑡$, so the LSTM’s outputs $(𝑠_𝑡,𝑢_𝑡)$\t​\n are taken as direct noisy measurements of the true latent pair $(𝑠_𝑡,𝑢_𝑡)$. Any encoder bias/miscalibration directly contaminates the update; there’s no learned/structured latent observation operator to absorb mismatch.\n\nDecoder-free assimilation hides reconstruction bias. Assimilation runs entirely in latent space (a strength), but it also means reconstruction errors don’t get corrected during the loop. If the decoder has bias, final physical-space fields can drift even when latent RMSE falls. (The paper itself emphasizes that decoding is only needed at the end, which is fast, but leaves this feedback gap.)\n\nNoise handling is crude in latent space. The latent observation noise 𝛾^_{𝑡} is estimated post-hoc and then treated as uniform across latent dimensions. That’s convenient but brittle when different latent directions have very different uncertainty.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5OLKb8s6Cs", "reviewer": "Reviewer_TvkW", "summary": "This paper proposes LD-EnSF (Latent Dynamics Ensemble Score Filter), a novel data assimilation method designed for high-dimensional dynamical systems with extremely sparse and irregular observations. The core idea is to avoid expensive full-dimensional simulations during assimilation by operating entirely in a learned latent space. The authors train an improved Latent Dynamics Network (LDNet) to serve as a surrogate model of the system’s dynamics, and a history-aware LSTM encoder to map past sparse observations into the latent space. Assimilation is then performed via the Ensemble Score Filter (EnSF) in the latent space, jointly updating the latent state and system parameters. The method yields orders-of-magnitude speedups (up to $10^5$–$10^6\\times$ faster in the experiments) compared to traditional approaches, while maintaining high accuracy and robustness. Extensive experiments on three challenging scenarios (Kolmogorov flow, tsunami propagation, and an atmospheric model) demonstrate that LD-EnSF consistently achieves the lowest estimation errors among strong baselines (LETKF, EnSF, Latent-EnSF), and remains stable even under extreme sparsity where other methods break down. The contributions include: (1) enhancing LDNet with a novel initialization, two-stage training, and a ResNet+Fourier architecture for accurate low-dimensional dynamics; (2) an LSTM-based observation encoder for irregularly-sampled, time-sequential observations; (3) the integration of these components into a fast latent-space score-based filtering framework, delivering real-time capable performance without sacrificing accuracy.", "questions": "Generality to Out-of-Distribution Dynamics: How well would LD-EnSF handle scenarios where the true dynamics deviate from the training data? For instance, if the system experiences an unforeseen regime or a parameter outside the trained range, would the assimilation degrade gracefully, or could it diverge? Did you observe any cases where the learned LDNet struggled when the truth lay outside its training distribution?\n\nAblation on Observation Encoder: Have you evaluated the impact of the LSTM-based observation encoder versus a simpler or non-temporal encoder? For example, how would Latent-EnSF perform if augmented with your LDNet but still using the original VAE observation encoding at each time step (ignoring history)? This would isolate the benefit of the history-aware LSTM. Similarly, what happens if observations are on a fixed grid – does the LSTM still offer advantages over a time-independent encoder?\n\nParameter Estimation Performance: The method jointly assimilates state and parameters. Can you provide more insight into how accurately the uncertain parameters (Re, initial bump location, forcing amplitude/spread) were estimated in your experiments? It would be useful to know, for example, the final parameter RMSE or if the filter consistently converges to near-true parameter values. This would demonstrate the effectiveness of treating $(s_t, u_t)$ together in the state vector.\n\nEnsemble Size and Filter Stability: You mentioned that increasing ensemble size beyond a point had minimal impact on LD-EnSF’s accuracy. Could you elaborate on this? Is LD-EnSF less sensitive to ensemble size because the score-based update effectively approximates the Bayesian posterior even with few samples? Any intuition on why, say, even 1 or 5 ensemble members can yield good results would be interesting – it’s an intriguing contrast to standard EnKF which typically benefits from larger ensembles.\n\nComputational Overheads: Table 2 shows dramatic speedups in the online phase. Could you comment on the offline costs (training LDNet and LSTM) relative to those gains? For a fair real-world assessment, one might consider how many assimilation cycles are needed to amortize the training cost. Do you envision scenarios (like repeatedly assimilating in the same system) where the upfront training is justified by many future uses of the model?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "Significant Practical Advance: The paper addresses a critical bottleneck in data assimilation – the computational cost under high-dimensional, sparse observation settings. By eliminating full-state simulations during filtering, LD-EnSF achieves massive speedups (e.g. 200,000× in one case), enabling applications (real-time forecasting, larger ensembles) that were previously infeasible. This practical improvement is highly valuable for the community.\n\nRobust Accuracy under Extreme Conditions: Empirical results show state-of-the-art accuracy and robustness. LD-EnSF outperforms LETKF, EnSF, and the prior Latent-EnSF by a clear margin in all tested scenarios. Notably, in an extremely sparse observation scenario (0.1% spatial and 0.2% temporal coverage with 10% noise), LD-EnSF attains about 5% RMSE, whereas LETKF and EnSF either diverge or fail to converge. The method also successfully estimates system parameters (e.g. Reynolds number, forcing amplitude) alongside the state, which is a strong plus.\n\nWell-Motivated Method Design: The integration of learned latent dynamics with score-based filtering is novel and well-justified. The authors identify the shortcomings of VAE-based latent filters (oscillatory latent trajectories, need to propagate in full space) and introduce targeted improvements. In particular, the improved LDNet architecture (with shifted initial latent state handling, two-stage training, and Fourier-featured ResNet decoder) yields a remarkably compact and smooth latent representation that outperforms both the original LDNet and a VAE+LSTM baseline in surrogate modeling accuracy. Similarly, the LSTM-based observation encoder is a sensible choice to capture temporal context and irregular spatial sampling, surpassing the static VAE encoder used in Latent-EnSF. Each design choice is supported by discussion or ablation (e.g. Table 1, Fig. 3 for LDNet vs VAE).\n\nThorough Experiments: The evaluation is comprehensive. The paper tests three diverse and complex systems, includes strong baseline comparisons (including a “Latent-EnSF-dyn” variant using a VAE-dynamics baseline), and reports detailed metrics. The authors also explore scenarios with varying observation noise (Appendix E.5) and demonstrate the method’s insensitivity to ensemble size (even a single-sample “ensemble” recovers reasonable estimates, akin to MAP). Such thoroughness increases confidence in the results.\n\nReproducibility: The inclusion of references to code for hyperparameter search (Weights & Biases) and discussion of computational setup (CPU/GPU times in Table 2) is appreciated.", "weaknesses": "Dependence on Offline Training and Generalization Limits: \nA potential concern is the heavy reliance on training a surrogate model (LDNet) on simulation data before deployment. Acquiring a comprehensive training dataset covering all relevant system behaviors and parameter ranges can be costly, and the method’s performance may degrade if the true system behavior deviates from the training distribution. The paper’s approach is essentially as good as its learned model – for scenarios with significant model uncertainty or evolving dynamics, one might need to frequently retrain or fine-tune the surrogate. This limitation is hinted at in the conclusion (need to retrain adaptively for long-term complex systems), but it remains a practical caveat: the method inherits the generalization limitations of data-driven surrogates.\n\nIncremental Novelty and Comparison with Variational Surrogate Approaches: \nWhile combining latent dynamics learning with score-based filtering is a creative and well-engineered contribution, the methodological novelty of LD-EnSF is largely incremental rather than foundational. The core components—EnKF/EnSF-style ensemble filtering, neural surrogate modeling (LDNet), and LSTM-based latent encoders—are all built upon established techniques. The innovation primarily lies in the system-level integration and practical realization of these elements, rather than in introducing fundamentally new theoretical insights or learning paradigms. Consequently, LD-EnSF may be viewed as an effective and elegant evolution of Latent-EnSF, replacing the VAE and full-state propagation with a better learned surrogate while maintaining the same overall Bayesian filtering framework.\n\nMoreover, when comparing LD-EnSF against variational methods such as 3D-Var or 4D-Var, it should be noted that those methods can also leverage surrogate models like LDNet to reduce computational cost, provided the surrogate is differentiable for adjoint-based optimization. Thus, the runtime advantage of LD-EnSF does not arise solely from operating in latent space, but also depends on whether comparable surrogates are incorporated into the variational baseline. Explicitly discussing this relationship would clarify that LD-EnSF’s main strength lies in its practical integration of latent surrogates with score-based filtering—an important step forward, but not a fundamentally new formulation of data assimilation.\n\nRelation to recent latent-space Bayesian filters:\nThe paper would benefit from discussing its relationship to other recent approaches that perform Bayesian filtering directly in learned latent spaces, such as the Deep Bayesian Filter (DBF, ICML 2025).\nWhile both LD-EnSF and DBF share the goal of combining learned latent dynamics with probabilistic filtering, they differ in philosophy: DBF integrates inference and dynamics in an end-to-end generative framework, whereas LD-EnSF decouples the surrogate dynamics learning (LDNet) from score-based filtering.\nPositioning LD-EnSF more explicitly within this broader family of latent-space Bayesian filters—highlighting differences in training objectives, inference mechanisms, and scalability—would clarify its contribution and increase its relevance to ongoing developments in this research area.\n\nComputational Complexity and Implementation Practicality: \nWhile the paper convincingly demonstrates that LD-EnSF achieves major runtime savings by performing data assimilation in a low-dimensional latent space, its computational efficiency still involves important trade-offs. Each assimilation cycle requires iterative reverse-time SDE integration for $N_e$ ensemble members over $T_{diff}$ diffusion steps, leading to a total cost of $O(N_e T_{diff} d_s)$. This iteration structure is conceptually analogous to the optimization loops in 3D-Var, with the distinction that LD-EnSF's iterations are fully parallelizable across ensemble members but sequential in diffusion time. Moreover, the ensemble size $N_e$ generally needs to grow with the latent dimension to maintain statistical accuracy, as analyzed in recent works (e.g., Oko et al., 2023). Discussing these trade-offs explicitly would help clarify when LD-EnSF offers practical computational advantages over variational baselines, especially since 3D-Var can also benefit from surrogate models such as LDNet if differentiable.\n\nIn terms of implementation, the overall pipeline remains quite complex: it integrates multiple neural components (latent dynamics network, reconstruction network, and LSTM-based observation encoder) with a non-trivial score-based assimilation algorithm. Implementing and tuning such a system—including hyperparameter searches, ensuring stable LDNet training, and discretizing the SDE solver—requires substantial expertise. While the methodology is sound, this complexity may pose barriers to adoption compared to simpler ensemble or variational filters. Providing open-source code or detailed pseudocode would greatly enhance reproducibility and accessibility for the community.\n\nLimitation and Clarity of Latent-State Initialization: \nThe paper initializes all latent trajectories with a fixed zero state ($s_{-1} = 0 $) and relies on the parameter input $u_t$ to encode differences among trajectories. While this design simplifies training and stabilizes the latent dynamics, it implicitly assumes that all variation in the initial conditions can be captured through the parameter space. In practice, however, many physical systems exhibit diverse and high-dimensional initial states that cannot be fully represented by a few parameters. As a result, this assumption may limit LD-EnSF’s generalization to systems with unseen or highly variable initial conditions.\n\nIn addition, the paper’s explanation of this mechanism is somewhat unclear. The statement that the initialization “flexibly accommodates varying initial conditions” is misleading, since all trajectories start from the same latent point. Figure 1 also omits how the initial latent state is introduced or how it connects to the LSTM encoder, which may confuse readers about how the model handles initial-state diversity. Clarifying this design choice—possibly with an explicit schematic or additional discussion—and considering a learned initial encoder (e.g., mapping initial full states to latent $s_{-1}$) would both improve clarity and broaden the applicability of LD-EnSF to more general dynamical regimes.\n\nEvaluation Scope: \nA minor weakness is that the experimental comparison could be broadened or analyzed further. The authors did not compare against particle filtering or variational methods (e.g., 4D-Var). It’s understandable given those struggle in such high-dimensional sparse settings, but discussing their expected performance or including them in a smaller-scale experiment would strengthen the positioning. Additionally, an ablation study on each novel component of LD-EnSF (for instance, using a static VAE-based encoder instead of LSTM, or using the ground truth dynamics vs. LDNet during assimilation) would isolate the contributions of each part. While the paper does compare LDNet vs VAE offline, it doesn’t explicitly show the effect of the LSTM encoder in the online phase versus a baseline. Such experiments (perhaps in an appendix) would provide deeper insight into how much each innovation (smooth latent dynamics, history encoder) contributes to the final performance.\n\nMinor Presentation Issues: \nThere are a few presentation details that could be improved. For example, the text’s statement of speedups (e.g. “$2\\times10^3$ times speedup”) seems slightly inconsistent with Table 2 values – clarifying these calculations would avoid confusion. Also, the results discussion could more explicitly highlight why LD-EnSF outperforms baselines (e.g., pointing out in text that EnSF fails due to vanishing likelihood gradients in unobserved dimensions, which LD-EnSF overcomes by using informative latent gradients). The figures are generally clear; still, adding a bit more explanation in the captions or main text for Figure 4 (e.g., explaining the behavior of “Latent-EnSF-dyn” curves, or noting that LETKF diverged in the hardest case) would be helpful. These are minor issues and easily addressable.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ef7NwBaorZ", "reviewer": "Reviewer_YcmD", "summary": "## Summary\nThis paper introduces LD-EnSF, a data-driven framework for data assimilation (DA) that aims to solve two major challenges: the high computational cost of numerical simulations and the difficulty of applying score-based methods to sparse observations. The core idea is to (1) learn a fast surrogate model that forwards the system dynamics in a low-dimensional latent space, and (2) use a LSTM encoder to map sparse observations into this latent space, where EnSF is then performed. The numerical experiments on three challenging high-dimensional benchmarks show that LD-EnSF achieves notable speedups while maintaining high accuracy even with sparse observations.", "questions": "## Questions\n\n- In Line 47, what is meant by *score becomes ill-posed*? This phrasing is imprecise. Could you clarify? The term *ill-posed* refers to problems (typically inverse problems) that fail to meet one or more of the following criteria: a solution exists, the solution is unique, and the solution depends continuously on the data. There is no such definition for a function itself, e.g., score function. Do you mean the estimation of the score from sparse data is high-variance, or simply that the likelihood gradient is zero in unobserved dimensions? \n- The authors claim the latent representation itself addresses observation sparsity, e.g., Line 48-50. But no detailed justification. Could authors elabroate on: (1) How can the latent projection itself solve the sparsity problem, given that the input observation (y_t) is indeed sparse? A latent projection does not create information for that is not already present. (2) Should this capability not be attributed almost entirely to the \" LSTM Encoder\"? By processing a sequence of sparse observations, this encoder is implicitly learning a time-delay embedding. This concept has been used in data assimilation, yet this is not discussed and cited.\n- In Line 50, the authors state that \"*latent representations enable more informative gradients*\". Could you please explain what \"more informative gradients\" means in this context? How exactly does working in the latent space make the gradients better for handling sparse observations compared to working in the original space? \n- It is a valuable strength to handle the irregular grid observation data which is commen in DA. Could the author include a numerical case study demonstrating the method’s performance on irregular grids?\n- In Line 210, the author state the random projection $B$ is trianbale. However, in Random Fourier Features/positional encodings, $B$ is typically fixed [Tancik et al., 2020]; optimizing it can change the random feature distribution across update steps and making the training unstable. This is an unconventional choice and its benefits are not inituitive. Could authors provide justify the reason why set $B$ as trainable and provide an ablation study: fixed versus trainable $B$, including training loss curves and DA accuracy.\n\n\n## References\n- Takens, Floris. \"Detecting strange attractors in turbulence.\" Lecture Notes in Mathematics, Berlin Springer Verlag 898 (1981): 366.\n- Noakes, Lyle. \"The Takens embedding theorem.\" International Journal of Bifurcation and Chaos 1.04 (1991): 867-872.\n- Gottwald, Georg A., and Sebastian Reich. \"Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations.\" Chaos: An Interdisciplinary Journal of Nonlinear Science 31.10 (2021).\n- Tancik, Matthew, et al. \"Fourier features let networks learn high frequency functions in low dimensional domains.\" Advances in neural information processing systems 33 (2020): 7537-7547.\n- Si, Phillip, and Peng Chen. \"Latent-EnSF: A Latent Ensemble Score Filter for High-Dimensional Data Assimilation with Sparse Observation Data.\" The Thirteenth International Conference on Learning Representations (2025).\n- Tarumi, Yuta, Keisuke Fukuda, and Shin-ichi Maeda. \"Deep Bayesian Filter for Bayes-Faithful Data Assimilation.\" Forty-second International Conference on Machine Learning (2025).\n- Yang, Yiming, et al. \"Tensor-Var: Efficient Four-Dimensional Variational Data Assimilation.\" Forty-second International Conference on Machine Learning (2025).", "limitations": "", "rating": 2, "confidence": 5, "soundness": 1, "presentation": 2, "contribution": 1, "strengths": "## Strengths\n- The authors provide a comprehensive numerical studies on three challenging, high-dimensional benchmarks and present solid comparisons of LD-EnSF against strong baselines. The results show the advantages of proposed methods in computational efficiency and assimilation accuracies.", "weaknesses": "## Weakness\n- The core idea of performing EnSF in a latent space was already achieved by Latent-EnSF [Si and Peng, 2025]. This paper's primary contributions are (1) replacing the VAE encoder from Latent-EnSF with an LSTM encoder and (2) replacing the full-space dynamics with an LDNet surrogate. While this is a successful *engineering* effort that yields speedups and imporved performances, the novlties in machine learning for DA is minimal. The paper does not provide sufficient theoretical justification for why this specific combination is fundamentally superior, beyond empirical performance.\n- The paper claims the latent representation addresses sparsity. However, the actual mechanism enabling the system to work is the \"*history-aware* LSTM encoder\". By processing a sequence of sparse observations $(y_{1:t})$, the LSTM is implicitly learning a time-delay embedding. It is a well-established concept (e.g., via Takens' embedding theorem [Takens et al. 1981; Noakes et al., 1991]). The paper does not discuss this mechanism entirely. And, It lacks a large and highly relevant recent advances of such works for DA, for example Gottwald et al., 2021; Tarumi et al, 2025; Yang et al, 2025.\n- Several key statements and claims regarding the method's motivation and mechanism are vague and require further clarification and justification, see Questions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)