Sparse Diffusion Autoencoder for Test-time Adapting Prediction of Complex Systems
Authors: Jingwen Cheng
Ruikun Li
Huandong Wang
Yong Li
Venue: NeurIPS 2025 poster ()
Keywords: Complex system
Diffusion
Primary area: machine_learning_for_sciences
TL;DR: We propose a novel Sparse Diffusion Autoencoder that efficiently predicts spatiotemporal dynamics and dynamically self-adjusts at test-time.

Abstract: Predicting the behavior of complex systems is critical in many scientific and engineering domains, and hinges on the model’s ability to capture their underlying dynamics.
Existing methods encode the intrinsic dynamics of high-dimensional observations through latent representations and predict autoregressively. 
However, these latent representations lose the inherent spatial structure of spatiotemporal dynamics, leading to the predictor's inability to effectively model spatial interactions and neglect emerging dynamics during long-term prediction.
In this work, we propose SparseDiff, introducing a test-time adaptation strategy to dynamically update the encoding scheme to accommodate emergent spatiotemporal structures during the long-term evolution of the system. 
Specifically, we first design a codebook-based sparse encoder, which coarsens the continuous spatial domain into a sparse graph topology. Then, we employ a graph neural ordinary differential equation to model the dynamics and guide a diffusion decoder for reconstruction.
SparseDiff autoregressively predicts the spatiotemporal evolution and adjust the sparse topological structure to adapt to emergent spatiotemporal patterns by adaptive re-encoding. 
Extensive evaluations on representative systems demonstrate that SparseDiff achieves an average prediction error reduction of 49.99\% compared to baselines, requiring only 1\% of the spatial resolution.

Sparse Diffusion Autoencoder for Test-time Adapting Prediction of Complex Systems
Jingwen Cheng" Department of Electronic Engineering Tsinghua University Beijing, China
Ruikun Li
Shenzhen International Graduate School Tsinghua University Shenzhen, China
Huandong Wang Department of Electronic Engineering BNRist, Tsinghua University Beijing, China
Yong Li Department of Electronic Engineering BNRist, Tsinghua University Beijing, China
Abstract
Predicting the behavior of complex systems is critical in many scientific and en- gineering domains, and hinges the model's ability capture their underlying dynamics. Existing methods encode the intrinsic dynamics of high-dimensional observations through latent representations and predict autoregressively. However, these latent representations lose the inherent spatial structure of spatiotemporal dy- namics, leading to the predictor's inability to effectively model spatial interactions and neglect emerging dynamics during long-term prediction. In this work, we pro- pose SpfrseDiff, introducing a test-time adaptation strategy to dynamically update the encoding scheme to accommodate emergent spatiotemporal structures during the long-term evolution of the system. Specifically, we first design a codebook- based sparse encoder, which coarsens the continuous spatial domain into sparse graph topology. Then, we employ a graph neural ordinary differential equation to model the dynamics and guide a diffusion decoder for reconstruction. SparseD- iff autoregressively predicts the spatiotemporal evolution and adjust the sparse topological structure to adapt emergent spatiotemporal patterns by adaptive re-encoding. Extensive evaluations on representative systems demonstrate that SparseDiff achieves average prediction error reduction of 49.99% compared baselines, requiring only 1% of the spatial resolution.
Introduction
Spatolemoora Dynamics
Probe Topalogy
The dynamics of complex systems are driven by the nonlinear interactions and co-evolution of numerous components, giving rise to rich emergent spatiotem- poral structures,
in fluid dynamics [. cli-
mate science [2]. and molecular dynamics [3]. Ac- curate long-term prediction of these systems cru- cial for many real-world applications 4 [5]. How-
Fresk-o
ever, complex systems are typically observed in high- dimensional spaces with unknown intrinsic dynam-
Figure Adapting probe topologies.
"qqual contribution (see $ Corresponning author (vanghaandong@ttsinguuaa edu. cn)
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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ics, making high-fidelity, long-term prediction a challenging and computationally expensive prob- lem 7
Data-driven methods offer an equation-free and computationally friendly approach. The core idea is to build reduced-order models of high-dimensional observations using encoder-decoders, which project observed states to a low-dimensional latent space for prediction, thereby improving computational efficiency 6 9
However, most existing work employs parameterized neural network encoders that implicitly encode spatial structure by compressing observational spatial information into a latent representation vector. This conversion of opatially-correlate data into latent vectors hinders predictors from effectively understanding and modeling the system's ' inherent spatial interactions and structural relationships [5]. For instance, in fluid convection, heat drives fluid to form rising plumes and sinking cold flows. These specific spatial structures, their relative positions, and interactions are critical for understanding and predicting convection patterns [3]. Considering that spatial interaction patterns and emergent structures continuously change during long-term evolution, losing such information progressively amplifies the misalignment between the encoder-decoder and predictor in long-tern predictions. This leads to core question: Can we construct reduced-order model that preserves crucial spatial structure and continuously adapts to newly emerging dynamics patterns during prediction?
This work is inspired by a recent finding that the system state in the entire spatial domain can be
effectively reconstructed from sparse observation points [14] [5] [6]. The data disparity between
sparse points and the full spatial grid strongly suggests a promising encoding paradigm: aggregating the full-space dynamic information onto much smaller set of sparse probes [17]. These probes and their spatial topology constitute the dynamical skeleton of the spatiotemporal dynamics, as shown in Figure[ [ In this paradigm, the model is capable of adapting to the system's latest spatiotemporal patterns during long-term predictions by dynamically adjusting probe positions and topology based on the re-encoded predicted states 16 (8) Nevertheless, achieving effective probe-based aggregation and adaptive updating in practice faces two key challenges: 1) A lack of theoretical understanding for aggregating rich spatiotemporal dynamics from continuous domain into small set of discrete probes; 2) Frequent re-encoding during long-term prediction introduces significant computational overhead and potential accumulation of prediction errors.
To address these challenges, we propose a novel Sparse Diffusion Autoencoder, SparseDifft consisting of a codebook-basee sparse encoder and an unconditional diffusion decoder. The sparse encoder learns a representative pattern codebook from historical spatiotemporal trajectories, enabling it to dynamically aggregate and project full-space observational data onto spatial topology formed by sparse probes, thus constructing a reduced-order model. On this probe topology, we model spatiotemporal dynamics using a graph neural ordinary differential equation and explicitly introduce diffusion term to capture spatial interactions and information propagation among probes. Finally, we pad the probe predictions to the full spatial domain to serve the initial state for the diffusion process, enabling rapid reconstruction of the full spatiotemporal field. SparseDiff's key innovation and strength lie in its test-time adaptation: during long-term prediction, it utilizes the learned codebook to re- encode the latest prediction, dynamically adjusting and constructing a probe topology better matched to the current system state to continuously adapt emerging spatiotemporal patterns. Experimental evaluation on simulated and real-world systems demonstrates that SparseDiff outperforms baselines by over 49. 9.99% in long-term predictions, and reliably predicts full-space spatiotemporal dynamics using less than 1 %% of grid points as probes.
The highlights of this work are summarized as follows:
We propose SparseDiff, a novel autoencoder that uses sparse probes to construct a reduced-order model that effectively preserves the spatial structure of spatiotemporal dynamics.
We introduce a test-time adaptation strategy via re-encoding, allowing the model dynamically adapt its probe-based representation to emerging dynamics patterns, significantly improving long- term prediction accuracy and computational efficiency.
Experiments demonstrate that Sparse significantly outperforms baselines in long-term predic- tion accuracy and achieves high efficiency by reliably predicting full-space dynamics using only approximately 1 1 of grid points as probes. Our code open-source: hhtpps: htt:://gttttt thub. com/ tsinghua- fib- lab/SparseDifff
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2 Preliminary
2.1 Problem Definition
In this work, we focus on spatiotemporal dynamics du - f(u, x, L, dz oe.,... on 2D regular spatial domain z C R2. Such systems can often be formalized as time-dependent partial differential equations, for example, the Navier-Stoker equations, which include Laplacian operator term acting on the spatial domain [9]. Without loss of generality, we address the problem of data-driven modeling of system dynamics from historical evolving trajectories U {u(r, t))², thereby extending our focus to real-world systems like climate dynamics where explicit equations are unknown, In practice, the spatial domain is discretized into an h w grid. Consequently, the full-domain observation state at each timestamp represented as a tensor u E x Given a historical observation trajectory of lookback steps, predict multiple future steps in autoregressive manner, forecasting one step a time.
8
2.2 Guided Diffusion for Sparse Reconstruction
Diffusion methods have recently been shown to reliably reconstruct the full spatial domain's 's system state, guided by sparse observations [16 15] [4]. Diffusion models aim to Iearn a probabilistic mapping from simple prior distribution, such standard Gaussian to complex target data distribution 22 21 [22]. This is achieved by defining a forward diffusion process that gradually adds noise data instances and training reverse process sequentially denoise them. We represent the original data point as Zy- The forward stage transforms Zy into a noisy version En, over n steps, governed by the equation vin v 20, vI En en and {ān) represent the Gaussian noise and noise schedule (23), respectively. The learned reverse diffusion process models the transition from noise back to data through sequence of conditional distributions given by
Po (zn- ( = (Z I.),
where 140 Va., (Zyy 1-0, co (n, (..nn n)) and { are step dependent constants. The term 6ø
vI-a.
represents the model's prediction of the added noise, typically implemented using parameterized neural network architecture like UNet Transformer. The optimization of this network's parameters
is performed by minimizing the objective function [23
Ln E Em,e...o an ||c, Co(V V1 Cn n2)13
This loss function is derived from the negative log-likelihood B 1- ( Once trained, the diffusion model progressively denoises from Gaussian noise yield high-fidelity data samples. To guide diffusion in reconstructing the full spatial domain state from K sparse observations M(xk), Bayes' rule guides the diffusion gradient direction [4 be
Z logp(zn|M) Re
CV 1ly M(xx)12
(3)
VI- a
where y represents the noise values at the sparse observation points and Ç = 1/0².
3 Methodology
In this section, we first introduce the proposed sparse diffusion autoencoder to discover the skeleton of spatiotemporal dynamics, namely the probe topology. Subsequently, we design a diffusion graph neural ordinary differential equation on the probe topology model spatiotemporal dynamics. Finally, we propose the test-time adaptation strategy to dynamically sense emerging spatiotemporal patterns in long-term prediction. The overall framework is illustrated in Figure
3.1 Sparse Diffusion Autoencoder
We introduce our approach for discovering the dynamical skeleton of complex systems, utilizing a codebook-based sparse encoder and guided diffusion decoder.
3

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Ob1-T
(o heste
Gurde:
CVa.lla -M(sx)
Encoder
Worky Wadey
Denoise
de
Probe Topology
Discrete
Pad
Accepiinn
Represeriaaioons
Codewords
Sparoterporal Dynamios
Starting points
Figure Overall framework of proposed Sparse Diffusion Autoencoder.
3.1.1 Codebook-basee Sparse Encoder
To discover the skeleton of spatiotemporal dynamics on the spatial domain, we first identify and record the rich dynamical patterns inherent in complex systems. Specifically, we maintain a codebook C' containing K codewords for spatiotemporal dynamics, where each codeword E IR is d- dimensional learnable vector. For given historical observation RRxxxx we employ an encoder E to encode along the time dimension, yielding a d-dimensional representation vector E Rdxhx for the full-domain states. We replace the representation vector with the most similar codeword c, which serves the decoder's input. The decoder then reconstructs the original observation ú. The encoder-decoder and the codebook are trained by minimizing the objective function [25]
L logp(ulc) + [[s][E(u)] - cl1 (l(E(u) sg[c]l2.
(4)
where sg indicates gradient detachment. Through this process, we effectively record the rich spa- tiotemporal dynamical patterns from the observation trajectories within the pretrained codebook.
In the inference stage, we first discretize the historical observation into set of hit codewords {c:) using pretrained encoder & and codebook C. We define the governing region of each hit codeword & as the set of spatial grid points that are mapped it, denoted as Ue.. Consequently. the full spatial domain is divided into k region types by the k hit codewords (a single region type might consist dispersed patches). These regions represent local-scale spatiotemporal units. Therefore, we accordingly construct probe set V = {u, E R j==1 to represent them, where / is the lookback window size.
For probe we select one spatial grid point from its corresponding codeword's governing region us its coordinate. We use random selection here instead of averaging, ensuring that the probe falls within patches even if the patches of its region are spatially dispersed. Subsequently, we average the historical observation sequences of all grid points within Ue, to aggregate them as probe states Vi R Finally, connect all probes to construct the topological structure g (V, E). and assign different weights Eij E E to each edge to characterize the spatial association strength. Specifically, edge weights ezy quantify the spatial association strength from probe o probe vy. We consider the governing region ue, associated with oy and spatial neighborhood of every grid point within uc We count how many grid points belonging to vy's governing region Ue appear within the combined neighborhoods of all points in de,, This count is normalized along all probes to determine the edge weight Cij. Thereby, we obtain the probe topology G of the spatiotemporal dynamics.
3.1.2 Guided Diffusion Decoder
We pretrain an unconditional diffusion model to capture the spatiotemporal patterns of the system state u. Given the set of probes V. we use these probe values U/ as sparse observations to guide the reconstruction by the diffusion decoder. Existing works [66] [5 [4] typically directly substitute probe values as the sparse observation M(*K) ) in Equation to guide reconstruction. However, our approach differs in that we guide the governing regions te, corresponding to the probes. That is, our probes represent the dynamical information of the entire governing region, not just single grid points. Therefore, we use probe value to guide diffusion in reconstructing its governing region.

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Specifically, we fill the probe value to each grid point within its governing region Ue serving as M(xx) in Equation 3 to guide reconstruction. Furthermore, considering the computationally intensive of the denoising process, we use the filled full-space state instead of Gaussian noise as the starting point for denoising accelerate the diffusion sampling process.
3.2 Probe-graph Diffusive Predictor
Given the probe topology G and probe states V1, we now predict the spatiotemporal dynamics of the system. Specifically, model this process neural diffusion process on the graph [26
hr = ho + l (A-I) hT,
ho Φ(v),
0/ = (h+),
where h2 refers to the representation at time 1, 6 and v are the parameterized encoder and decoders, and A is the time-varying diffusion coefficient between probes, calculated by parameterized attention network. To explicitly model the spatial association strength between probes, we introduce the edge weights into the calculation of the diffusion coefficient
(W kh)) T Wah + WEEL)
a(h,, h2) = softmax(
d2
where h; denotes the representation of the ith probe, W Q. W K. and WE are learnable matrices. We employ multi-head attention mechanism & Ex A to capture complex dynamical mechanisms.
3.3 Test-time Adapting Prediction
During long-term prediction, complex system dynamics exhibit continuous emergence of spatiotem- poral patterns, implying tire-ngrying dynamical spatial interactions. Therefore, continuously update the predictor during testing to adapt to the latest spatiotemporal dynamics patterns. For T T-step prediction, the prediction window is divided into N aab-windows {w,) N We assume each window possesses specific spatial interactions, with a corresponding prohe topology sequence (Gw., J.M. We update the probe topology at the beginning of each window through re-encoding. Specifically, at a window transition time, we decode the probe states back to full space. Following the method in Section 3.1.I then re-encode the probe topology & for the latest dynamics within the current window using encoder & and codebook C. This topology is updated again at the next transition time.
We design a dynamic update strategy to dynamically determine the window transition time. Instead of using fixed- fixed-length sub-windon nub-windows, continuously monitor whether the recent evolution of each probe remains well-aligned with its assigned codeword in the latent space of the encoder E. Specifically, after each re-encoding, governing region. For each codeword, a probe v; is randomly selected to represent its region and forms probe graph g for prediction. At each prediction step, compute the latent consistency score Xt by encoding the past T-step trajectory of each probe and evaluating its cosine similarity with the associated codeword G::
obtain hit codewords {c))/_ where each c; Rd corresponds
1 Σ cos (E(1)- (ool-Ti)...))
X
When Xt drops below a predefined threshold 5, it indicates that many probe representations have drifted away from their original latent codewords, suggesting mismatch between the current probe partition and the evolving system dynamics. In response, we decode the probe graph of the past T steps back to the full space using the diffusion decoder and recompute the probe topology via re-encoding. This strategy enables the model to adaptively reallocate codewords and update the graph structure in accordance with the varying speed and patterns of spatiotemporal evolution.
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Table 1: Average performance of the trajectories predicted from different initial conditions with standard deviation from 10 runs. The best results are highlighted bold.
Systems
RMSE
Lamada Onieaa
SSIM x10
RMSb x30
Nasier Stokes
XSUM x
KMSE
Swift Hobenberg
x10
SSSM XIU
RMSE y010
Cylinder Flww
SSIM X30
RMSSI
Real world
SSIM x310
5.03 a 0.507
9:224 2: 0.125
30,674 6: 2.7M4
6,606 a: UOI
9,508 0,029
1.200 d: 0,005
11,08 & 0.330
5.253 0,344
13082 de 2.777 7:268
3.195
PNO
5:138
0.557
15:90
7.00 6:
5.501
9431 0.009
19.55
1243H
0:775
1:355
7.022
3.789
GILED
6,200
0.305
12,334
0,455
8008
5.214
0.211
10021
=
0.291
10,304
Ouss
2 2.012:1
6.187
M.607
012218
11. 130
241
8,002 0.763
7.228
9:676
192
2200
7.392 0.371
7.867 4: 1:307
7.781 1.8144
$3.905
18.645
73.675
57.479
1.89%
62.6713
17.999
97.80%
15.51%
4 Experiments
In this section, we validate the accuracy and efficiency of SparseDiff on simulated PDE systems and real-world datasets. Furthermore, we evaluate its robustness and generalization ability and examine the specific contribution of its components through ablation studies.
4.1 Experimental Setup
We conduct experimental validation on four PDE systems, including: 1) Lambda-Omega; 2) Navier- Stokes; 3) Cylinder Flow; and 4) Hoitt-eohenberg systems. They encompass complex diffusion effects, convection terms, and higher-order spatial interaction terms, among other nonlinear dynamic components. Additionally, we also evaluate SparseDiff's performance in real-world applications on an open-source climate record dataset [27]. All models are trained on trajectories with different initial conditions and predict long-term (more than 100 steps) on new trajectories. Details on the equations, data generation, and training settings are in Appendix
We compare our method against a set of state-of-the-art spatiotemporal forecasting models, including operator learning methods, recurrent architectures, and generative frameworks. Specifically, we consider Fourier Neural Operator (FNO) (28), ConvLSTM [29]. UNet [30], and
Baselines
G-LED B1
A detailed description of each baseline is provided in Appendix
4.2 Main Results
PDE systems Table T shows the long-term prediction performance of all models. SpfrseDiff achieves the optimal prediction quality almost all scenarios. This indicates that SparseDiff's test- time adaptation enhances long-term prediction by timely sensing of new spatio-temporal dynamics. Furthermore, compared to other baselines, another characteristic of SparseDiff is that it performs dynamic prediction on the probe topology. Other methods predict in the original or uniformly downsampled grid space and, unlike SparseDiff, fail discover the spatial interactions of complex systems and structure them into an explicit topology to enhance the predictor's ' representational capacity.
Real-world dataset We use the SEVIR dataset [27] for real-world weather forecast- ing. Specifically, we select the GOES-16 Chan- nel (99 (6.9pm) infrared imagery ir069, which captures mid-level water vapor and is widely used for storm tracking. Each frame covers 384 x 384 km area at 2 km resolution, yielding 192 192 grid, with a temporal resolution of 5 minutes. Experimental results are shown in Figure 3, where SparseDiff achieves the most faithful prediction of complex climate patterns with the lowest error. This indicates that the proposed probe-based encoding scheme is of great value for applications in real-world com- plex systems, especially in large-scale observational data scenarios in geoscience. SparseDiff encodes
SperseDif
17.5
68
12.66
RN4DEE (x10-3)
-2.
10.90
7.74
FNO
G-LED
7.5
86
Chan
Figure 3: Real- world dataset. (a) Prediction visualiza- tion of different models. (b) RMSE comparison of dif- ferent models.

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b
SparseD
SparsoD
0.8
Ours Unweightd
11,55
SSIM
0.6
0.0
Emrer
04
02
32
512
1024
32
64
128
250
612
1024
LG
hs System
Climate
Partitin Size
Partition Size
Figure 4: Ablation studies. Prediction performance with (a) uniform and (b) random probe selection. (c) Impact of probe topology edge weights prediction.
high-dimensional observations into sparse probe topology, thus reducing computational overhead while ensuring accuracy.
4.3 Ablation Study
We capture a compact low-dimensional skeleton spatiotemporal dynamics through codebook Iearn- ing, thereby improving prediction performance. The contrasting version is uniform downsampling random selection of probes. We also establish heuristic calculation rules for the edge weights of the probe topology. The contrasting version is unweighted graph (where all weights set to 1). In the following, we ablate these two key components.
Selection of Probes Probes are the most critical components the SparseDiff framework, as they serve the information carriers that represent the coarse-grained structure of the spatiotemporal dynamics. Their selection fundamentally determines the prediction quality of the entire model. Figures D and b illustrate the prediction performance of SparseDiff on the Hohen-eo system when the probes are selected via spatially uniform and random sampling, respectively. We observe that even when the number of probes is increased to 1024, these two variants still perform significantly worse than our method. In contrast, the original version achieves high-quality predictions with as few 150 probes. This highlights that our proposed codehook-based sparse encoder not only compresses the spatial domain effectively but also selects informative and dynamically representative probes. demonstrates strong ability extract the low-dimensional spatiotemporal skeleton where the intrinsic dynamics reside.
Probe-Graph Edge Weights When the edge weight feature is disabled, the edge-feature-awar attention mechanism described in Section degenerates dedgeents into standard inter-node attention that treats all probe connections equally. Without incorporating the spatial association strength Cij. the model loses the ability capture relative distances and directional relationships between probes, which essential in spatiotemporal systems. As shown in Figure 4 this leads to a notable performance drop on the Navier-Stokes system, where accurate modeling spatial interactions critical for capturing fine-grained dynamics.
4.4 Robustness
Here, we evaluate the robustness of SparseD- iff data observation noise and codebook size settings. Specifically, we first test the predic- tion performance with different preset codebook sizes on the Navier-Stolee system. The code- book size is the upper limit on the number of activated codewords. It determines the maxi- mum number of probes SifrseDiff can select to perceive spatiotemporal dynamics. Experi- mental results are shown in Figure 5p. where the prediction accuracy (SSIM) converges after the codebook size reaches 150. For the Navier- Stokes system with 128x 128 resolution, this is
0.78
0.55
0.70
0.74
077
0.9
0700
0.2
100
150 200
259
200 0.%
1.0%
Codaboon
Nekee Strengh
Figure 5: Robustness experiments. (a) Impact of code- book size on SparseDiff's performance on the Navier- Stokes system. (b) Impact of the noise SparseDiff's performance on the real climate dataset.
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quite small number. This indicates that SparseDiff utilizes probes efficiently and achieves superior prediction with very low computational overhead.
Next, we examine SparseDiff's robustness to real-world noise. We apply Gaussian noise with varying percentage intensity to real-world climate dataset (relative the original data amplitude). Experimental results are shown in Figure 5p, where SparseDiff's performance deteriorates as the noise increases, but moderate noise levels, still outperforms the baseline without noise.
4.5 Generalization
We examine SparseDiff's out-of-distribution generalization ability. Specifically, using the Cylinder Flow system the experimental sub- ject, we evaluate SparseDiff's ability to pre- dict flows with Reynolds numbers not included in the training set. For turbulent systems, the Reynolds number influences the viscous coeffi- cient, thereby affecting collision frequency. For fluids with high Reynolds numbers, the flow hibits chaotic tendencies and is thus difficult to predict over long periods. We train SparseDiff on Reynolds numbers less than 500 and predict on larger Reynolds numbers. Experimental re- sults are shown in Figure [ where the baseline shows rapid performance degradation in out-of- distribution scenarios, whereas SparseDiff's s performance consistently outperforms the baseline and exhibits fluctuations. The reason may be that a rich set of spatiotemporal dynamic patterns recorded in SparseDiff's pretrained codebook. Although there are differences in the turbulent dynamic behav- iors at different Reynolds numbers, local small-scale dynamic patterns share similarities and are thus recognized by SparseDiff and accurately generalized.
SperseCill
Seymsiice
Figure 6: Generalization experiments. Visualized (a) ground truth and (b) SparseDiff prediction of turbulent at Re = 660. (c) SSIM as a function of Reynolds number.
4.6 Trade-off between Accuracy & Efficiency
Here, we analyze how SparseDiff' trades off between accuracy and efficiency. Specifically, during testing. SpfrseDiff can adjust the probe topology through re-encoding, thus construct- ing the most suitable form for newly emerging spatiotemporal dynamics. Frequent adjustments allow for sufficient perception of real-time dy-
b
0.08
RMSE (xx10-)
Neercaiing tre
He-encainng tran
i
namies, but the corresponding computational overhead increases. We conducted tests at dif- ferent update intervals on the Navier-Stokes sys- tem, and the results are shown in Figure 7 where Rollout Step refers to the time index the au- toregressive prediction sequence, and each step corresponds to one forward prediction in the rollout process. According Figure accuracy and time overhead exhibit a negative correlation.
0.00
Time Gost (5)
Step
Figure 7: Tradc-off of accuracy efficiency, Circles represent SparseDiff, and crosses represent GLED,
We also compare with the best-performing baseline, G-LED. accuracy by decoding and re-encoding at intervals, but its accuracy ceiling is lower than SparseDiff, and its time overhead is much greater than SparseDiff. Our adaptive re-encoding strategy proposed in Section 3.3 achieves a balance between accuracy and efficiency, as shown by the red star in Figure 7h.
enhance long-term prediction
4.7 Comparison of Graph Construction Schemes
To validate our proposed region-aware edge weighting scheme, we compare it against two primary baseline categories: (a) Learnable Graph Construction: The Graph Kernel Network (GKN) [32. method designed for learnable graph construction; and (b) Heuristic Alternatives: Using the same GRAND predictor but replacing learned eij with static graph structures, including k-nearest

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neighbors (kNN) based on Euclidean coordinates (k 20% of probe count), and spectral clustering followed by kNN in the embedding space.
Table 2: Comparison of different graph
We report the average RMSE over a 100-step autoregressive prediction on the L.0 system. As shown in Table our region- aware edge scheme yields significantly better performance than both the learnable GKN and the heuristic variants. This demon- strates that explicitly encoding region-level adjacency improves the expressiveness of the model for probe-baue based dynamics.
construction methods.
Method
RMSE 10
Ours
2.873
GKN
4.210
GRAND KNN
3.287
GRAND spectral
3.019
5 Related Work
5.1 Encoding Dynamics of Complex Systems
Discovering the -i-dimeeniooal latent space containing the intrinsic dynamics of complex systems represents a core challenge in long-term prediction. To accelerate spatiotemporal dynamical system prediction, some earlier studies focus on approximating solutions on coarse grids. Lee al. 7 utilize Gaussian processes and diffusion maps to coarse-grain high-resolution microscopic observations into coarse-scale PDEs. Bar-Sinai al. [К]] employ uniform grid downsampling of the continuous spatial domain for specific nonlinear partial differential equations. This coarse-grid approach has inspired several recent works (3). With advancements in representation learning, autoencoders have been employed to data-drivenly uncover the low- -iwddmensiooal latent space of high-dimensional observations [6 (3]. Prediction models such as recurrent neural networks [6] and neural operators [)] operate directly within the diw-dimeniional space constructed by autoencoders. To ensure the latent space aligns with physical intuition, existing research suggests integrating autoencoders with physical priors [36.. Li et al. (II] restrict the output semantics of the encoder by specifying timescales and intrinsic dimensions. Wu et al. (TO) combine delay embeddings with feature embeddings to discover the -wwdimmensoonal manifold of high-dimensional observations. Additionally, some work (37 learns the latent coordinates of partial differential equations through autoencoders to reveal governing equations. In contrast these methods, our proposed probe topology adaptively senses spatial structure, without enforcing a regular grid. This explicitly preserves the tight spatial correlations of spatiotemporal dynamics with a high compression ratio.
5.2 Generative Modeling for Complex Systems
Diffusion models have performed remarkably well in generative tasks, with their significant achieve- ments in video synthesis [38 339 and time series modeling 44 41 inspiring numerous studies in complex system dynamics prediction. Some works [42] 15] have utilized large-scale pre-trained
diffusion models to reconstruct high-fidelity data from lower-fidelity samples sparse measure- ment data. Physics knowledge is also integrated. For example, known partial differential equations provide physical constraints (PDE loss) for the denoising step, improving accuracy [43] Beyond reconstruction, diffusion models have also been applied to generate specific dynamical data. Li et al. [44] propose machine learning approach for generating single-partich trajectory data in high Reynolds number three-dimensional turbulence. Lienen et al. [13 treat turbulence simulation itself generative task, employing diffusion models capture the distribution of turbulence induced by unseen objects and generate high-quality samples for downstream applications. Other strategies focus on directly embedding dynamics or prediction processes into the diffusion mechanism. Cachay et al. [45) align the temporal evolution axis of spatiotemporal dynamics with diffusion process steps, replacing noise injection with temporal interpolation and the denoising operation with prediction. G-LED [1I, a more direct approach, incorporates system states as prediction targets and utilizes predicted future frames as conditions to guide the diffusion model in reconstructing the original high- fidelity states. Li et al. [5] embed multiscale features as conditioning for Diffusion modeling, while 46 dynamically adjust the number of denoising steps based on dynamical timestamps. Compared to these methods, we introduce novel sparse encoder working in tandem with a diffusion decoder. This combination not only reveals the low-dimensional spatial structure of long-term dynamics but also enables time adaptation to emerging spatiotemporal dynamics.

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6 Conclusion
Modeling the long-term dynamics of complex systems a challenging problem. Data-driven methods often constrained by the difficulty in explicitly preserving the inherent spatial interactions of spatiotemporal dynamics in latent vectors. This information loss is exacerbated by new patterns that continuously emerge during the system's long-term evolution. Therefore, we propose novel Sparse Diffusion Autoencoder, SparseDiff, a prediction framework that preserves the system's spatial structure and adapts test-time. SparseDiff coarsens the full spatial domain state into probe topology by means of discrete codebook learning to construct a compact skeleton of spatiotemporal dynamics. The pretrained codebook is able capture the rich dynamic patterns of complex systems. Once training is complete, SparseDiff' can adjust the probe topology to adapt to emerging dynamic patterns during testing and without updating weights. On the probe topology, we design an edge- weight-aware graph neural diffusion ordinary differential equation to model spatiotemporal dynamics, thereby predicting the future states of the probes and guiding the diffusion model to efficiently reconstruct back to the full spatial domain. Experiments on simulated and real-world systems show that SparseDiff can achieve optimal long-term prediction and exhibits excellent robustness and generalization ability.
Limitation & Future Work SparseDiff's 's spatiotemporal dynamics module relies on the perception of edge weights, therefore the rules for calculating weights will affect long term-termmm prediction quality, Furthermore, the state initialization of probes is achieved through average aggregation, which may lose high-frequency information a few grid points. Future research will focus on end-to-end graph structure learning and probe initialization strategies.
Author Contributions
Ruikun Li, Huandong Wang and Yong Li launched this research and provided the research outline. Ruikun Li, Jingwen Cheng and Huandong Wang designed the research methods. Jingwen Cheng performed the experiments and prepared the figures & technical appendix. Ruikun Li wrote the original manuscript. Huandong Wang and Yong Li provided critical revisions.
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under grant 62171260 and 92270114.
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Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There many potential societal consequences of our work, none which we feel must be specifically highlighted here.
A Data Generation
Below, we provide an overview of the dynamics and data generation process for each complex system. Lambda-Omega (I.O) system is governed by
dt Au + (1 - x² yu 3(1² + v²)u
(5)
8z = u. (1 - u² -²) - B(u² + v²) J²),, where the Laplacian operator. Here, we set Mo and uo to 0.5, while 3 1.0. Navier-Stokes (NS) system is governed by:
We + (u
V lw vAw + f
(6)
Vi w
(7)
O
Ou
(u, v)
(8)
Oy
0r
dr
du
V u) 2 dr Dy
(9)
where f A (sin (2 y + s)) + cos (2 + y + s))) is the driving force, and is the viscosity
coefficient. Parameter values used in this simulation are set as follows: v = 1.0, forcing amplitude 0.1, and phase shift s 0.
Swift-Hohenberg (SH) system is simulated as:
du = ru - 2Au - A + gu² u²,
(10)
ôt where r the linear instability parameter, g controls the strength of the quadratic nonlinearity, and A denotes the biharmonic operator. In this system we set to 0.7 and 1.0 separately.
Cylinder flow (CY) system is governed by:
8 Vp
i
Vu
Au,
&
a 3
(11)
1
Vp
Av.
a
where a is set to 1.0, and 3 corresponds to the dynamic viscosity 1a as defined in Equation 12
The systems under study are simulated across a diverse range of initial conditions. Temporal downsampling by a factor 10 and 25 are applied to the LO and NS systems separately to reduce redundancy, while all the systems spatially rescaled uniform 128 x 128 resolution.
The Cylinder flow system modeled via the lattice Boltzmann method (LBM) 6 capturing complex vortex shedding phenomena governed by the Savier-Stokes equations in the presence of cylindrical obstacle. The simulation operates on lattice velocity grid, where relaxation dynamics are dictated by the kinematic viscosity and the Reynolds number. To ensure data quality, we begin recording after the flow reaches statistically steady turbulent regime. The outputs are resampled in time by factor of 300 and spatially interpolated to 128 x 64 grid. We generate 50 training and 20 testing trajectories across varying flow conditions, with Reynolds numbers uniformly sampled: 10 training samples in the range Re E [100, 500) and 10 out-of-distributiin (OOD) samples in Re E [500, 1000).
Based on the relation
D D
P
(12)
where p = 1, U, = 0.08, and = 0.2, the corresponding dynamic viscosity spans E [3.2 x 10-5 1.6 10-2) for training and μ [1.6 10-3 3.2 10-51 for OOD cases.
To standardize input scales and facilitate stable training, we apply min-max normalization indepen- dently across all channels.
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B Model Architecture
Our model is composed of three key modules: Codebook-basee Sparse Encoder for spatiotemporal discretization, Probe-graph Diffusive Predictor for latent dynamics modeling, and Guided Dif- fusion Decoder for reconstructing full-resolution system states. Below, we detail the architectural settings used in each component.
Sparse Encoder hidden_ddal dim = 1024
# hidden dimension of NLP
embedding. dim 512
# Latent dimension (d)
num embeddingg M
# Codebook size, hyperparameter
Diffusive Predictor
input_steps 10
number of lookback steps
feature_din 256
Feature dimension
num num_heads
# Attention heads for calculating Matriz A
ODE_method = 'r44'
# Numerical solver for ODE integration
Unconditioned Diffision
n_ channels 128
Base number of channels
ch_ th_mults [1, 2, 2] Channel mul multipl tipplerr for each resolution level
is _s_attn = [False, False, True] Whether to apply self-attention
dropout 0.1
Dropout rate in residual blocks Number of residual blocks per resolution level
n blocks 2
Table 3: Trainable parameter counts (in millions).
SparseDiff'
Model
FNO
ConVLSTM
UNet
G-LED
Encoder
Predictor
Diffusion
Params (M)
2.3 x 10
1.32
25.8
23.90
5.47
10.91
26.3
C Baseline Implementation
We provide a brief description of the baseline methods used for comparison in our experiments. These methods represent diverse set of state-of-the-art approaches for modeling spatiotemporal dynamics. Their corresponding trainable parameter counts are summarized in Table
FNO 28
Fourier Neural Operator leverages fast Fourier transforms to model spatially continuous operators, enabling efficient learning of solution mappings for partial differential equations. is widely adopted for learning complex physical dynamics.
ConvLSTM integrates convolutional structures into recurrent networks, allowing spatial correlations be preserved while capturing temporal dependencies. It is standard baseline for video and sequence-based spatial forecasting tasks.
ConyLSTM 29
UNet employs encoder-decoder structure with skip connections to effectively combine global context and local details. particularly suitable for dense prediction tasks involving structured outputs.
UNet 33
G-LED 31 G-LED a generative latent evolution model that models temporal dynamics using autoregressive attention in latent space, and reconstructs full-resolution outputs using a Bayesian diffusion model. achieves strong performance on high-dimensional physical systems.
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D Additional Results
D.I Application to Irregular or Sparse Spatial Domain
While our current implementation operates on 2D regular grids, SparseDift can be applied to irregular or sparse spatial data by first transforming the inputs into regular grid through interpolation or zero-filling enabling processing within the same framework.
To validate this, we conduct two experiments:
Sparse sampling Navier-Stokes system: On the Navier-Stokes system, randomly sample only 10% of grid points. These sparse samples are then interpolated onto a 128x 128 regular grid before being fed into SparseDiff. Despite the sparsity, SparseDiff outperforms baselines that given full-resolution inputs, as shown in Table
Table 4: Sparse sampling on Navier-Stokes system. report average metrics of 100 prediction steps.
Method
RMSE x10-²
SSIM x x10-11
FNO (full grid)
15.99
7.26
G-LED (full grid)
12.33
8.10
SparseDiff (full grid)
11.39
8.36
SparseDiff (10% observations)
12.27
8.19
We also evaluate SparseDiff' on the 2D wave equation with observations sampled on irregular meshes. These inputs are first completed onto regular grids using interpolation within observed regions and sero-filling excluded regions before being processed by the model. Compared to FNO, SparseDiff maintains stable prediction accuracy (Table 5).
b. Irregular 2D wave equation dataset 47
Table 5: Results the irregular 2D wave equation datasct 47 of 100 prediction steps.
Method FNO
RMSE x10-²1
SSIM x10
23.76
7.81
SparseDiff
15.58
8.25
While these results demonstrate practical applicability irregular sparse inputs, we acknowledge certain limitations of this approach:
For highly non-uniform spatial distributions, interpolating uniform grid be problem- atic. Also, the sparse observation may be too made-sd to allow reliable interpolation, leading unnecessary computational cost significant reconstruction errors.
If the distribution spatial inputs shifts between training and testing, performance may degrade, as our current model lacks coordinate-querying capabilities (unlike operator- learning models such as neural operators), and cannot generalize across arbitrary spatial layouts.
Nevertheless, the core design SparseDiff is not inherently tied to regular grids. The key innovation of SparseDiff lies in its latent probe representation, which aggregates and models regional dynamics through sparse, learnable units. This representation is inherently flexible and does not require regular spatial structures.
The main constraint stems from the UNet-based diffusion decoder, which requires regular-grin -iiid input. However, this is design choice rather than a fundamental limitation of the method. In future work, the diffusion decoder could be replaced with architectures that naturally support irregular domains, such as Graph Neural Networks. Alternatively, it could be substituted with other super-resolution models, including transformer- based approaches and operator-learning models, further extend DifrseDiff to irregular spatial settings.
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D.2 Visualization
We provide long-term prediction visualizations on three representative systems: NS, SH, and CY. For each case, the left column shows the ground truth, the middle column presents the prediction reconstructed via the diffusion decoder, and the right column shows the vanilla reconstruction by directly filling each codeword region with its corresponding probe value.
GT
Pred
Vanilla Reconstruct
GT
Pred
Vanilla Reconstruct
Rolicut
Rollout
Rolicut
(a) CY system
(b) NS system
Vanilla Reconstruct
Rollout = 20
Rollout
(c) SH system
Figure Long-term prediction results on three representative systems.
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Reviews:
[{"review_id": "2ze8M5uVge", "reviewer": "Reviewer_sBsw", "summary": "This paper proposes SparseDiff, a novel framework for long-term prediction of complex spatiotemporal systems. SparseDiff uses a codebook-based sparse encoder to construct a “probe topology” capturing low-dimensional spatial structures, and a guided diffusion model to reconstruct full-domain states. The key innovation lies in the test-time adaptation mechanism that dynamically updates the encoding based on the current system state. The approach is validated on both synthetic PDE systems and real-world climate data, with strong quantitative and qualitative performance.", "questions": "-\tIn Section 3.1.2, you highlight the differences between your method and previous works [14, 13, 12], but these methods are not included in the experimental comparisons. Could you clarify why they were omitted?\n\n-\tHow is the value of k, described in Section 3.1.1, chosen, and how does it affect the final performance and effiency?\n\n-\tThe trainable parameter of SparseDiff are larger than all of other baseline, could you also compare  training GPU hours for SparseDiff and other baselines?\n\n\n## Typos:\n\n-\tIn line 101, \\epolsion should be \\epolsion_n? Keep consistent with Equation (2), where the subscript denotes the specific noise term at step n.\n\n-\tFigure 7(b) lacks explanation; it is unclear what the term 'Rollout step' refers to.", "limitations": "Yes.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n\n- The Test-Time Adaptation  mechanism based on latent similarity offers a practical solution for capturing emerging spatiotemporal structures, which help improve long-term prediction accuracy and inference efficiency.\n\n- Extensive experiments on diverse PDE benchmarks and real-world data demonstrate clear advantages over baselines, including FNO and G-LED.\n\n- The method that uses sparse probes to construct a reduced-order model is very interesting. However, I'm not very familiar with this field, so I'm not sure about its novelty.\n\n## Weeknesses :\n\n- The number of trainable parameters in SparseDiff is larger than that of all other baselines, and the training efficiency compared to the other methods is not clearly discussed.\n\n- The experimental configurations for the 'Robustness,' 'Generalization,' and 'Trade-off between Accuracy & Efficiency' sections are not consistent. For example, the Robustness experiment compares against G-LED on the Navier-Stokes system, while the Generalization experiment compares against FNO on the Cylinder Flow system. Could you clarify the rationale behind these choices, or consider adding experiments on additional PDE systems with more consistent baseline comparisons?", "was_revised": true, "final_justification": "I will maintain my positive score!", "rebuttal": "We sincerely thank you for your thoughtful and constructive feedback! We truly appreciate the time and effort you devoted to reading our paper and offering valuable insights.\n\n> ***Weakness 1 & Question 3: ...the training efficiency compared to the other methods is not clearly discussed; ...could you also compare training GPU hours for SparseDiff and other baselines?***\n\nWe sincerely thank the reviewer for raising this important point. After the initial submission, we revisited the model complexity of SparseDiff and experimented with a smaller version containing ~**25.8M** parameters, which is of similar scale to G-LED and FNO.\n\nNotably, this lighter model still achieves competitive or even better performance across various systems (RMSE × 10⁻²↓ ):\n\n| System      | 25.8M SparseDiff   | 55M SparseDiff    | G-LED|     \n|-------------|--------------------|-------------------|--------------------|\n| LO          | 2.912 ± 0.187      |2.873 ± 0.237     |6.506 ± 0.395      |\n| NS          | 11.130 ± 3.241     |11.390 ± 5.359     |12.334 ± 0.485     |\n| SH          | 7.628 ± 3.102      |7.700 ± 2.540      |8.214 ± 0.937      |\n| CY          | 8.544 ± 0.239      |8.012 ± 0.655      |10.021 ± 0.873      |\n| Real-world  | 7.957 ± 1.207      |7.742 ± 2.237      |10.304 ± 2.009      |\n\nInterestingly, on the NS and SH systems, the smaller model outperforms the original 50M-parameter version. We suspect this is due to improved optimization dynamics and stronger generalization.\n\nRegarding training efficiency, we conducted timing benchmarks on a single A100 GPU:\n\n- SparseDiff: ~500 minutes (400 epochs to converge)\n- G-LED: ~3100 minutes (400 epochs to converge)\n- FNO: ~30 minutes (20 epochs to converge)\n\nWhile diffusion-based models typically require more epochs to converge, SparseDiff is much more efficient than G-LED in both training and inference. This efficiency results from its sparse design and lightweight predictor.\n\nWe will include these comparisons in the final version to emphasize that SparseDiff’s strong performance is due to its structural advantages and effective design, not just a larger parameter count.\n\n> ***Weakness 2: The experimental configurations for the 'Robustness,' 'Generalization,' and 'Trade-off between Accuracy & Efficiency' sections are not consistent...***\n\nWe intentionally chose different PDE systems across sub-experiments to show that SparseDiff performs robustly across various settings, not just a single task.\n\nFor baseline selection, we compared against the **strongest-performing** method on each system to highlight SparseDiff’s advantages. For example:\n\n* On Navier–Stokes, G-LED performs best among baselines, so we use it for robustness experiments.\n* On Cylinder Flow, FNO has the highest SSIM, making it the most competitive baseline for generalization tests.\n\nTo improve consistency, we add the following experiments:\n\n**(a) Robustness to codebook size:**\nIn the robustness experiment on codebook size (Fig. 4a), we compare against G-LED on the Navier–Stokes system and report FNO’s performance, which yields an SSIM of 0.726. This confirms that SparseDiff outperforms multiple baselines when the codebook size is properly chosen and remains robust across various sizes.\n\n**(b) Robustness to noise:**\nWe report SparseDiff’s performance on the Navier–Stokes system with noisy inputs, in addition to results on a real-world system:\n\n| Noise Strength (%) | SSIM × 10⁻¹↑    |\n|-------------|--------------------|\n| 0.0          | 8.364 ± 0.992      |\n| 0.2        | 8.299 ± 0.328     |\n| 0.4        | 8.181 ± 0.623      |\n| 0.6        | 8.092 ± 0.837      |\n| 0.8        | 7.907 ± 0.477      |\n| 1.0        | 7.710 ± 1.402      |\n\nThis demonstrates that SparseDiff maintains strong performance under noise, with SSIM only slightly below G-LED’s when the noise exceeds 0.6%. This demonstrates the robustness of our model to input perturbations.\n\n**(c) Generalization:**\nWe evaluate SparseDiff’s generalization on the Navier–Stokes system, extending beyond the Cylinder-Flow system. The generalization parameter is the coefficient of the external forcing term. During training, the model is exposed to forcing coefficients within the in-distribution (ID) range $[0, \\pi]$, while out-of-distribution (OOD) testing uses coefficients in $(\\pi, 2\\pi]$. We use SSIM (×10⁻¹ ↑) as the evaluation metric, and the results are summarized below:\n\n| data distribution | SparseDiff    |G-LED|\n|-------------|--------------------|--------------------|\n| ID         | 8.173 ± 0.491      | 7.945 ± 0.832     |\n| OOD        | 8.002 ± 1.197     | 7.153 ± 1.970      |\n\n\nThis demonstrates that SparseDiff maintains strong performance under noise, with SSIM only slightly below G-LED’s when the noise exceeds 0.6%. This demonstrates the robustness of our model to input perturbations.\n\n> ***Question 1: In Section 3.1.2, you highlight the differences between your method and previous works [14, 13, 12], but these methods are not included in the experimental comparisons...***\n\nThe cited works [12, 13, 14] explore using sparse observations as conditioning inputs to guide diffusion models, aiming to reconstruct or complete solutions from partial measurements.\n\nFor instance, S³GM [13] focuses on recovering the full spatiotemporal field from sparse sensor data, while DiffusionPDE [14] addresses inverse problems like solving for unknown inputs from limited observations. [12] compares different conditioning strategies in diffusion frameworks for sparse-observation settings.\n\nIn contrast, our method doesn't rely on external sparse measurements but learns to extract sparse internal representations (key probes) that capture essential dynamics for prediction. This makes our approach fundamentally different.\n\nHowever, we agree these works are relevant and we conduct a direct comparison with DiffusionPDE [14] on the SH dataset, as it can serve as a representative example of the three methos applying diffusion models under sparse condition. The results are as follows (SSIM ×10⁻¹ ↑):\n\n| Predict Step | SparseDiff     | DiffusionPDE     |\n|-------------|--------------------| -----|\n| 10          | 9.87      | 9.52     |\n| 20        | 9.83    | 9.21  |\n| 40        | 9.80      |  8.70    |\n\nWe also compare inference time (s):\n\n| Predict Step | SparseDiff     | DiffusionPDE     |\n|-------------|--------------------| -----|\n| 10          |   20    | 3000     |\n| 20        | 22    | 6000  |\n| 40        | 26      |  12000    |\n\nAs shown, SparseDiff significantly outperforms DiffusionPDE in both accuracy and inference efficiency, especially in long-term prediction. This aligns with our assessment that DiffusionPDE is better suited for solving inverse or sparse-observation tasks, rather than for long-term predicting.\n\n\n> ***Question 2: How is the value of $K$, described in Section 3.1.1, chosen, and how does it affect the final performance and effiency?***\n\nThe $K$ mentioned in Section 3.1.1 refers to the codebook size, a tunable hyperparameter that determines how many codewords are available to represent system dynamics.\n\nThe optimal $K$ value varies across datasets depending on the complexity of the underlying patterns. For example:\n\n- On NS system, which exhibits highly complex dynamics, we use $K$ = 150 to ensure sufficient expressiveness.\n\n- On SH system, the patterns are simpler, and a smaller codebook ($K$ = 30) already achieves strong performance.\n\nIn general, the choice of $K$ involves a trade-off between reconstruction quality and codebook utilization:\n\nA larger $K$ offers greater representational capacity, potentially improving accuracy. However, an excessively large $K$ may lead to many unused codewords, reducing utilization efficiency, increasing memory overhead, and even slightly degrading performance due to undertraining of rarely selected codes.\n\nThe result is illustrated in Section 4.4, Figure 5(a), where we vary the codebook size $K$ on the NS system. The results show that performance plateaus around $K$ = 150, and further increasing $K$ leads to a slight performance drop—highlighting the importance of dataset-specific tuning. Notably, the plateau spans a fairly wide range of $K$ values, indicating that the method is robust to the choice of codebook size within a reasonable interval.\n\n> ***Typo Corrections:***\n\nWe thank the reviewer for catching these issues:\n\n- We have corrected the typo in line 101: `\\epolsion` is now consistently written as `\\epolsion_n` to match the notation in Equation (2), where the subscript denotes the specific noise term at step \\( n \\).\n\n- The term *\"Rollout step\"* in the caption of Figure 7(b) refers to the time index in the autoregressive prediction sequence, where each step corresponds to one forward prediction in the rollout process.\n\n> ***Strength 3: ...is very interesting ... not sure about its novelty.***\n\nThank you for your positive feedback. The sparse probe scheme indeed represents a breakthrough compared to traditional encoding methods [1, 2]. Its novelty and practical utility have been recognized by multiple reviewers. For example, E8yK highlighted the novel idea of test-time adapting prediction, and X3mc pointed out its advantages in capturing emergent spatiotemporal dynamics.\n\n## Reference\n\n[1] Yu, D., Li, X., Ye, Y., Zhang, B., Luo, C., Dai, K., ... & Chen, X. (2024). Diffcast: A unified framework via residual diffusion for precipitation nowcasting. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 27758-27767).\n\n[2] Gao, H., Kaltenbach, S., & Koumoutsakos, P. (2024). Generative learning for forecasting the dynamics of high-dimensional complex systems. Nature Communications, 15(1), 8904."}, {"review_id": "fccRUyYtJz", "reviewer": "Reviewer_X3mc", "summary": "This paper proposes SparseDiff, a Sparse Diffusion Autoencoder for efficient long-term prediction in spatiotemporal systems. It encodes global dynamics via a sparse set of probes using a codebook-based encoder, models them with a graph neural ODE, and reconstructs the full state with a diffusion decoder. The probe topology adapts at test time, enabling dynamic tracking. Experiments on PDE benchmarks and a climate dataset show superior accuracy and efficiency compared to state-of-the-art baselines.", "questions": "1. Is the topological structure present in every frame? And is it just the probe states that incorporate information from all previous frames up to the current one? If so, it would be helpful to clarify this distinction in the paper, as it’s not immediately clear whether the topology itself evolves over time or remains static while only the probe representations are temporally informed.\n2. Why is the dimension of $v_i$ given as $h \\times w$ ? Typically, one would expect $v_i$ to represent a vector or feature at a probe, which would usually have a fixed channel or embedding dimension. If $h\\times w$ refers to spatial dimensions, this seems inconsistent with the idea of sparse probe representations. Could the authors clarify the meaning and rationale behind this shape?\n3. In the first equation of Section 3.2, it’s recommended not to use $t$ as the integration variable, as it may cause confusion with the timestep index. Additionally, it's unclear whether $h_t$ and $h_i$ share the same meaning or dimensionality—does $h_t$ refer to the representation at time $t$, while $h_i$ denotes the representation of the $i$-th probe? Clarifying this distinction would help avoid ambiguity in notation.\n4. In line 177, should the superscript be $N$ instead? \n5. Are the Codebook Encoder and Diffusion Decoder trained separately, or is the entire model trained end-to-end?", "limitations": "See Weakness", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "### Strengths\n1. The manuscript introduces a practical strategy for updating probe selections and topologies dynamically at test time via codebook re-encoding. This design is argued to provide an advantage for capturing emergent spatiotemporal dynamics compared to conventional static encoders and predictors.\n2. The experiments are thorough and comprehensive. On PDE benchmarks and the SEVIR climate dataset, SparseDiff reduces prediction error by an average of 49.99% compared to leading baselines (Table 1), while only using about 1% of the input spatial grid. Such results suggest the framework can maintain accuracy with substantial efficiency gains. Moreover, they conduct ablation studies  to emphasize the necessity of the codebook-driven probe encoder and edge-aware attention.\n\n### Weaknesses\n1. Notation and method description are occasionally ambiguous. For example, Section 3.1.1 is overly difficult to follow, with dense explanations and somewhat confusing notation (though not incorrect). I recommend revising this section for clarity—streamlining the writing and organizing the symbols more clearly would greatly improve readability.\n2. The edge weighting scheme for probe topology (Section 3.1.1) is domain-agnostic and not learned end-to-end, as acknowledged under limitations. While the ablation (Figure 6c) shows this is better than unweighted edges, it leaves open whether optimal graph connectivity could be learned or data-driven; alternatively, comparisons to other heuristic graph construction methods (e.g., k-nearest neighbors, spectral approaches) are not included.", "was_revised": true, "final_justification": "My initial concerns have been addressed in the authors' response. I have therefore revised my score to 4.", "rebuttal": "We deeply appreciate the time you took to review our work and your meticulous comments for improvement. Below we address the concerns raised in your review:\n\n> ***Question 1: Is the topological structure present in every frame? And is it just the probe states that incorporate information from all previous frames up to the current one?...***\n\nWe appreciate the reviewer’s question. The probe topology is not fixed throughout prediction. Instead, it is periodically updated during test-time via re-encoding. Specifically, the prediction horizon is divided into adaptive sub-windows. At the start of each sub-window, a new probe graph is constructed using the encoder and codebook.\n\nIn contrast, probe states are continuously updated at each prediction step, as their trajectories evolve over time. These changes reflect the system’s ongoing dynamics even when the underlying graph remains fixed within a window. The topology is updated only when a latent mismatch is detected between current probe states and their assigned codewords. As detailed in **Section 3.3**, this mismatch is quantified by the latent consistency score $\\chi_t$, which measures how well the current probe state aligns with its nearest codeword in latent space. When $\\chi_t$ falls below a threshold $\\tau$, the system triggers re-encoding and reconstructs a new topology, allowing adaptive alignment with evolving spatiotemporal patterns (see Section 3.3 for implementation details and Figure 4 for illustration).\n\nAdditionally, probe states are not derived from all previous frames but from a fixed-length lookback window. As noted in **Appendix B**, we use a 10-frame history in all experiments to balance memory efficiency and temporal context.\n\nWe will revise the paper to clarify these distinctions.\n\n\n> ***Question 2: Why is the dimension of $v_i$ *given as* $h \\times w$? Typically, one would expect $v_i$ to represent a vector or feature at a probe, which would usually have a fixed channel or embedding dimension. If $h \\times w$ refers to spatial dimensions, this seems inconsistent with the idea of sparse probe representations...***\n\n\nWe appreciate the reviewer’s observation. The expression $v_i \\in \\mathbb{R}^{h \\times w}$ was a notational error. Here, $v_i$ is intended to indicate the spatial location of the $i$-th probe, not a high-dimensional tensor.\n\nThe actual temporal feature vector associated with each probe is denoted as $\\mathbf{v}_i \\in \\mathbb{R}^l$, where $l$ is the lookback window size (10 in our experiments). This will be clarified and corrected in the revised version.\n\n> ***Question 3: In the first equation of Section 3.2, it’s recommended not to use* $t$ as the integration variable... Additionally, it’s unclear whether $h_t$ and $h_i$ share the same meaning or dimensionality—does $h_t$ refer to the representation at time $t$, while $h_i$ denotes the representation of the $i$-th probe?...***\n\nWe thank the reviewer for the helpful suggestion. We agree that using $t$ as the integration variable in Section 3.2 may cause confusion, and we will revise it to a different symbol (e.g., $\\tau$) to avoid overlap with the timestep index.\n\nThe reviewer is also correct that $h_t$ refers to the representation at time $t$, while $h_i$ denotes the representation of the $i$-th probe. We chose not to write $h_i^t$ uniformly because different equations emphasize different aspects: some focus on temporal evolution without distinguishing probes, while others highlight probe interactions without indexing time. We will clarify this in the final version.\n\n> ***Question 4: In line 177, should the superscript be N instead?***\n\nYes, the superscript should indeed be $N$ rather than $n$. We will correct it in the final version.\n\n\n> ***Question 5: Are the Codebook Encoder and Diffusion Decoder trained separately, or is the entire model trained end-to-end?***  \n\nWe appreciate the reviewer’s question. The codebook-based Sparse Encoder, probe-graph diffusive predictor, and guided Diffusion Decoder are trained as separate modules.\n\nEnd-to-end training is not feasible in our framework because the probe selection and graph construction processes involve discrete operations (e.g., nearest-neighbor lookup, subgraph assignment) that block gradient flow. Additionally, during the diffusion decoding stage, we apply explicit guidance (Equation 3) by injecting probe values into the denoising process step-by-step, rather than relying on implicit conditioning. This guidance mechanism is inherently non-differentiable, further limiting the feasibility of end-to-end optimization.\n\nWe experimented with combining both implicit and explicit conditioning for the diffusion model. However, adding implicit conditioning significantly increased the number of trainable parameters without providing obvious performance gains. This supports our decision to rely on explicit guidance for accurate reconstruction.\n\n| Model | Parameters (M) | RMSE × 10⁻² ↓ |\n| --------------------- | ---------- | ------------- |\n| Ours | 26  | 2.912 |\n| Implicit conditioning | 74  | 2.809         |\n\nWhile the model is not end-to-end trainable, modular training offers several practical advantages. It allows each component to be developed, pre-trained, or replaced independently, facilitates better interpretability, and reduces the memory and compute requirements during training. This modularity also makes it easier to adapt the framework to new physical systems or plug in more advanced predictors and decoders in future work.\n\n> ***Weakness 1: ...Section 3.1.1 is overly difficult to follow, with dense explanations and somewhat confusing notation (though not incorrect). I recommend revising this section for clarity...***\n\nWe appreciate the reviewer’s feedback regarding clarity and notation in Section 3.1.1. We agree that this section can be better structured and will revise it to improve readability. We hope that our responses to Q2–Q4 above have helped clarify some of the core notational choices and methodological distinctions. We will incorporate these clarifications into the paper to reduce potential confusion.\n\n> ***Weakness 2: The edge weighting scheme for probe topology (Section 3.1.1) is domain-agnostic and not learned end-to-end... alternatively, comparisons to other heuristic graph construction methods (e.g., k-nearest neighbors, spectral approaches) are not included.***\n\nWe appreciate the reviewer’s comments regarding the edge weighting scheme in our probe topology. Our edge weighting scheme is **partially heuristic, partially learnable**, and the resulting topology is **fully differentiable** during training. While the graph structure is not constructed end-to-end due to the discrete nature of probe grouping, the diffusion weights used by the predictor are trainable, allowing data-driven adaptation over a spatially informed backbone.\n\n\nSpecifically, our predictor is based on Graph Neural Diffusion (GRAND) [1], which models information propagation on a graph as a continuous diffusion process:\n\n\n$$\nh_t = h_0 + \\int_0^t A  h_\\tau \\, d\\tau,\n$$\n\nwhere $A$ defines the diffusion strength, and each $a_{ij}$ encodes how much node $j$ contributes to node $i$’s evolution over virtual time.\n\nIn the original GRAND framework, diffusivity $ a_{ij} $ is computed via attention over learnable probe features. In our case, each node represents a region-aggregated embedding constructed by averaging features of all grid points within the codeword region $ u_{c_i} $. Therefore, probe similarity should reflect not only feature proximity but also region-level spatial adjacency.\n\nTo reflect this, we introduce an additional edge weight term $e_{ij}$, which quantifies the degree of spatial adjacency between probe regions. Specifically, $e_{ij}$ measures the fraction of non-internal neighboring points in $u_{c_i}$ that are adjacent to points in $u_{c_j}$. This term is integrated into the attention-based diffusivity calculation as:\n$$\na(h_i, h_j) = \\text{softmax}\\left( \\frac{(W_K h_i)^T W_Q h_j + W_E e_{ij}}{d_k} \\right),\n$$\n\nwhere $W_E$ is a learnable weight matrix. This allows $e_{ij}$ to serve as an additional input feature rather than a fixed multiplicative factor, making the final diffusion strength $a_{ij}$ still trainable via backpropagation.\n\nMoreover, we do not predefine a fixed graph. Instead, we assume full connectivity and let the attention-based matrix $A$ learn effective interactions. When $a_{ij} \\approx 0 $, it implies no effective connection, enabling the topology to adapt during training.\n\nTo assess our edge weighting scheme, we compare against:\n\na. Graph Kernel Network (GKN): A learnable graph construction method [2];\n\nb. Heuristic alternatives using the same GRAND predictor but replacing $e_{ij}$ with:\n- k-nearest neighbors based on Euclidean coordinates ($k = 20\\% $) of probe count);\n- Spectral clustering followed by KNN in embedding space.\n\n\nResults are as follows (average RMSE over 100-step autoregressive prediction on LO system):\n\n| Method                                 | RMSE × 10⁻² |\n|----------------------------------------|--------|\n| Ours    |    2.873    |\n| GKN      |   4.210       |\n| GRAND + kNN           |   3.287     |\n| GRAND + spectral       |   3.019     |\n\n\nAs shown, our region-aware edge scheme yields better performance than both heuristic variants and learnable graph construction methods, demonstrating that encoding region-level adjacency improves the expressiveness of probe-based dynamics modeling.\n\n## Reference\n[1] Chamberlain, B. P., Rowbottom, et. al. (2021). GRAND: Graph neural diffusion. arXiv preprint arXiv:2106.10934.\n\n[2] Li, Z., Kovachki, N., Azizzadenesheli, et. al. (2020). Neural operator: Graph kernel network for partial differential equations. arXiv preprint arXiv:2003.03485."}, {"review_id": "lWgj8W9sHb", "reviewer": "Reviewer_E8yK", "summary": "The paper presents SparseDiff, a novel autoencoder framework for predicting complex spatiotemporal dynamics. The model first encodes the system's state into a sparse probe topology using a learned codebook of dynamic patterns. It then employs a graph neural ODE to model the dynamics on this graph and guides a diffusion decoder for reconstruction. A test-time adaptation strategy is adopted to dynamically update the probe topology, ensuring the model remains accurate during long-term prediction. The authors validate the accuracy and efficiency of SparseDiff on various PDE systems and real-world datasets.", "questions": "1. The current framework is limited to 2D regular spatial domain. Many real-world systems, such as those in climate science or sensor networks, involve data sampled from irregular or sparse spatial domains. Can you comment on the applicability of SparseDiff to such irregular or sparse data? \n\n2. l. 143 $v_i \\in R^{t\\times h\\times w}$. l.38  $v_i \\in R^{h\\times w}$. There seems to be a notational inconsistency. Can you further clarify the precise algorithm for computing the probe features? \n\n3. How were the two hyperparameters (number of codewords $K$, and the threshold $\\tau$ ) selected in practice for the experiments? Can you provide some insight into the influence of the hyperparameters on the performance? \n\n4. It is unclear to me why the test-time adapting prediction improves the performance. Can you provide a quantitative explanation for why this re-encoding step leads to a better-aligned probe topology and improved performance? It will be good to do an ablation study that isolates the effect of adaptation. To do this, can you compare the following two models on the Navier-Stokes experiment and report the result as in Table 1? \n    - SparseDiff without test-time adaptation\n    - G-LED with test-time adaptation", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- novel idea of test-time adapting prediction\n- Strong performance over baselines across multiple PDE systems and a real-world dataset.\n- Ablation studies on probe selection and edge weighting\n\n\n**Weaknesses**\n\nsee questions", "was_revised": true, "final_justification": "I maintain my positive score.", "rebuttal": "Thank you so much for your thoughtful review and your suggestions for improvement. Below we address the concerns raised in your review:\n\n> ***Question 1: The current framework is limited to 2D regular spatial domain... Can you comment on the applicability of SparseDiff to such irregular or sparse data?***\n\nThanks for the thoughtful question. While our current implementation operates on 2D regular grids, SparseDiff can be applied to irregular or sparse spatial data by first transforming the inputs into a regular grid through interpolation or zero-filling, enabling processing within the same framework.\n\nTo validate this, we conduct two experiments:\n\n**a. Sparse sampling on Navier-Stokes system:**\nOn the Navier-Stokes system, we randomly sample only 10% of grid points. These sparse samples are then interpolated onto a 128×128 regular grid before being fed into SparseDiff. Despite the sparsity, SparseDiff outperforms baselines that are given full-resolution inputs.\n\n| Method                        | RMSE × 10⁻² ↓ | SSIM × 10⁻¹ ↑ |\n| ----------------------------- | ------ | ------ |\n| FNO (full grid)               | 15.99      | 7.26     |\n| G-LED (full grid)             | 12.33     | 8.10      |\n| SparseDiff (full grid)             | 11.39     | 8.36      |\n| SparseDiff (10% observations) | 12.27  | 8.19  |\n\n**b. Irregular 2D wave equation dataset [1]:**\nWe also evaluate SparseDiff on the 2D wave equation with observations sampled on irregular meshes. These inputs are first completed onto regular grids using interpolation within observed regions and zero-filling in excluded regions before being processed by the model. Compared to FNO, SparseDiff maintains stable prediction accuracy.\n\n| Method                        | RMSE × 10⁻² ↓ | SSIM × 10⁻¹ ↑ |\n| ----------------------------- | ------ | ------ |\n| FNO               | 23.76      | 7.81      |\n| SparseDiff        | 15.58      | 8.25      |\n\nWhile these results demonstrate practical applicability to irregular or sparse inputs, we acknowledge certain limitations of this approach:\n\n- For highly non-uniform spatial distributions, interpolating to a uniform grid can be problematic. Also, the sparse observation may be too under-sampled to allow reliable interpolation, leading to unnecessary computational cost or significant reconstruction errors.\n\n- If the distribution of spatial inputs shifts between training and testing, performance may degrade, as our current model lacks coordinate-querying capabilities (unlike operator-learning models such as neural operators), and cannot generalize across arbitrary spatial layouts.\n\nNevertheless, the core design of SparseDiff is not inherently tied to regular grids. The key innovation of SparseDiff lies in its latent probe representation, which aggregates and models regional dynamics through sparse, learnable units. This representation is inherently flexible and does not require regular spatial structures.\n\nThe main constraint stems from the UNet-based diffusion decoder, which requires regular-grid input. However, this is a design choice rather than a fundamental limitation of the method. In future work, the diffusion decoder could be replaced with architectures that naturally support irregular domains, such as Graph Neural Networks. Alternatively, it could be substituted with other super-resolution models, including transformer-based approaches and operator-learning models, to further extend SparseDiff to irregular spatial settings. \n\n\n> ***Question 2: l.143 $v_i \\in \\mathbb{R}^{t \\times h \\times w}$. l.138 $v_i \\in \\mathbb{R}^{h \\times w}$. There seems to be a notational inconsistency...***\n\nWe appreciate the reviewer’s careful attention to notation. The expression $v_i \\in \\mathbb{R}^{h \\times w}$ was a notational error—$v_i$ is intended to denote the spatial location of the $i$-th probe, not a high-dimensional tensor.\n\nThe temporal feature vector should be written as \\(\\mathbf{v}_i \\in \\mathbb{R}^l\\), where \\(l = 10\\) is the lookback window size. The previous notation \\(\\mathbb{R}^{t \\times h \\times w}\\) was a mistake. We will clarify and correct both notations in the revised version.\n\n> ***Question 3: How were the two hyperparameters (number of codewords $K$, and the threshold $\\tau$) selected in practice for the experiments?...***\n\n**Choice of $K$ (Codebook Size):**\n\nThe hyperparameter $K$ controls the number of discrete codewords used to represent latent dynamics and is selected based on dataset complexity.\n\nFor the Navier–Stokes system which exhibits highly chaotic and multi-scale dynamics, we set $K = 150$ to ensure sufficient capacity. For the more regular Swift–Hohenberg system, a smaller size $K = 30$ can already perform well.\n\nIn general, choosing $K$ involves a trade-off: larger values offer more expressiveness but may lead to under-utilized codewords and inefficiency; smaller values reduce capacity and risk underfitting.\n\nAs shown in Section 4.4, Figure 5(a), performance improves with increasing $K$ up to ~150, then plateaus or slightly degrades. Overall, SparseDiff remains robust across a broad range of $K$, highlighting its flexibility in codebook sizing.\n\n**Choice of $\\tau$ (Topology Update Threshold):**\n\nThe threshold $\\tau$ controls how sensitively the topology is updated based on latent mismatch. A smaller $\\tau$ triggers more frequent updates, improving accuracy by promptly correcting structural drift, but increases computational cost. A larger $\\tau$ reduces update frequency, improving efficiency but risking misalignment with evolving dynamics.\n\nIn Figure 7(a), we compare the adaptive strategy (governed by $\\tau$) with fixed-interval updates. The adaptive method strikes a balance:\nIt achieves higher accuracy than low-frequency updates (e.g., every 5–10 steps) and lower runtime than high-frequency updates (e.g., every 1–3 steps). This shows that the threshold-based adaptation offers a practical trade-off between precision and efficiency.\n\nThe table below quantifies this balance across different $\\tau$ values:\n\n| $\\tau$ | Avg. Update Frequency (steps) | RMSE × 10⁻² ↓ | Runtime (s) ↓ |\n| -------------- | ----------------------------- | ------ | ------------- |\n| 0.001           | 1.25                           | 8.77  | 157           |\n| 0.003           | 2.17                           | 10.03  | 114           |\n| 0.005           | 3.89                           | 11.91  | 93         |\n| 0.008           | 6.15                           | 15.43  | 79           |\n| 0.012           | 10.73                           | 25.68  | 70           |\n \nAs shown, $\\tau$ controls a trade-off between accuracy and efficiency and can be tuned based on task needs. The adaptive mechanism flexibly updates topology more often during rapid dynamics and less often when the system is stable, enabling efficient yet accurate long-term prediction.\n\n> ***Question 4: ...Can you provide a quantitative explanation for why this re-encoding step leads to a better-aligned probe topology and improved performance? ...can you compare the following two models on the Navier-Stokes experiment and report the result as in Table 1?***\n> - ***SparseDiff without test-time adaptation***  \n> - ***G-LED with test-time adaptation***\n\nIn spatiotemporal PDEs, global patterns evolve through structural transitions, such as mode aggregation, symmetry breaking, and wavefront propagation. Therefore, a fixed encoder, set at initialization, can quickly become misaligned, especially in long-term prediction. Without updates, the misaligned latent space leads to inaccuracies in prediction and decoding.\n\nThis motivates our test-time adaptation strategy, which dynamically updates the topology based on emerging dynamics. Frequent updates help maintain alignment with dominant structures, ensuring that the decoder receives accurate, structure-aware conditioning. In contrast, a static topology can introduce bias and compounding errors.\n\nWe further test the performance of SparseDiff without test-time adaptation using a fixed topology throughout prediction and G-LED with adaptation:\n\n- **SparseDiff w/o adaptation** suffers from severe degradation on Navier–Stokes, with RMSE exceeding 100, indicating a substantial divergence from the ground truth trajectory due to the static and increasingly misaligned probe structure.\n\n- **G-LED w/ adaptation**: results are included in Figure 7(a) (marked with cross symbols) and further summarized in the Table below. Although G-LED uses simple uniform downsampling to define its latent space, its periodic updates help capture temporal variations. As a result, it achieves moderate prediction accuracy, albeit lower than SparseDiff, and incurs higher computational cost due to the lack of adaptive locality.\n\nFor reference, SparseDiff takes only **163s** at interval 1 and **73s** at interval 10, which are substantially faster than G-LED under comparable settings.\n\n| Re-encoding Interval | RMSE × 10⁻² ↓   | Runtime (s) ↓ |\n| -------------------- | -------- | ------------- |\n| 1             | 12.59 | 6950      |\n| 2          |  12.87   | 4875          |\n| 3      | 13.05     | 3537         |\n| 4      | 13.29     | 3007          |\n| 5      | 13.72     | 2642          |\n| 6      | 14.46     | 2460          |\n| 7      | 16.60     | 2214          |\n| 8      | 17.92     | 2179          |\n| 9      | 18.58    | 2003          |\n| 10      | 19.85     | 1877          |\n\n\nIn conclusion, test-time adaptation is essential for PDE prediction tasks under encoding-decoding scheme. It ensures that the low-dimensional representation remains aligned with the evolving system state, preventing error accumulation and enhancing long-term prediction stability.\n\n\n\n## Reference\n[1] Zhao, Q., Lindell, D. B., & Wetzstein, G. (2022). Learning to solve PDE-constrained inverse problems with graph networks. arXiv preprint arXiv:2206.00711."}, {"review_id": "zrsgy7Zfrz", "reviewer": "Reviewer_WcE9", "summary": "The paper “Sparse Diffusion Autoencoder for Test-time Adapting Prediction of Complex Systems” introduces SparseDiff, a framework for long-term forecasting of high-dimensional spatiotemporal dynamics (e.g., fluid flow, climate fields). SparseDiff compresses each high-dimensional frame into a handful of learned “probes” with a codebook-based sparse encoder, advances those probes through an edge-aware graph neural ODE, and then uses a guided diffusion decoder to regenerate the full field, thus preserving spatial structure while working in an extremely low-dimensional space.  Evaluated on PDE benchmarks and the SEVIR weather dataset, this test-time adaptation cuts long-range RMSE by roughly 50 % versus state-of-the-art baselines while requiring fewer than 1 % of grid points and remaining robust to noise and out-of-distribution Reynolds numbers.", "questions": "Please see the “Weaknesses” section. If the authors can supply adequate theoretical justification, I would be willing to consider raising my score.", "limitations": "Please see the “Weaknesses” section.", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. Robustness and generalization: maintains performance under noisy inputs and extrapolates to unseen Reynolds numbers without retraining. \n\n2. Test-time adaptive encoding automatically refreshes the probe graph during inference, preventing drift in very long forecasts and preserving accuracy. \n\n3. State-of-the-art long-horizon accuracy, reducing RMSE by about 50 % versus FNO, UNet and other strong baselines on four PDEs and a weather dataset.\n\nWeaknesses:\n\n1.  The introduction points out “a lack of theoretical understanding for aggregating rich spatiotemporal dynamics from a continuous domain into a small set of discrete probes”; yet the paper itself provides no such theoretical analysis—most notably, it does not quantify how prediction accuracy scales with the forecast horizon t, leaving readers without guidance on the temporal range over which the method’s outputs remain reliable.\n\n2. The Methods section introduces a “Sparse Encoder,” yet it never defines what “sparse” specifically refers to—is it merely the step of replacing the latent vector z with its nearest codeword c before decoding? In addition, the paper offers no theoretical justification for why sparsity is needed or how it benefits the model.\n\n3. In the PDE context $\nh_t = h_0 + \\int_{0}^{t} A\\,h_t \\, dt,\n$, we also care about estimating and inferring the parameter A, as it is crucial for stability analysis and control design. Could your method be extended to provide reliable inference of A, rather than focusing solely on long-term state prediction?", "was_revised": false, "final_justification": "", "rebuttal": "> ***Weakness 1: The introduction points out “a lack of theoretical understanding for aggregating rich spatiotemporal dynamics from a continuous domain into a small set of discrete probes”... it does not quantify how prediction accuracy scales with the forecast horizon t...***\n\nWe sincerely thank the reviewer for raising this important question. We provide additional clarification from both a theoretical and empirical perspective.\n\n**a. Theoretical Support:**\n\nMany spatiotemporal dynamical systems, such as those in fluid dynamics or reaction-diffusion processes, evolve over time along a low-dimensional manifold after an initial transient phase. This has been formalized under slow manifold and center manifold theory [1, 2], and further supported by recent works that show data-driven models can recover effective latent dynamics in reduced spaces [3, 4].\n\nWhile these manifolds capture dominant macroscopic behavior, it has also been shown that global system states can be approximated from a sparse set of spatial observations, as analysed in [5]. Combining these insights, we construct a sparse latent representation by placing probes along the system’s manifold structure to efficiently capture and propagate global dynamics.\n\nOur probe topology serves this purpose by selecting informative locations and modeling their evolution through a graph neural network, allowing the model to track long-term dynamics over a compact, structure-aware representation. This aligns with the principles of modal compression and reduced-order modeling, ensuring both physical interpretability and computational efficiency.\n\n**b. Empirical Evidence: Fixed topologies rapidly lose expressiveness during long-term prediction**\n\nSparseDiff dynamically updates the probe topology by perceiving recent states within the previous forecast horizon. Therefore, a smaller forecast horizon implies more frequent updates. As shown in Figure 7(a), we evaluate how prediction accuracy evolves with different re-encoding intervals in the Navier-Stokes experiment. When the probe topology is updated at every step, the model maintains stable and accurate long-term predictions. As the interval increases, prediction error grows more rapidly. At every 10 steps, the MSE exceeds acceptable levels, indicating accuracy degradation.\n\nWe further test a fixed-topology variant where no updates are performed throughout prediction. In this limiting case, the MSE quickly rises to ~100 within 50 steps, showing that the model fails to track system evolution. This underscores the importance of regularly updating the macroscopic structure.\n\nFor reference, we report G-LED’s MSE over increasing horizons. As shown in the table below, G-LED accumulates errors faster and performs worse overall than SparseDiff.\n\n| Forecast Horizon (steps) | G-LED | SparseDiff (Adaptive)|\n| ------------------------ | ---------- |---------- |\n| 20                       | 0.012      |0.004      |\n| 40                       | 0.026      |0.007      |\n| 60                       | 0.039      |0.005      |\n| 80                       | 0.062      |0.011      |\n\nWe will include the full G-LED error curves in the camera-ready version as a comparative reference to our method.\n\n> ***Weakness 2: The Methods section introduces a “Sparse Encoder,” yet it never defines what “sparse” specifically refers to... In addition, the paper offers no theoretical justification for why sparsity is needed or how it benefits the model.***\n\nTo avoid ambiguity, we clarify the definition of the Sparse Encoder and the motivations for introducing sparsity into the representation.\n\nThe term “Sparse Encoder” in our paper refers not merely to the step of replacing the continuous latent vector $\\mathbf{z}$ with its nearest codeword $\\mathbf{c}$ before decoding. More fundamentally, it denotes the use of a finite codebook to compress the system’s dynamic representations into a discrete set of sparsely activated latent patterns. Specifically, the encoder outputs a continuous vector $\\mathbf{z}_i$, which is then discretized by mapping it to its nearest codeword $\\mathbf{c}_j \\in \\mathcal{C}$, where $\\mathcal{C}$ is a codebook with a predefined size $K$. Since every latent vector is constrained to fall within this finite set, only a small subset of representative modes is activated across the representation space, thereby introducing structured sparsity into the model.\n\nThis sparse design is motivated by two key principles for modeling high-dimensional spatiotemporal systems:\n\n- Computational Efficiency and the Manifold Theory.\n\n  High-dimensional PDE systems are computationally expensive to model directly. However, their intrinsic dynamics often evolve on a much lower-dimensional manifold, an idea formalized in center manifold theory [1, 3, 4, 8]. Our sparse probe topology is explicitly designed to identify and approximate this low-dimensional manifold. This allows the predictor to operate in a highly efficient, reduced-order space that still captures the system's core dynamics.\n\n- Robust Long-term Prediction.\n\n  Sparsity is crucial for capturing emergent patterns in long-term forecasting. By forcing the model to represent the system state using a few key probes, the encoder must learn to capture the most essential dynamical components while filtering out redundant or noisy information. This focus on the dynamical skeleton makes the predictor more robust and better able to adapt to new structures as they emerge over time [8].\n\n> ***Weakness 3: In the PDE context... we also care about estimating and inferring the parameter A... Could your method be extended to provide reliable inference of A, rather than focusing solely on long-term state prediction?***\n\nWhile our primary objective is long-term trajectory prediction, the diffusion operator $A$ is not purely learned from data. It is explicitly designed based on physically inspired priors that reflect the spatial organization of the system.\n\nSpecifically, our predictor is based on Graph Neural Diffusion (GRAND) [1], which models information propagation on a graph as a continuous diffusion process:\n\n$$\nh_t = h_0 + \\int_0^t A  h_\\tau \\, d\\tau,\n$$\n\nwhere $A$ defines the diffusion strength, and each $a_{ij}$ encodes how much node $j$ contributes to node $i$’s evolution over virtual time.\n\nIn the original GRAND framework, diffusivity $a_{ij}$ is modeled purely through an attention mechanism over learnable probe features. In contrast, our approach represents each probe as a region-aggregated feature, computed by averaging the grid points within the corresponding codeword region $u_{c_i}$. Thus, we argue that similarity between probes should account not only for feature embeddings but also for how region-level adjacencies affect spatial interactions.\n\nTo enhance the physical interpretability and spatial accuracy of the graph, we introduce an edge weight term $e_{ij}$ to measure the spatial adjacency between probe regions. Specifically, $e_{ij}$ quantifies the fraction of neighboring points in $u_{c_i}$ adjacent to those in $u_{c_j}$. This term is integrated into the attention-based diffusivity calculation:\n\n$$\na(h_i, h_j) = \\text{softmax}\\left( \\frac{(W_K h_i)^T W_Q h_j + W_E e_{ij}}{d_k} \\right),\n$$\n\n\n\nwhere $W_E$ is a learnable weight matrix. This allows $e_{ij}$ to be an input feature, not a fixed multiplicative factor, ensuring the final diffusion strength $a_{ij}$ remains trainable via backpropagation. This design incorporates domain knowledge into the construction of $A$, reflecting the physical meaning of a diffusion operator rather than being arbitrarily fitted for improved prediction.\n\nTo validate that our operator $A$ captures meaningful, physically grounded structure, we perform an ablation study comparing different graph construction methods, each defining a distinct form of $A$.\n\nSpecifically, we compare:\n\na. **Graph Kernel Network** [4], which learns the connectivity pattern end-to-end from data;\nb. **Heuristic alternatives** using the same GRAND predictor but replacing $e_{ij}$ with:\n\n* K-nearest neighbors based on Euclidean coordinates ($k = 20\\%$ of probes);\n* Spectral clustering followed by KNN in the embedding space.\n\nThe results, reported as average RMSE over 100-step prediction on the LO system, are shown below:\n\n| Method                                 | RMSE × 10⁻² |\n|----------------------------------------|--------|\n| Ours    |    2.873    |\n| GKN      |   4.210       |\n| GRAND + kNN            |   3.287     |\n| GRAND + spectral       |   3.019     |\n\nIn summary, while our main goal is trajectory prediction, the probe graph defines a physically meaningful approximation of $A$. Instead of brute-force fitting, it utilizes spatial and latent structure. Empirical results show it outperforms both heuristic and learned methods, highlighting its potential for broader applications beyond prediction.\n\n## Reference\n[1] E Knobloch, K Wiesenfeld, Bifurcations in fluctuating systems: The center-manifold approach.\n\n[2] N Fenichel, Geometric singular perturbation theory for ordinary differential equations.\n\n[3] Vlachas, P. R., et. al. Multiscale simulations of complex systems by learning their effective dynamics.\n\n[4] Floryan, D., & Graham, M. D. (2022). Data-driven discovery of intrinsic dynamics.\n\n[5] Brunton, B. W., et. al. Sparse sensor placement optimization for classification.\n\n[6] Chamberlain, B. P., et. al. GRAND: Graph neural diffusion.\n\n[7] Li, Z., Kovachki, et. al. Neural operator: Graph kernel network for partial differential equations.\n\n[8] Li, R., Wang, H., et. al. Predicting Long-term Dynamics of Complex Networks via Identifying Skeleton in Hyperbolic Space."}]

Author rebuttal:

Meta-review: Summary: This paper proposes SparseDiff, an adaptive model for spatiotemporal forecasting that dynamically updates a sparse graph representation to capture emergent patterns. Using a graph neural ODE and diffusion decoder, it reduces prediction error by 49.99% compared to baselines with a 99% reduction in spatial resolution.

Strengths: 1. The test-time adaptation mechanism provides a practical approach for capturing emerging spatiotemporal structures. 2. Extensive experiments across diverse PDE benchmarks and real-world data show clear advantages over baseline methods. 3. The technique of using sparse probes to construct a reduced-order model is highlighted as particularly interesting.

Weaknesses: 1. The paper suffers from poor writing quality and occasionally ambiguous descriptions of notation and methods. 2. Inconsistencies are noted in the experimental configurations. 3. The paper's theoretical contributions are considered overclaimed. 4.The edge weighting scheme for the probe topology is criticized for being domain-agnostic and not learned in an end-to-end manner. 5. The proposed framework is limited in applicability to 2D regular spatial domains.

During the rebuttal, the authors adequately addressed the major concerns. Reviewer X3mc acknowledged that the training pipeline, while not fully end-to-end, is a reasonable and effective approach. The sole negative score came from Reviewer WcE9, who raised a valid concern regarding the lack of a clear explanation for how prediction accuracy evolves over time (t)—a key point in the paper's theoretical claims. However, as Reviewer WcE9 did not engage in the rebuttal discussion, and based on the authors' proposed response, I believe this weakness can be tractably addressed by modifying the relevant statements without diminishing the paper's core novelty and interestingness. I therefore recommend borderline acceptance, strongly urging the authors to integrate all suggestions into their final version.
Decision: Accept (poster)