Title: FlowNet: Modeling Dynamic Spatio-Temporal Systems via Flow Propagation
Venue: NeurIPS 2025
Accepted: yes
Keywords: Spatio-Temporal Data Mining

Dynamic System Modeling
Abstract: Accurately modeling complex dynamic spatio-temporal systems requires capturing flow-mediated interdependencies and context-sensitive interaction dynamics. Existing methods, predominantly graph-based or attention-driven, rely on similarity-driven connectivity assumptions, neglecting asymmetric flow exchanges that govern system evolution. We propose Spatio-Temporal Flow, a physics-inspired paradigm that explicitly models dynamic node couplings through quantifiable flow transfers governed by conservation principles. Building on this, we design FlowNet, a novel architecture leveraging flow tokens as information carriers to simulate source-to-destination transfers via Flow Allocation Modules, ensuring state redistribution aligns with physical laws. FlowNet dynamically adjusts the interaction radius through an Adaptive Spatial Masking module, suppressing irrelevant noise while enabling context-aware propagation. A cascaded architecture enhances scalability and nonlinear representation capacity. Experiments demonstrate that FlowNet significantly outperforms existing SOTA approaches on seven metrics in the modeling of three real-world systems, validating its efficiency and physical interpretability. We establish a principled methodology for modeling complex systems through spatio-temporal flow interactions.

FlowNet: Modeling Dynamic Spatio-Temporal Systems via Flow Propagation

Abstract

Accurately modeling complex dynamic spatio-temporal systems requires capturing flow-mediated interdependencies and context-sensitive interaction dynamics. Existing methods, predominantly graph-based or attention-driven, rely on similaritydriven connectivity assumptions, neglecting asymmetric flow exchanges that govern system evolution. We propose Spatio-Temporal Flow, a physics-inspired paradigm that explicitly models dynamic node couplings through quantifiable flow transfers governed by conservation principles. Building on this, we design FlowNet, a novel architecture leveraging flow tokens as information carriers to simulate source-todestination transfers via Flow Allocation Modules, ensuring state redistribution aligns with conservation laws. FlowNet dynamically adjusts the interaction radius through an Adaptive Spatial Masking module, suppressing irrelevant noise while enabling context-aware propagation. A cascaded architecture enhances scalability and nonlinear representation capacity. Experiments demonstrate that FlowNet significantly outperforms existing state-of-the-art approaches on seven metrics in the modeling of three real-world systems, validating its efficiency and physical interpretability. We establish a principled methodology for modeling complex systems through spatio-temporal flow interactions.

1 Introduction

Accurately modeling the evolution of complex dynamic spatio-temporal systems remains a fundamental challenge in domains ranging from urban mobility planning [1, 2] to environmental monitoring [3]. We identify a critical yet understudied paradigm governing such systems: the mutual interactions between distributed entities through directed information flow exchanges [4, 5]. These flow-driven dynamics not only modulate the system states [6] but also serve as primary catalysts for emergent spatio-temporal patterns [7, 8].

For decades, extensive research has focused on effectively predicting spatio-temporal systems [9, 10] and has demonstrated remarkable success by jointly modeling spatial dependencies and temporal dynamics [11, 12]. Graph-based architectures, particularly spatio-temporal graph neural networks (STGNNs) [13, 14, 15], leverage structural priors by constructing adjacency matrices from predefined spatial relationships like geographic proximity or semantic connections. Attention-based methods, especially Transformers [16, 17, 18, 19], introduce complementary advantages through dynamically computed node affinity matrices, enabling adaptive modeling of non-stationary dependencies while maintaining favorable scalability. Current approaches further bridge this framework with the foundation models [20, 11], enabling the extraction of spatio-temporal dependencies from various data streams [21, 22, 23, 24].

Observed similar patterns is the result of Flows.

Inflow

Outflow

R

Residential

C

Commercial

R1

C

R2

time

flow

Observed

time

flow

Observed

Commuting Flow

R1

C

R2

Weekday Morning

R1

C

R2

Weekday Evening

R1

E

R2

Weekends

E

Entertainment

E

NovenaR

Similar?

Inherent Flow

Mechanism

Although effective, existing approaches remain constrained by a critical oversight: they primarily capture surface-level observational correlations while fundamentally neglecting the flow-mediated interdependencies that govern intrinsic system dynamics. Specifically, prevalent operators like graph convolution [25, 26] and spatial attention [27, 28, 16, 19] implicitly assume system dynamics emerge from static node attribute similarities, whereas a range of real-world interconnected systems evolve through asymmetric flow exchanges that transcend mere statistical correlation [29, 30, 31, 32]. Consider an urban system comprising residential and commercial zones as shown in Figure 1, where directional population movement creates distinct spatio-temporal patterns: morning peaks exhibit concentrated flows from residential to commercial areas, while evening peaks reverse this directional pattern. Crucially, while spatially distant residential zones may exhibit similar traffic volumes through observational metrics, the underlying system dynamics derive from directional flow interactions between functionally complementary regions. The critical modeling imperative lies in capturing how transit flows actively reconfigure system states across different time, rather than correlating static attribute similarities among different zones.

Furthermore, the inherent complexity of modeling spatio-temporal systems arises from the contextsensitive dynamics governing flow propagation [33, 34]. These dynamics exhibit dual heterogeneity: temporal variations in flow magnitude and directionality during peak hours alongside spatial shifts in functional destinations during holidays. As previously discussed, the flows that form the morning and evening peaks within the city show an opposite direction. During holiday periods, leisure-oriented mobility redirects flows toward entertainment venues instead of workplaces. Conventional approaches attempt to resolve this complexity through two restrictive strategies. Predefined connectivity graphs [25, 26] enforce static spatial priors that cannot adapt to contextually evolving interaction ranges. Conversely, pairwise similarity computations [35, 36, 16, 19] dynamically estimate node affinities but inherently aggregate noise from irrelevant connections.

The intrinsic limitations of existing methodologies necessitate a paradigm shift in modeling dynamic spatio-temporal systems. Observing that transportation networks, urban mobility, or hydrological systems share the common intrinsic structure of latent information flow propagation, we propose a new paradigm termed Spatio-Temporal Flow . This physically inspired framework departs from traditional approaches constrained by static or similarity-based connectivity assumptions by explicitly modeling dynamic node coupling through mobile information carriers. Additionally, information transfer adheres to explicit conservation laws where outflow operations deplete source node states while inflow operations proportionally augment destination states. Finally, interaction ranges adaptively adjust based on contextual system states, dynamically reconfiguring propagation pathways without relying on fixed thresholds. This framework fundamentally contrasts with previous schemes that propagate information through merely blending node features while ignoring the state redistribution mechanisms inherent to flow-mediated systems.

Contribution. Building upon this paradigm, we propose FlowNet , a novel prediction model based on spatio-temporal flow. Unlike conventional message passing through feature blending (e.g., attention or graph convolution), FlowNet introduces flow tokens as quantifiable information carriers that explicitly model source-to-destination transfers. These tokens are generated and redistributed between nodes through Flow Allocation Modules (FAM) , ensuring compliance with the conservation principle. To address the dynamic propagation ranges across spatio-temporal contexts, FlowNet utilizes an Adaptive Spatial Masking (ASM) module that learns a node-specific adaptive interaction radius where information exchange occurs only between nodes within this radius, eliminating noise from irrelevant distant nodes. Furthermore, we design a cascaded architecture with hyper-connection [37] and Mixed Multi-Layer Perceptron (M-MLP), enabling dynamic adaptation of inter-layer connectivity strengths and branch interactions. Our experiments demonstrate that FlowNet achieves statistically significant improvements over existing state-of-the-art approaches across seven key metrics when modeling three challenging real-world systems.

2 Similarity vs. Intrinsic Flow: Which is better?

In this section, we formalize the modelling of the dynamic spatio-temporal systems and the key concepts underlying our proposed paradigm. Let a spatio-temporal system be represented as a graph G = ( V , E ) , where V = { v 1 , . . . , v N } denotes N nodes such as traffic sensors or geographic regions, and E defines edges. The system's state at time t is characterized by node observations X t ∈ R N , which capture measurable attributes such as traffic speed or water level. Given historical observations { X t -T , . . . , X t -1 } , the task aims to predict future states { X t , . . . , X t + τ } .

Traditional similarity-driven methods, such as graph convolution and spatial attention, fundamentally operate by blending node features based on predefined or dynamically inferred similarity metrics. While these approaches effectively capture surface-level correlations in observational data, they conflate statistical associations with the intrinsic mechanisms driving system evolution. By prioritizing feature proximity over directional flow dynamics, these methods fail to disentangle observational correlations from the flow-mediated interdependencies that actively redistribute system states. In contrast, our flow-based framework adopts a first-principles perspective, modeling system dynamics through intrinsic flow interactions. Specifically, we propose three guidelines for the framework:

Flow-Centric System Dynamics : All interactions between nodes across varying spatio-temporal contexts are mediated by flow tokens ϕ ∈ R d , which serve as quantifiable information carriers. A node's state is determined by its accumulated flow tokens, and system evolution emerges from the directed redistribution of ϕ across nodes and time steps.

Conservation Laws : Information transfer between nodes follows a source-destination conservation law. If v j transmits a flow token ϕ t j → i to v i , the source token Φ j depletes by ϕ t j → i while the destination token Φ i increases proportionally.

Adaptive Propagation Domain : For each node v i , its interaction neighborhood N t i ⊆ V at time t is determined by a learnable, context-aware radius r t i , discarding interactions beyond r t i .

3 Methodology

We present FlowNet , a physics-inspired architecture that re-formulates the modeling of spatiotemporal dynamics through flow-based information propagation, as shown in Figure 2. FlowNet first transforms the system state into latent representations via a patchify module, then progressively processes these features through cascaded modules to capture the spatio-temporal evolution of the system, and ultimately decodes the refined patterns through a projection layer to generate future state predictions. Our design dynamically constrains node interactions via Adaptive Spatial Masking (ASM), which prevents non-physical long-range dependencies by restricting communication to physically plausible regions. Central to the FlowNet are learnable flow tokens ϕ ∈ R d that serve as adaptive carriers for spatio-temporal information exchange. Specifically, FlowNet first embeds each node's observation into a high-dimensional flow space R d through patching. Each node generates and distributes its own flow tokens through the Flow Allocation Modules (FAMs). Based on these, different FAMs are cascaded with each other via hyper-connection and M-MLP to enhance the nonlinearity of the model.

×

L

#Linears = 3

N

×

P

×

d

N

×

P

×

d

ASM

Patchify

Project

Historical Series

FAM

M-MLP

Predicted Series

× N

Mask

Masked Node

Learnable

Perceptual Radius

Norm & Activation

Activate Linear

Deactivate Linear

P

×

N

×

N

N

×

T

N

×

τ

Destination

FEM

N

×

h

×

P

×

d'

Origin

N

×

h

×

P

×

d'

h

×

N

×

N

×

P

OD-Matrix

Flow-token

N

×

h

×

P

×

d'

N

×

P

×

d

unfold

×

N

×

P

×

d

Time

Value

Time

Value

Router

Hyper-connection

Data-pipeline

000000

0

3.1 Preprocess

We initially construct the latent state of the spatio-temporal system by structured sequence tokenization, a process designed to capture temporal patterns. Given an input sequence X ∈ R N × T representing N nodes over T historical timesteps, we first partition each node's univariate series into P localized patches via a sliding window operation:

where M denotes the patch length and S is the stride size controlling overlap between adjacent patches ( S < M for overlapping regimes). Each patch is then projected into the flow space via a parameterized embedding:

where W e ∈ R P × d and b e ∈ R d are learnable parameters. The patching process reduces the number of input tokens to the model by a factor of S and significantly increases the model runtime performance in comparison with standard point-wise approaches. For convenience, we utilize the subscript t to refer to different patches in the next subsections.

3.2 Adaptive Spatial Masking

Spatio-temporal systems inherently follow Tobler's First Law of Geography [38] - nearby locations tend to share similar characteristics through spatial interactions. However, real-world observations reveal that the effective interaction range between locations dynamically changes across space and time. We propose the Adaptive Spatial Masking (ASM) mechanism to address this gap by enabling each node to automatically determine its context-aware interaction range, adapting to both geographical constraints and transient system states. For each node v i , we first augment its temporal features X ′ i ∈ R P × d with a learnable node-specific embedding E i ∈ R d , capturing static attributes (e.g., geographic coordinates or infrastructure properties):

where ⊙ denotes broadcasting and 1 P is a ones vector replicating E i across all P patches for aligning the dimension. The perception radius r t i ∈ R + for node v i at patch t is dynamically predicted via:

where W h ∈ R 2 d × 1 and b e ∈ R 1 × 1 are learnable weights. The Softplus operation ensures a non-negative radius while maintaining gradient stability. A time-varying spatial mask M t ∈ R N × N is then constructed based on geographic distances d ij between node v i and v j :

We use a Sigmoid function to make the mask a differentiable operation, which in turn can be optimized by gradient descent and backpropagation during training. The measure of d ij can be determined according to the specific spatio-temporal system. For instance, the Manhattan distance can be chosen to measure the distance between nodes in an urban context, while the Euclidean distance can be utilized in a spatially isotropic natural system. By choosing different distance measurements and introducing the learnable perceptual radius, ASM is able to dynamically decide the propagation range of the flow tokens of each node based on the spatio-temporal context, thus filtering out non-physical long-distance dependent interference.

3.3 Flow Allocation Modules

To model spatio-temporal dependencies arising from flow token exchange between nodes, we propose Flow Allocation Modules (FAMs). At their core, FAM incorporates two Flow Estimation Modules (FEMs) with identical architectures but distinct learnable parameters. One FEM estimates the initial tokens Φ o retained by nodes, while the other predicts the allocated tokens Φ a distributed by nodes.

Notably, while standard attention mechanisms show limitations in capturing spatial dependencies through our analysis, they remain effective for temporal modeling when applied to local temporal patches. Building on this insight, we implement each FEM using Causal Temporal Multi-head Self-Attention ( CTMSA ) [3] to process time-ordered patch sequences. Formally, given augmented node features ˜ X = [ ˜ X 1 , · · · , ˜ X N ] ∈ R N × P × d , CTMSA produces:

where W f , b f are learnable parameters of affine transformation, h denotes attention heads and d ′ = d/h . Here, ˆ Φ corresponds to the head-folded representations of Φ o and Φ a , and we utilize this superscript to denote the tensor reshaped into multiple heads in the subsequent part. The FEMs operate in a channel-independent manner, while we retain the vanilla multi-head design to enhance the characterization of token.

Given the estimated tokens Φ o (retained) and Φ a (allocated) from the FEMs, we next compute the flow distribution between neighboring nodes. For node v i at patch t , we first transform its augmented features ˜ X t i through an affine projection followed by node-wise layer normalization, yielding ˙ X t i ∈ R d . From this, we derive two dynamic representations via linear transformations:

Origin vector O t i ∈ R d encoding v i 's role as a token source.

Destination vector D t i ∈ R d encoding v i 's role as a token recipient.

These vectors are augmented with static node properties through learnable head-folded embeddings E i,O , E i,D ∈ R d ′ , yielding multi-head representations ˆ O t i and ˆ D t i . The flow logit from v i to neighbor v j ∈ N t i is then computed as follows:

where α = 1 / √ d ′ is the scaling factor that stabilizes gradient magnitudes, and M t [ i, j ] is the element in the spatial mask learned by ASM. The normalized transfer probability is obtained via:

Thus, the final tokens ˆ ϕ t i ∈ R h × d ′ that v i owns can be formulated as:

where ˆ ϕ t i,a is the tokens sent out by v i , and ˆ ϕ t k,a → i = p t ki · ˆ ϕ t k,a is the tokens received by v i . Subsequently, the matrix expression for the flow allocation process at t can be formulated as:

where ˆ Φ t ∈ R h × N × d ′ is the flow for all nodes at t . Λ t ∈ R N × N is the allocation matrix at t where Λ t [ i, j ] = p t ij . By separating retained and allocated tokens, FAM enforces source depletion and destination enhancement during transfers.

3.4 Cascading of Modules

We introduce hyper-connection [37] instead of normal residual connections to optimize the way modules are stacked. The hyper-connection introduces depth-connection to assign weights to the connections between the inputs and outputs of each module, and includes width-connection to allow information exchange within the same layer. Unlike rigid residual connections, these depth- and width-aware connections adaptively scale information flow, ensuring that multi-scale flow dynamics are hierarchically aggregated within deep network structures. Additionally, We intersperse MLP units between different FAM modules. After directional exchange via FAM, flow tokens pass through an MLP that fuses information across their multi-head representations. We replace vanilla linear layers with a Mixture of Linears (MoL). Unlike Mixture of Experts (MoE), which treats entire blocks as experts, MoL treats each linear projection as an independent expert. For an MLP with L layers, this design creates a combinatorial parameter space of L × E distinct linear transformations compared to only E transformations in equivalently sized MLP-MoE designs.

4 Experiments

In this section, we conduct experiments to evaluate FlowNet's performance and reveal its underlying mechanisms. Our experiments are designed around the following Research Questions (RQ):

RQ1 : How does FlowNet perform compared to existing Spatio-Temporal forecasting approaches?

RQ2 : How does each component in the flow dynamics contribute to improving model performance?

RQ3 : How efficient is FlowNet compared to other models?

RQ4 : What distribution do the node degree and perceptual radius learned through ASM obey?

RQ5 : What are the differences between the allocation matrix Λ learnt through FAM and the attention map learnt through the spatial attention-based methods?

4.1 Experimental Setups

Datasets. We evaluate our approach on three datasets (PEMS04F [36], DeepBase [39], SINPA [40]) of dynamic spatio-temporal systems, spanning transportation, hydrology, and urban mobility domains. To compare the forecasting effectiveness of FlowNet with other baselines on different temporal forecasting scales, we set up both short-term forecasting and long-term forecasting tasks on each dataset. The historical and predicted step sizes utilised were aligned across all tasks. Further details are listed in Appendix B.

Baselines. We include ten state-of-the-art methods for forecasting performance comparisons, including Autoformer [41], PatchTST [42], Crossformer [43], iTransformer [44], FEDformer [45], STGCN [25], GWNET [35], SCINet [46], STTN [16], and STAEformer [19].

Implementation Details. We conduct all experiments on one NVIDIA A100 80GB GPU. The Adam optimizer is utilized to train our model, and the batch size is 8. The learning rate starts from 1 × 10 -3 , halved every 20 epochs until the 60th epoch, and we start early stopping at the 20th epoch of training. For FlowNet, we stack 2 layers of FAM for FlowNet and set up 16 experts inside the M-MLP. the Flow token uses 4 heads, and all hidden layer dimensions are set to 64. We leverage Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) for evaluation, where a smaller metric means better performance. More details can be found in Appendix B.

4.2 Model Comparison (RQ1)

In this section, we perform a model comparison in terms of MAE and RMSE. We run each method five times with different random seeds and report the average metric of each model. According to the results in Table 1, our proposed FlowNet model demonstrates remarkable improvements over the baseline models across all datasets and metrics. This outcome validates the effectiveness of our model in handling spatio-temporal data derived from diverse domains and varying spatial scales. Notice that FlowNet has taken the lead on both long-term and short-term prediction tasks, whereas none of the previous methods (STGNN or Transformer-based models) have been able to maintain a consistent advantage on a particular dataset or task. We identify that there are two potential reasons for FlowNet to perform well. Firstly, the flow mechanism is a more intrinsic modeling of the system compared to similarity capture, allowing FlowNet to accurately predict the short- and long-term evolution of the system. Secondly, the deployment of ASM allows FlowNet to dynamically determine the propagation range of the flow based on the spatio-temporal context, which allows it to be able to adapt to diverse spatio-temporal systems.

4.3 Ablation Study (RQ2)

Effects of Retained Flow . We remove the flow tokens Φ o retained by each node, and each node's tokens are obtained only by the other nodes sending to themselves in this experiment. According to the results presented in Table 2, eliminating the retained flow has the least impact on the short-term forecasting task compared to eliminating the other items. However, when it comes to the long-term forecasting task, eliminating this item has a greater impact on FlowNet. Given that retained flow complements the flow tokens of the nodes, this suggests that the spatio-temporal system as a whole remains closed and that the total amount of information essentially remains unchanged in the short term. On longer time scales, however, the spatio-temporal system may experience additional inflows and outflows of information.

Effects of Allocation Flow . We remove the flow tokens Φ a that nodes send to each other, and the tokens for each node are only generated by the retained flow, which means that no more information is exchanged between nodes in this experiment. Eliminating this item has the greatest impact on the model, as can be observed in Table 2. This suggests that, in complex spatio-temporal systems, the exchange of information between nodes cannot be ignored, and that it is undesirable to use node independence alone to predict the system's evolution.

Effects of Conservation Laws . We break the conservation law of flow exchange in this experiment, where each node no longer subtracts the flow tokens it sends after sending them. As shown in Table 2, breaking conservation laws has less impact on the model in long-term forecasting than eliminating retained flow. However, it has more impact in short-term forecasting. Combining this with previous analyses leads to an interesting conclusion: For short-term prediction, the spatio-temporal system is relatively closed. Therefore, maintaining information conservation

throughout the system, as well as in the exchange of information between nodes, is more important for predicting the system's evolution. For long-term prediction, however, information flowing into and out of the system cannot be ignored. It is more important to consider the fluctuation of the total amount of information in the system as a whole for accurate prediction.

4.4 Efficiency Analysis (RQ3)

Train time (s)

Memory (GB)

Autoformer

PatchTST

Crossformer

iTransformer

Fedformer

SCINet

STGCN

GWNET

STTN

STAEformer

FlowNet

46.13

0.05

12.01

0.46

39.96

0.09

9.68

2.98

112.07

2.98

124.83

2.98

29.53

0.13

37.49

0.47

40.54

3.19

38.62

50.81

65.74

1.50

PEMS04F

Train time (s)

Memory (GB)

Autoformer

PatchTST

Crossformer

iTransformer

Fedformer

SCINet

STGCN

GWNET

STTN

STAEformer

FlowNet

59.36

0.11

53.91

1.72

76.33

1.13

45.02

20.18

251.23

20.18

838.45

64.08

94.22

3.59

172.05

8.50

934.39

63.85

862.40

63.23

434.00

23.70

DeepBase

In this experiment, we analyze the training efficiency and GPU memory consumption of FlowNet versus baseline models on PEMS04F and DeepBase datasets for short-term forecasting tasks. As shown in Figure 3, FlowNet achieves intermediate computational efficiency between Transformer-based and STGNNbased models, balancing both training time and memory requirements. This efficiency profile stems from FlowNet's hybrid architecture that strategically integrates flow propaga- tion mechanism with adaptive spatial masking, effectively balancing the computational intensity of full-sequence transformers against the memory overhead inherent in graph neural operations. Meanwhile, the advanced cascading method of hyper-connection does not significantly increase memory overhead. We skillfully achieve a balance between training efficiency and accuracy.

4.5 Distribution Analysis (RQ4)

In this experiment, we count the distribution of the degree and the perceptual radius of the nodes learned by ASM on the PEMS04F dataset. If node A is within the perceptual radius of node B, then we consider that there exists a directed edge from node B pointing to node A. As shown in Figure 6, for the long-term forecasting task, both the degree and perceptual radius distributions of the nodes are more concentrated compared to the



















































short-term forecasting task. It is worth noting that the distributions of both the degree and radius of the nodes within the short-term prediction show a multi-peaked distribution. Based on this observation, we can reveal the reason for the sub-optimal performance of graph-based and attention-based models: graph-based models are unable to dynamically take into account dependencies between nodes according to diverse spatio-temporal systems and different tasks, while attention-based models compute extensive pseudo-dependencies, but dependencies exist among only some but not all nodes. Additionally, based on the previous analyses, both approaches mistake superficial proximity for intrinsic dependency, limiting the upper bound of the model's representation.

4.6 Visualization of Allocation Matrix (RQ5)

In Figure 5, we visualized the allocation matrix Λ learned by FAM and compared it with the attention maps of transformer-based models such as iTransformer and STTN in the PEMS04F dataset. For FlowNet, we can observe that the allocation matrix of nodes is more centralized and clearer in the long-term prediction task than in the short-term prediction task. Based on the learning from FlowNet, some nodes receive the vast majority of the flow converged from the source nodes when they act as target nodes. In contrast, the attention maps of iTransformer and STTN have larger values for the attention coefficient, and the similarities captured be-

0.025

0.08

0.010

0.020

0.008

0.06

0.015

0.006

0.04

0.010

0.004

0.02

0.005

- 0.002

(a) FlowNet (short)

(b) iTransformer (short)

(c) STTN (short)

0.035

0.04

0.030

0.10

0.025

0.03

0.08

0.020

0.02

0.06

0.015

0.04

0.010

0.01

0.02

0.005

(d) FlowNet (long)

(e) iTransformer (long)

(f) STTN (long)

tween nodes are much denser. We speculate that this is due to the adaptive radius of the ASM filtering out many spurious dependencies. According to the statistics in Table 1, FlowNet outperforms both transformer-based models on all datasets, especially on DeepBase and SINPA, which are two large spatio-temporal systems. This suggests that filtering out spurious correlations is important to improve the performance of the model.

5 Related Works

Classical statistical methods [47, 48] model temporal dependencies through linear or probabilistic formulations, but fail to capture nonlinear spatio-temporal couplings. Recent innovations attempt to bridge these gaps through adaptive graph learning and hybrid architectures. Modern deep learning advances [22, 49] decompose temporal patterns via specialized mechanisms. Methods such as STSGCN [26] and DCRNN [50] introduce dynamic graph structure adjustments while retaining heuristic similarity metrics. Transformer-based architectures like Crossformer [43] and iTransformer [44] advance cross-variable interaction modeling through dimension-aware attention or inverted embedding strategies. Concurrent work like STAEformer [19] incorporates graph information bottlenecks to improve interpretability. PDFormer [51] introduces a novel propagation delay-aware dynamic longrange dependency modeling approach through gated self-attention mechanisms, achieving advanced performance across multiple traffic flow prediction benchmarks. UniST [52] introduces a unified spatio-temporal learning framework that integrates adaptive multi-scale attention and task-agnostic representation learning, achieving superior accuracy and generalization across diverse spatio-temporal forecasting tasks. However, these approaches still neglect the directional flow dynamics inherent in transportation networks [53, 54] and hydrological systems [55, 56, 57]. Parallel progress in physicsinspired paradigms [58] prioritizes transfer mechanisms through spatio-temporal dynamic processes. However, such methods lack modular frameworks for adaptive propagation that could systematically model context-dependent information flows.

6 Conclusion and Future Work

In this work, we introduce FlowNet, a novel paradigm for modeling dynamic spatio-temporal systems that rethinks interactions in complex systems through the lens of directional flow dynamics. By shifting from static or similarity-driven representations to adaptive spatio-temporal flows, FlowNet overcomes critical limitations of existing methods. Experiments demonstrate FlowNet's superior accuracy and physical plausibility. Looking ahead, extending FlowNet to incorporate domain-specific conservation laws and applying it to flow-driven systems like supply chains or social networks presents promising directions. By harmonizing data-driven learning with principles of physical realism, this work advances the development of spatio-temporal models that better align with the dynamic, directional nature of real-world systems.

Acknowledgments and Disclosure of Funding

This work is mainly supported by the Guangdong Basic and Applied Basic Research Foundation (No. 2025A1515011994). This work is also supported by the National Natural Science Foundation of China (No. 62402414), Tencent (CCF-Tencent Open Fund, Tencent Rhino-Bird Focused Research Program), Didi (CCF-DiDi GAIA Collaborative Research Funds), Guangzhou Municipal Science and Technology Project (No. 2023A03J0011), Huawei Industrial Funds, and the Guangzhou Industrial Information and Intelligent Key Laboratory Project (No. 2024A03J0628).

References

Z. Pan, Y. Liang, W. Wang, Y. Yu, Y. Zheng, and J. Zhang, 'Urban traffic prediction from spatio-temporal data using deep meta learning,' in Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining, 2019, pp. 1720-1730.

Y. Liang, K. Ouyang, L. Jing, S. Ruan, Y. Liu, J. Zhang, D. S. Rosenblum, and Y. Zheng, 'Urbanfm: Inferring fine-grained urban flows,' in Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining, 2019, pp. 3132-3142.

Y. Liang, Y. Xia, S. Ke, Y . Wang, Q. Wen, J. Zhang, Y. Zheng, and R. Zimmermann, 'Airformer: Predicting nationwide air quality in china with transformers,' in Proceedings of the AAAI conference on artificial intelligence, vol. 37, no. 12, 2023, pp. 14 329-14 337.

D. Helbing, D. Armbruster, A. S. Mikhailov, and E. Lefeber, 'Information and material flows in complex networks,' Physica A: Statistical Mechanics and its Applications, vol. 363, no. 1, pp. xi-xvi, 2006.

U. Harush and B. Barzel, 'Dynamic patterns of information flow in complex networks,' Nature communications, vol. 8, no. 1, p. 2181, 2017.

R. Li, L. Dong, J. Zhang, X. Wang, W.-X. Wang, Z. Di, and H. E. Stanley, 'Simple spatial scaling rules behind complex cities,' Nature communications, vol. 8, no. 1, p. 1841, 2017.

Z. Zheng, 'An introduction to emergence dynamics in complex systems,' Frontiers and Progress of Current Soft Matter Research, pp. 133-196, 2021.

J. Gao and B. Xu, 'Complex systems, emergence, and multiscale analysis: A tutorial and brief survey,' Applied Sciences, vol. 11, no. 12, p. 5736, 2021.

G. Atluri, A. Karpatne, and V. Kumar, 'Spatio-temporal data mining: A survey of problems and methods,' ACM Computing Surveys (CSUR), vol. 51, no. 4, pp. 1-41, 2018.

S. Wang, J. Cao, and S. Y. Philip, 'Deep learning for spatio-temporal data mining: A survey,' IEEE transactions on knowledge and data engineering, vol. 34, no. 8, pp. 3681-3700, 2020.

Y. Liang, H. Wen, Y. Xia, M. Jin, B. Yang, F. Salim, Q. Wen, S. Pan, and G. Cong, 'Foundation models for spatio-temporal data science: A tutorial and survey,' arXiv preprint arXiv:2503.13502, 2025.

A. Goodge, W. S. Ng, B. Hooi, and S. K. Ng, 'Spatio-temporal foundation models: Vision, challenges, and opportunities,' arXiv preprint arXiv:2501.09045, 2025.

Z. A. Sahili and M. Awad, 'Spatio-temporal graph neural networks: A survey,' arXiv preprint arXiv:2301.10569, 2023.

G. Jin, Y. Liang, Y. Fang, Z. Shao, J. Huang, J. Zhang, and Y. Zheng, 'Spatio-temporal graph neural networks for predictive learning in urban computing: A survey,' IEEE Transactions on Knowledge and Data Engineering, vol. 36, no. 10, pp. 5388-5408, 2023.

Y. Li, D. Yu, Z. Liu, M. Zhang, X. Gong, and L. Zhao, 'Graph neural network for spatiotemporal data: methods and applications,' arXiv preprint arXiv:2306.00012, 2023.

M. Xu, W. Dai, C. Liu, X. Gao, W. Lin, G.-J. Qi, and H. Xiong, 'Spatial-temporal transformer networks for traffic flow forecasting,' arXiv preprint arXiv:2001.02908, 2020.

Q. Wen, T. Zhou, C. Zhang, W. Chen, Z. Ma, J. Yan, and L. Sun, 'Transformers in time series: A survey,' arXiv preprint arXiv:2202.07125, 2022.

X. Liu, J. Liu, G. Woo, T. Aksu, Y. Liang, R. Zimmermann, C. Liu, S. Savarese, C. Xiong, and D. Sahoo, 'Moirai-moe: Empowering time series foundation models with sparse mixture of experts,' arXiv preprint arXiv:2410.10469, 2024.

H. Liu, Z. Dong, R. Jiang, J. Deng, J. Deng, Q. Chen, and X. Song, 'Spatio-temporal adaptive embedding makes vanilla transformer sota for traffic forecasting,' in Proceedings of the 32nd ACM international conference on information and knowledge management, 2023, pp. 41254129.

J. Huang, Y. Xu, Q. Wang, Q. C. Wang, X. Liang, F. Wang, Z. Zhang, W. Wei, B. Zhang, L. Huang et al., 'Foundation models and intelligent decision-making: Progress, challenges, and perspectives,' The Innovation, 2025.

Y. Yan, Q. Zeng, Z. Zheng, J. Yuan, J. Feng, J. Zhang, F. Xu, and Y. Li, 'Opencity: A scalable platform to simulate urban activities with massive llm agents,' arXiv preprint arXiv:2410.21286, 2024.

X. Liu, J. Hu, Y. Li, S. Diao, Y. Liang, B. Hooi, and R. Zimmermann, 'Unitime: A languageempowered unified model for cross-domain time series forecasting,' in Proceedings of the ACM Web Conference 2024, 2024, pp. 4095-4106.

Z. Li, L. Xia, X. Ren, J. Tang, T. Chen, Y. Xu, and C. Huang, 'Urban computing in the era of large language models,' arXiv preprint arXiv:2504.02009, 2025.

C. Hou, F. Zhang, Y. Li, H. Li, G. Mai, Y. Kang, L. Yao, W. Yu, Y. Yao, S. Gao et al., 'Urban sensing in the era of large language models,' The Innovation, vol. 6, no. 1, 2025.

B. Yu, H. Yin, and Z. Zhu, 'Spatio-temporal graph convolutional networks: A deep learning framework for traffic forecasting,' arXiv preprint arXiv:1709.04875, 2017.

C. Song, Y. Lin, S. Guo, and H. Wan, 'Spatial-temporal synchronous graph convolutional networks: A new framework for spatial-temporal network data forecasting,' in Proceedings of the AAAI conference on artificial intelligence, vol. 34, no. 01, 2020, pp. 914-921.

Y. Liang, S. Ke, J. Zhang, X. Yi, and Y. Zheng, 'Geoman: Multi-level attention networks for geo-sensory time series prediction.' in IJCAI, vol. 2018, 2018, pp. 3428-3434.

C. Zheng, X. Fan, C. Wang, and J. Qi, 'Gman: A graph multi-attention network for traffic prediction,' in Proceedings of the AAAI conference on artificial intelligence, vol. 34, no. 01, 2020, pp. 1234-1241.

W. Yao, Y. Sun, A. Ho, C. Sun, and K. Zhang, 'Learning temporally causal latent processes from general temporal data,' arXiv preprint arXiv:2110.05428, 2021.

Y. Liu, R. Cadei, J. Schweizer, S. Bahmani, and A. Alahi, 'Towards robust and adaptive motion forecasting: A causal representation perspective,' in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 17 081-17 092.

B. Tian, Y. Cao, Y. Zhang, and C. Xing, 'Debiasing nlu models via causal intervention and counterfactual reasoning,' in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 36, no. 10, 2022, pp. 11 376-11 384.

Y. Zhao, P. Deng, J. Liu, X. Jia, and J. Zhang, 'Generative causal interpretation model for spatiotemporal representation learning,' in Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, 2023, pp. 3537-3548.

W. Liu, Y. Zheng, S. Chawla, J. Yuan, and X. Xing, 'Discovering spatio-temporal causal interactions in traffic data streams,' in Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining, 2011, pp. 1010-1018.

S. Yanchuk and G. Giacomelli, 'Spatio-temporal phenomena in complex systems with time delays,' Journal of Physics A: Mathematical and Theoretical, vol. 50, no. 10, p. 103001, 2017.

B. Xu, H. Shen, Q. Cao, Y. Qiu, and X. Cheng, 'Graph wavelet neural network,' arXiv preprint arXiv:1904.07785, 2019.

S. Guo, Y. Lin, N. Feng, C. Song, and H. Wan, 'Attention based spatial-temporal graph convolutional networks for traffic flow forecasting,' in Proceedings of the AAAI conference on artificial intelligence, vol. 33, no. 01, 2019, pp. 922-929.

D. Zhu, H. Huang, Z. Huang, Y. Zeng, Y. Mao, B. Wu, Q. Min, and X. Zhou, 'Hyperconnections,' arXiv preprint arXiv:2409.19606, 2024.

H. J. Miller, 'Tobler's first law and spatial analysis,' Annals of the association of American geographers, vol. 94, no. 2, pp. 284-289, 2004.

P. Ghaneei and H. Moradkhani, 'Deepbase: A deep learning-based daily baseflow dataset across the united states,' Scientific Data, vol. 12, no. 1, p. 25, 2025.

H. Zhang, Y. Xia, S. Zhong, K. Wang, Z. Tong, Q. Wen, R. Zimmermann, and Y. Liang, 'Predicting carpark availability in singapore with cross-domain data: A new dataset and a data-driven approach,' in Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence, 2024, pp. 7554-7562.

H. Wu, J. Xu, J. Wang, and M. Long, 'Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting,' Advances in neural information processing systems, vol. 34, pp. 22 419-22 430, 2021.

Y. Nie, N. H. Nguyen, P. Sinthong, and J. Kalagnanam, 'A time series is worth 64 words: Long-term forecasting with transformers,' arXiv preprint arXiv:2211.14730, 2022.

Y. Zhang and J. Yan, 'Crossformer: Transformer utilizing cross-dimension dependency for multivariate time series forecasting,' in The eleventh international conference on learning representations, 2023.

Y. Liu, T. Hu, H. Zhang, H. Wu, S. Wang, L. Ma, and M. Long, 'itransformer: Inverted transformers are effective for time series forecasting,' arXiv preprint arXiv:2310.06625, 2023.

T. Zhou, Z. Ma, Q. Wen, X. Wang, L. Sun, and R. Jin, 'Fedformer: Frequency enhanced decomposed transformer for long-term series forecasting,' in International conference on machine learning. PMLR, 2022, pp. 27 268-27 286.

M. Liu, A. Zeng, M. Chen, Z. Xu, Q. Lai, L. Ma, and Q. Xu, 'Scinet: Time series modeling and forecasting with sample convolution and interaction,' Advances in Neural Information Processing Systems, vol. 35, pp. 5816-5828, 2022.

G. E. Box and D. A. Pierce, 'Distribution of residual autocorrelations in autoregressiveintegrated moving average time series models,' Journal of the American statistical Association, vol. 65, no. 332, pp. 1509-1526, 1970.

J.-M. Montero, G. Fernández-Avilés, and J. Mateu, Spatial and spatio-temporal geostatistical modeling and kriging. John Wiley & Sons, 2015.

H. Wu, T. Hu, Y. Liu, H. Zhou, J. Wang, and M. Long, 'Timesnet: Temporal 2d-variation modeling for general time series analysis,' arXiv preprint arXiv:2210.02186, 2022.

Y. Li, R. Yu, C. Shahabi, and Y. Liu, 'Diffusion convolutional recurrent neural network: Data-driven traffic forecasting,' arXiv preprint arXiv:1707.01926, 2017.

J. Jiang, C. Han, W. X. Zhao, and J. Wang, 'Pdformer: Propagation delay-aware dynamic long-range transformer for traffic flow prediction,' in Proceedings of the AAAI conference on artificial intelligence, vol. 37, no. 4, 2023, pp. 4365-4373.

Y. Yuan, J. Ding, J. Feng, D. Jin, and Y. Li, 'Unist: A prompt-empowered universal model for urban spatio-temporal prediction,' in Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, 2024, pp. 4095-4106.

T. Li, 'stability of conservation laws for a traffic flow model.' Electronic Journal of Differential Equations (EJDE)[electronic only], vol. 2001, pp. Paper-No, 2001.

T. Seo and T. Kusakabe, 'Probe vehicle-based traffic state estimation method with spacing information and conservation law,' Transportation Research Part C: Emerging Technologies, vol. 59, pp. 391-403, 2015.

S. G. Dobrovolski, V. P. Yushkov, T. Y. Vyruchalkina, and O. V. Sokolova, 'Are there fundamental laws in hydrology?' Pure and Applied Geophysics, vol. 179, no. 4, pp. 1475-1484, 2022.

M. L. Kavvas, 'Nonlinear hydrologic processes: Conservation equations for determining their means and probability distributions,' Journal of Hydrologic Engineering, vol. 8, no. 2, pp. 44-53, 2003.

P. Y. Lu, R. Dangovski, and M. Soljaˇ ci´ c, 'Discovering conservation laws using optimal transport and manifold learning,' Nature Communications, vol. 14, no. 1, p. 4744, 2023.

Y. Feng, Q. Wang, Y. Xia, J. Huang, S. Zhong, and Y. Liang, 'Spatio-temporal field neural networks for air quality inference,' in Proceedings of the Thirty-Third International

Joint Conference on Artificial Intelligence, IJCAI-24, K. Larson, Ed. International Joint Conferences on Artificial Intelligence Organization, 8 2024, pp. 7260-7268, aI for Good. [Online]. Available: https://doi.org/10.24963/ijcai.2024/803

Y. Wang, H. Wu, J. Dong, Y. Liu, M. Long, and J. Wang, 'Deep time series models: A comprehensive survey and benchmark,' 2024.

X. Liu, Y. Xia, Y. Liang, J. Hu, Y. Wang, L. Bai, C. Huang, Z. Liu, B. Hooi, and R. Zimmermann, 'Largest: A benchmark dataset for large-scale traffic forecasting,' Advances in Neural Information Processing Systems, vol. 36, pp. 75 354-75 371, 2023.

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: We propose FlowNet, a novel prediction paradigm based on spatio-temporal flow. We pioneer this data-driven approach to modeling complex spatio-temporal systems from first principles in a physically driven manner.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We explore the limitations of this work in the Appendix.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [NA]

Justification: This is an application-oriented work, and we do not provide any theoretical result.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: We provide detailed experimental settings in section 4.1 and Appendix to facilitate reproducibility.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: All the datasets we use are open source. We provide links to our anonymised code in the Appendix.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so 'No' is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: We provid detailed experimental settings in section 4.1 and Appendix to facilitate reproducibility.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: We conducted t-tests in the experiment section to illustrate the level of significance at which our method outperforms other methods on each metric.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: We provid detailed experimental settings in section 4.1 and Appendix to facilitate reproducibility.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: We follow the NeurIPS Code of Ethics in this paper.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: We state that our goal is to offer the machine-learning community a fresh perspective and inspire further research on the essence of dynamic spatio-temporal system modeling in section 6.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: The datasets chosen in this paper are common and open-sourced datasets for dynamic spatio-temporal system modeling.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: Yes, we have.

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [Yes]

Justification: This paper follows CC 4.0, and the code is in an anonymized URL.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: This paper does not involve crowdsourcing nor research with human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: This paper does not involve crowdsourcing nor research with human subjects. Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

16. Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.

Answer: [NA]

Justification: We only use LLM for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research.

Guidelines:

The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.

Please refer to our LLM policy ( https://neurips.cc/Conferences/2025/LLM ) for what should or should not be described.

A More Details of Model Implementation

A.1 Initialization

For the linear layers in M-MLP, we employed Kaiming initialization to set their initial parameters. All other linear layers in the model were initialized using Xavier initialization. GeLU is employed as the activation function in M-MLP. Specifically, for linear layers in the ASM module, we initialized the weight parameters w to zero and set the bias terms b to the average distance matrix for each corresponding dataset. Through this initialization method, each node can have a sufficiently large and identical receptive field at the beginning of training, and can optimize its own perception radius through gradient descent and backpropagation.

B More Details of Experiments

B.1 Datasets

PEMS04F. The PEMS04 [36] dataset, constructed from the Caltrans Performance Measurement System (PeMS), captures traffic flow dynamics across 307 sensor nodes in California over a twomonth period (January 1 to February 28, 2018) with 5-minute granularity (16,992 timesteps). The original dataset includes three key traffic metrics (traffic flow, lane occupancy, and average speed), and we focus on the traffic flow (denoted as PEMS04F) attribute as the prediction target.

DeepBase. The DeepBase [39] dataset is a hydrological dataset providing daily baseflow estimates for 1,661 basins across the contiguous United States (CONUS) from 1981 to 2022. This dataset captures the slow-varying groundwater contributions to streamflow at a daily temporal resolution.

SINPA. The SINPA dataset [40] captures large-scale parking availability dynamics across Singapore, covering 1,687 parking lots over a one-year period (July 2020 - June 2021). It provides high-frequency spatio-temporal observations recorded at 15-minute intervals, where each node represents the available parking spaces at a specific lot. While the original dataset integrates diverse urban features, we focuses on modeling parking vacancy counts as the primary prediction target.

B.2 Preprocess

For the graph-based baseline, we construct a weighted adjacency matrix by applying a thresholded Gaussian kernel to pairwise Euclidean distances between nodes:

where edge weight W ij ∈ [0 , 1] encodes proximity between stations v i and v j , σ controls the kernel width, and κ sparsifies connections beyond local neighborhoods.

We standardize the spatiotemporal dataset X ∈ R N × T ( N nodes, T timesteps) via z-score normalization:

where µ ∈ R N and σ ∈ R N denote per-node training means and standard deviations. During evaluation, predictions on validation/test sets are inversely transformed ˆ x i,t = ˜ x i,t · σ i + µ i before computing evaluation metrics.

B.3 Training & Validation

We configure prediction horizons based on temporal granularity and dominant frequencies across datasets. For PEMS04F and SINPA, short-term forecasting uses 12-step input/output sequences (1h/3h), while long-term forecasting employs 288-step windows (1d/3d). DeepBase adopts 32-step (monthly, ensuring divisibility for patching) and 360-step (yearly) horizons, respectively. The datasets are partitioned as follows: PEMS04F (70% train, 10% validation, 20% test), DeepBase (pre-2021 train, 2021-2022 validation/test), and SINPA (pre-May-2021 train, May-June 2021 validation/test). We implement sliding window sampling with stride 1 during training, aligning window size with output horizon during inference.

Following standard evaluation protocols in machine learning, we quantify the predictive performance of regression models through two widely adopted metrics: the Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE). Formally, let Y i ∈ R represent the ground-truth value of the i -th data instance and ˆ Y i ∈ R denote its corresponding predicted value. These error metrics are computed as

where n denotes the total number of test samples. During training, we employ MAE as the loss function for FlowNet and all baselines. Our Early Stopping mode protocol monitors the MAE metric at the validation set and terminates training when no improvement is observed for 10 consecutive epochs relative to the best recorded value, indicating potential overfitting. We retain the model parameters, achieving the lowest validation MAE for final evaluation on the test set.

C Details of Baselines

Autoformer [41]: Autoformer integrates decomposition architecture into Transformers, replacing self-attention with an auto-correlation mechanism to capture periodic dependencies. It progressively decomposes trend and seasonal components, achieving efficient long-term forecasting with linear complexity.

PatchTST [42]: By segmenting time series into independent patches and adopting channelindependent modeling, PatchTST enhances local semantic extraction and reduces computational costs. It outperforms traditional Transformers in long-term forecasting tasks by leveraging vision transformer-inspired patch processing and self-supervised learning.

Crossformer [43]: Crossformer employs a two-stage attention mechanism to model crossdimension dependencies in multivariate time series. Its hierarchical encoder-decoder structure and dimension-segment-wise embedding efficiently capture interactions between time steps and variables.

iTransformer [44]: iTransformer inverts the standard architecture by treating time points as tokens and applying attention across variables.

FEDformer [45]: Combining frequency-domain transformations with seasonal-trend decomposition, FEDformer reduces computational overhead through random frequency component selection. Its linear complexity and frequency-enhanced blocks make it effective for long-term forecasting in energy and weather datasets.

SCINet [46]: SCINet uses a binary tree structure with interactive convolution blocks to hierarchically decompose time series. This architecture captures multi-resolution temporal dependencies and mitigates information loss, outperforming RNN and Transformer models in efficiency and accuracy.

STGCN [25]: STGCN integrates graph convolutional networks (GCNs) and gated temporal convolutions to model traffic networks. Its fully convolutional design processes large-scale spatiotemporal data efficiently.

GWNET [35]: This model introduces an adaptive adjacency matrix to learn hidden spatial dependencies and dilated convolutions for long-range temporal patterns. It addresses incomplete graph structures in traffic forecasting and achieves linear complexity with stable multi-step predictions.

STTN [16]: Spatial-Temporal Transformer Network replaces GCNs with dynamic spatial attention to capture time-varying node relationships. Its non-autoregressive multi-step prediction framework avoids error accumulation, significantly improving long-horizon forecasting.

STAEformer [19]: By incorporating spatiotemporal adaptive embeddings into vanilla Transformers, STAEformer dynamically adjusts to traffic patterns without complex architectural modifications by preserving intrinsic chronological information and spatial heterogeneity through lightweight adaptive components.

All baseline code implementations are based on the time-series-library [59] and LargeST [60]. Specifically, to ensure the fairness of the comparison and guarantee that most models can run on our existing computing resources, we set the hidden dimension of all models to 64. For encoder-only models, we stacked 2 encoder layers. For encoder-decoder architecture models, we stacked 1 encoder layer and 1 decoder layer.

D Further Analysis

We analyze task-specific differences in node degree and perceptual radius distributions for shortand long-term predictions on the SINPA dataset. Short-term predictions exhibit a sharp node degree distribution peaking, indicating reliance on highly connected nodes to capture dense local interactions. In contrast, long-term predictions show a flatter distribution peaking below 500, suggesting sparser node sampling to prioritize broader contextual patterns over fine-grained dynamics. Short-term predictions concentrate around moderate radii, balancing localized state changes and mid-range dependencies. Long-term predictions, however, favor smaller radii with greater dispersion, reflecting adaptive trade-offs: smaller radii suppress noise from distant nodes, while occasional long-range connections capture systemic shifts, such as holiday-driven destination changes. These divergences reveal that the model dynamically tailors its spatial aggregation strategy to temporal scope: short-term tasks emphasize resolution-rich local dynamics, whereas long-term tasks hierarchically integrate multi-scale dependencies at lower resolution. The distinct distributions further validate the necessity of task-aware spatial perception mechanisms in spatio-temporal forecasting.



















































E More Discussion

Limitations. While FlowNet achieves state-of-the-art prediction accuracy, its computational efficiency is constrained by pairwise operations. Specifically, the Flow Allocation operation incurs an O ( N 2 ) complexity due to node-wise flow redistribution, and the Flow Evolution Module (FEM) scales quadratically with the prediction horizon ( O ( T 2 ) ). For systems with large node counts or long-term forecasting tasks, FlowNet remains less efficient than STGNN-based methods, though it outperforms Transformer-based models in both speed and memory usage. We argue that the accuracy

gains justify this trade-off in many real-world applications. Future work will explore sparse flow tokenization and hierarchical grouping to mitigate these bottlenecks.

Societal Impacts. FlowNet's ability to model flow-driven dynamics can benefit urban planning, environmental protection, and disaster response. The framework enhances interpretability by explicitly quantifying flow exchanges, enabling policymakers to design targeted regulations. Furthermore, its conservation-law alignment promotes physically plausible predictions, reducing risks of harmful decisions based on spurious correlations.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents FlowNet for learning dynamics of spatio-temporal systems. Modeling such a system as a graph, this work presents two main contributions: \n\n1) Flow-centric system dynamics: While the other related work did not consider interactions between the nodes in the graph and focuses on the observed node features, this work points out that such node features are the results of interactions, or flows, between the nodes. Also, this work proposes to apply the conservation laws for these flows -- that is, the total sum of flows should be maintained over the entire graph.\n\n2) Adaptive propagation domain: To model the change of each node's effective spatial range over time, this work proposes to predict each node's effective radius based on the context. Therefore, for different temporal window, the radius changes and thus the relationship between the nodes change. It is similar to dynamic graph where connectivity changes.\n\nBased on these innovations, this work shows that it outperforms the other various baseline methods in predicting the dynamics of several spatio-temporal systems, including traffic system.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n\n- This work focuses on the interactions between the nodes and applies the conservation law on such interactions, which sounds reasonable.\n- Its analysis about model components (Section 4.3) and distributiona analysis (Section 4.5) aligns with the motivation of this work. \n\nWeakness:\n\n- Even though I'm not an expert in this area, the number of datasets seems to be fairly small (there are only 3 datasets). In several other works like STAEformer, I can see there are more datasets that the authors could use for comparison. This lack of comparison is the main reason why I rate this work as borderline reject in the current stage. I recommend the authors to consider do more experiments for other datasets, if possible. \n- The mathematical notations in Section 3 are confusing. For instance, in Equation 2, I think the notations are wrong -- it should be changed to: $X_p \\in R^{N \\times P \\times d}, W_e \\in R^{M \\times d}$. Also, in line 141, I think $X'_i$ should be changed to $X_i$. I could be wrong, but the notations and the formulations are too confusing to follow right now. The writing could be improved to address this, I believe.\n\nQUESTIONS:\n- I'm wondering why the authors used radius-based spatial masking. It sounded reasonable at first, because the authors are mainly dealing with the nodes distributed in the real world. However, I became to think that this approach would be only effective for physical things like traffic. If it goes beyond the physical realm, I think the nodes that are situated far away could affect each other, and in this case, I suspect this radius-based approach could not be very effective. If it is true, I'm worried that it could be one of the limitations of this work.\n\nLIMITATIONS:\nThe authors discussed the limitation about computational cost in the appendix, but I think they have to consider more limitations, such as the one that I pointed above about the radius-based approach.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-25"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper claims that the literature for Spatio-Temporal (ST) models is missing information from some so-called flows. Such flows are justified with the example of people flowing in and out from a city for commuting. It propose an architecture to integrate such information for improving ST modeling.\n\nSTRENGTHS AND WEAKNESSES:\n**Weaknesses** \n\nI found several statements vague and not modivating well enogh the model. see in questions.\n\nExperimental results from see Table 1 are missing several recent works that achieve lower MAE. The proposed method achieves a MAE of `18.48`, while many of the missing baselines, not limited to, Gao et al (2023),  Dai et al (2024), Chen et al (2025), report better results.\nOther resuts on PeMS04 data are available on paperswithcode: https://paperswithcode.com/sota/traffic-prediction-on-pems04?p=time-series-is-a-special-sequence-forecasting\n\n**References** \n\nGao et al (2023). Spatial-Temporal-Decoupled Masked Pre-training for Spatiotemporal Forecasting\n\nDai et al (2024). A novel hybrid time-varying graph neural network for traffic flow forecasting\n\nChen et al (2025). Dynamic Trend Fusion Module for Traffic Flow Prediction\n\nQUESTIONS:\n**Question 1**\n> line 116: central to flownet are learnable flow tokens that serve as adaptive carriers for spatio-temporal information exchange\n\nunclear, could you elaborate and give intuition of what happens with this learnable part? \n\n**Question 2**\n>line 37: prevalent operators like graph convolution [24,25] and spatial attention [26,27,16,19] implicitly assume system dynamics emerge from static node attribute similarities,\n\n why? can you elaborate? give intuitive but specific ideas on how your tool differs?\n\n\n**Question 3**\n> line 44: Crucially, while spatially distant residential zones may exhibit similar traffic volumes through observational metrics, the underlying system dynamics derive from directional flow interactions between functionally complementary regions. \n \nthis sentence does not explain much, very vague. please elaborate and give intuition.\n\n**Question 4**\n> line 394: By prioritizing feature proximity over directional flow dynamics, these methods fail to disentangle observational correlations from the flow-mediated interdependencies that actively redistribute system states\n\nnot clear what that means, please elaborate and give intuition.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 1, "IMPACT": 1, "ORIGINALITY": 1, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nthe authors here propose FlowNet, a physics-inspired deep learning architecture for modeling dynamic spatio-temporal systems via explicit flow interactions governed by conservation principles. The key idea is that rather than relying on similarity-driven or static graph structures, FlowNet introduces flow tokens and Flow Allocation Modules (FAMs) to model information exchange. Empirical results on three real-world datasets show FlowNet outperforms state-of-the-art STGNNs and Transformer models in both short- and long-term forecasting tasks.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n- Novelty: I am not an expert in this field but from what I know  the idea of modeling spatio-temporal dependencies via directional flow exchanges is novel and presents a compelling shift in paradigm with physical interpretability.\n\n- Strong Empirical Results: FlowNet shows consistent improvements over competitive baselines, including Autoformer, STGCN, and STTN, across multiple datasets and tasks.\n\n**Weaknesses**\n\n- Narrow Evaluation Scope: The model is only tested on three datasets, all in related domains (transport, hydrology). No results are shown in settings like social networks, or language tasks, limiting claims of generality.\n\n- Scalability Not Demonstrated: Despite its design, the authors do not test FlowNet on large-scale graphs (e.g., >1000 nodes) or long sequence.\n\n- The claim that most prior work models dependencies solely via similarity is an oversimplification. Recent variants [1-4] variants already incorporate directional and structured modeling to some extent.\n\n\n[1] Li, M. and Zhu, Z., 2021, May. Spatial-temporal fusion graph neural networks for traffic flow forecasting. In Proceedings of the AAAI conference on artificial intelligence (Vol. 35, No. 5, pp. 4189-4196).\n\n[2] Wang, C., Tsepa, O., Ma, J. and Wang, B., 2024. Graph-mamba: Towards long-range graph sequence modeling with selective state spaces. arXiv preprint arXiv:2402.00789.\n\n[3] Yuan, H., Sun, Q., Wang, Z., Fu, X., Ji, C., Wang, Y., Jin, B. and Li, J., 2025, April. Dg-mamba: Robust and efficient dynamic graph structure learning with selective state space models. In Proceedings of the AAAI Conference on Artificial Intelligence (Vol. 39, No. 21, pp. 22272-22280).\n\n[4] Nanbo, L., Laakom, F., Xu, Y., Wang, W. and Schmidhuber, J., 2024. FACTS: A Factored State-Space Framework For World Modelling. arXiv preprint arXiv:2410.20922.\n\nQUESTIONS:\nSee weakness section above\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nFlowNet is a spatio-temporal neural network containing architectural innovations meant to adapt information flow between nodes according to the complex spatio-temporal dynamics in physical data. These innovations include adapting the radius mask of each node according to time and splitting node information into propagation and retention tokens subject to a conservation law. FlowNet further includes hyperconnections to mitigate gradient issues and make the information flow even more flexible. FlowNet demonstrates superior performance to several other spatio-temporal networks on three datasets, and ablation studies show the usefulness of the added components\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths:\n\n- The concept of flow tokens in addition to the conservation constraint appear to be novel. While the ablation study didn’t necessarily highlight the conservation constraint well, I believe this is an interesting concept worthy of being explored further in future work.\n\n- Methodologically, FlowNet appears to be sound, and the components interact well together. While Figure 2 appeared slightly convoluted to me and might benefit from being broken up into a high level diagram and an additional low diagram for each component, the architecture makes sense overall. It is understandable that the figure is a bit cramped due to severe space constraints in conference proceedings articles.\n\n- The benchmarking appears to be thorough, although it is not as important as the methodological innovation in my view.\n\n- The paper overall is organized well, and the description of the method is well-written, except for the part about hyper-connections. I found it about as straightforward to read as one could hope for a technical section.\n\n- Overall, I find this paper a good addition to the literature and impactful enough for this venue.\n\n## Weaknesses:\n\n- It could be argued that the impact of time dynamics on existing spatio-temporal networks is a bit overstated since one can easily modify these networks to condition on time, e.g. via a time dimension or time token, but I won’t harp on that here. It’s still beneficial to explore networks with stronger inductive biases towards controlling changes in spatial dynamics across time.\n\n- I didn’t find any further mention of hyper-connections in the appendix. I feel it would be beneficial to add some sort of discussion on hyper-connections either in the main article or appendix due to it being a lesser known tool with few citations. It’s not an innovation on the part of the authors, so I understand not wanting to waste space in the main article on someone else’s work, but I would’ve appreciated a brief technical explanation of what it precisely does rather than dig into the cited paper.\n\n- It would add impact to see the performance on a large scale dataset such as Liu et. al’s “LargeST: A Benchmark Dataset for Large-Scale Traffic Forecasting” or some other dataset, but the paper is not really claiming any scaling behavior. Perhaps the authors are constrained by GPU resources.\n\nQUESTIONS:\nMy score would stand even if these questions are answered barring some significant change in the manuscript.\n\n1. How would you assess this method's ability to scale beyond these small to medium sized datasets? If I understand correctly, these datasets have node-time features represented by scalars, so it would be interesting to see how these components interact on higher dimensional raw features.\n\n2. Related to question 1, how does scale impact the conservation constraint? I could imagine have orders of magnitude more nodes can create vanishing gradient issues due to small magnitudes in propagating flow tokens.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a meaningful delta of improvement in previously cited works in spatio-temporal pattern prediction. The authors motivate the work with examples of interesting prediction problems such as traffic flow, hydrological, and parking lot dynamics time-series prediction across several hundred to thousands of geographic node locations. In contrast to existing prediction architectures which either a) do not incorporate graph-like structure among geographically adjacent node pairings or b) incorporate graph-like structure primarily via graph convolution / spatial attention modules, the authors present FlowNet which builds upon works such as STGCN [1] by introducing flow tokens as the primary method of internal representation. The authors provide an alternative message passing scheme which obeys a proposed conservation law, yielding physically plausible information flows as opposed to more localized spatio-temporal correlations from recent observations. The authors also introduce an Adaptive Spatial Masking strategy, which consists of a learned neighborhood radius for which flow tokens are propagated motivated by the desire to allow the model to capture long-range spatial dependencies when necessary for the prediction task without implicit introducing inductive biasing towards close-proximity neighbors sharing similar statistical correlations. Based on experiments evaluating FlowNet against relevant baselines on PEMS04 [2], DeepBase [3], and SINPA [4] datasets, the authors demonstrate on-par or better performance with state-of-the-art methods for both short-term and long-term sequence prediction with proper statistical significance testing. The authors also give analysis for how computation is spread across shorter vs longer range spatial dependencies in order to solve short-window vs long-window time horizon prediction tasks, and demonstrate their method is able to adaptively vary the information flow spatially when useful to the model.\n\n[1] B. Yu, H. Yin, and Z. Zhu, “Spatio-temporal graph convolutional networks: A deep learning framework for traffic forecasting,” arXiv preprint arXiv:1709.04875, 2017.\n\n[2] S. Guo, Y. Lin, N. Feng, C. Song, and H. Wan, “Attention based spatial-temporal graph convolutional networks for traffic flow forecasting,” in Proceedings of the AAAI conference on artificial intelligence, vol. 33, no. 01, 2019, pp. 922–929.\n\n[3] P. Ghaneei and H. Moradkhani, “Deepbase: A deep learning-based daily baseflow dataset across the united states,” Scientific Data, vol. 12, no. 1, p. 25, 2025.\n\n[4] H. Zhang, Y. Xia, S. Zhong, K. Wang, Z. Tong, Q. Wen, R. Zimmermann, and Y. Liang, “Predicting carpark availability in singapore with cross-domain data: A new dataset and a data-driven approach,” in Proceedings of the Thirty-Third International Joint Conference on\nArtificial Intelligence, 2024, pp. 7554–7562.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n- Table 1 presents strong evidence that the method in question is on-par or better than state-of-the-art approaches, testing across multiple trials and ensuring statistical significance on relevant and sizeable datasets. \n- Figures 1 and 2 are very helpful and intuitive for understanding the motivations for and solution to achieve flow based modeling of spatio-temporal dependencies in FlowNet.\n- The approach applies a fresh message-passing approach to the task of long/short temporal sequence prediction where geographic neighborhoods give some correlation in state predictions.\n- The authors have given great attention to detail the reproduction steps from datasets used, hyperparameters, baselines, preprocessing, initialization, hardware used, etc. with a properly shared anonymous code link in the appendix with well-structured and clear Pytorch code.\n- Section 3 gives a clear mathematical foundation for the flow token approach and manipulation.\n\nWeaknesses\n- The ablations in Table 2 are useful as a reader, though the prose in section 4.3 does not shed much light on the near order of magnitude difference in allocation flow vs retained flow vs conservation law delta reductions in performance and why these are expected. The authors claim each of these components is useful in modeling spatio-temporal graph state and yet the ablations suggest retained flow and conservation law removal yield little impact to the final result. Further ablation may have also proven useful in understanding the Adaptive Spatial Masking, hyperconnection layers, etc. \n- The Efficiency Analysis Conducted in Section 4.4 is thorough and very useful. The authors recognize the computational complexity of the approach yields a more resource taxing algorithm than graph convolutional methods, though given the added resource usage one would expect more competitive results in Table 1 when comparing against methods such as STGCN which are quite a bit less taxing on time complexity.\n- While figures 4 and 5 give good intuition for qualitative differences in how the model is utilizing larger neighborhood information flow to influence predictions, it is still a bit hard to interpret how at the microscopic level a single geographic node’s time series predictions are improved via this approach as compared to methods which doesn’t allow for flow based internal representations. A form of analysis which could answer these questions via interesting exemplar sampling and comparison with graph convolution models might make the value of greater computational complexity clearer.\n\nQUESTIONS:\n- Many times throughout the paper statements that proximity-based similarity can be problematic are made, though as a reader it is not apparent in the data given Table 1 suggests models that make this assumption are competitive with FlowNet (occasionally tying). It would be helpful if the reader can see a concrete example in data of a particular small neighborhood’s time sequence for which flow modeling is particularly helpful.\n \n- It is still a bit unclear in Section 3.4 why certain layers are introduced and replaced as compared to baseline architectures with respect to the larger goals of this paper. Ablation results might be able to shed better light on the choice of certain architecture changes beyond just the plot comparing the number of experts in the appendix.\n \n- Section 2 is titled “84 2 Similarity vs. Intrinsic Flow: Which is better?”. The section might answer that question more strongly given a clear example in hypothetical or data where similarity alone cannot hold up.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}