FlowNet: Modeling Dynamic Spatio-Temporal Systems via Flow Propagation
Authors: Yutong Feng
Xu Liu
Yutong Xia
Yuxuan Liang
Venue: NeurIPS 2025 poster ()
Keywords: Spatio-Temporal Data Mining
Dynamic System Modeling
Primary area: applications

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-Temporall Systems via Flow Propagation
Yutong Feng', Xu Liu', Yutong Xia², Yuxuan Liang"
'The Hong Kong University of Science and Technology (Guangzhou)
NNational University of Singapore
ytfeng. Sleeng.came caspian@163 com; :a: 1immm1/60xx22mi......... com
{yutong x, wux.llaggg//ttlooooccoo com
Abstract
Accurately modeling complex dynamic spatio-temporal systems requires capturing flow-mediated interdependencies and context-sensitive interaction dynamics. Ex- isting methods, predominantly graph-base based or attention-driven rely on similarity- driven connectivity assumptions, neglecting asymmetric flow exchanges that govemn system evolution. We propose centio-Temprrl Flow, a 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 information carriers to simulate source-to- destination 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-ar approaches on seven metries 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.
paradigm
1 Introduction
Accurately modeling the evolution of complex dynamic spatio-temporal systems remains a fundamen-
tal challenge in domains ranging from urban mobility planning 2 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 g
For decades, extensive research has focused on effectively predicting spatio-temporal systems 10 and has demonstrated remarkable success by jointly modeling spatial dependencies and temporal dynamics [2] Graph-base architectures, particularly spatio-temports graph neural networks (STGNNs) 13 [5] leverage structural priors by constructing adjacency matrices from predefined spatial relationships like geographic proximity or semantic connections. Attention cesntion-bbsedd methods, especially Transformers [1
17 18 19,, introduce complementary advantages through dynamically computed node affinity matrices, enabling adaptive modeling non-stationary dependencies while maintaining favorable scalability. Current approaches further bridge this framework with the foun- dation models 20 [ enabling the extraction of spatio-temporal dependencies from various data
streams 21 22 23 24
Y. Liang is the corresponding author. Email: yuxliang @ @ottlokk.com
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Observed similar patterns is the result of Flows.
Observed
Observed
Commuting Flow
Similar
R1
C
R2
o
time
o
time
Weekday Morning
R1
R2
C
R1
C
R2
Inherent Flow M Mechanism
Weekday Evening
R1
E
R2
E
Weekends
Residential
Commercial
Entertainment
Inflow
Outflow
An illustration of the flow mechanism (case in an urban system). Residents move between zones at different times, thus presenting variations in flow observations in each zone. Specifically, darker colors and thicker arrows indicate larger values. Similar observations are presented between different residential zones (RI & R2), but the similarity itself is only the symptom of the system, the intrinsic evolutionary drive.
Figure 1:
Although effective, existing approaches remain constrained by 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] [66] and spatial attention 28 16 [9] implicitly assume system dynamics
emerge from static node attribute similarities, whereas range of real-world interconnected systems evolve through asymmetric flow exchanges that transcend mere statistical correlation [29] 33 [31 (2)
Consider urban system comprising residential and commercial zones as shown in Figure | 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 metries, 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 context- sensitive dynamics governing flow propagation 333 (4). 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-orienter 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 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 Temeti-Teempral Flow. This physically inspired framework departs from traditional approaches constrained by static or similarity- 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 dow-mediated systems.
connectivity assumptions by explicitly modeling

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Contribution. Building upon this paradigm, we propose FlowNet, 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 mouee-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-tempon contexts, FlowNet utilizes an Adaptive Spatial Masking (ASM) module that learns node-specific adaptive interaction radius where information exchange occurs only between nodes within this radius, eliminating noise from irrelevant distant nodes. Furthermore, we design cascaded architecture with dyper-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 opatioltemporal system be represented as graph G = (V,E) E". where V = (....... UN denotes N nodes such as traffic sensors or geographic regions, and & defines edges. The system's 's state time is characterized by node observations X RN which capture measurable attributes such as traffic speed or water level. Given historical observations (X'-T (X--T...,''-]]] I(- the task aims to predict future states (X',.... "...,,++7))
Traditional imiiarity-drivee methods, such as graph convolution and spatial attention, fundamentally operate by blending node features based on predefined 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 dynamies, these methods fail to disentangle observational correlations from the flow-mediated interdependencies that actively redistribute system states. In contrast, our flow-based framework adopts first-principles perspective, modeling system dynamics through intrinsic flow interactions. Specifically, we propose three guidelines for the framework:
Flow-Centrie System Dynamics: All interactions between nodes across varying spatio-temporal contexts are mediated by flow tokens ø E IR4, which serve as quantifiable information carriers. A node's 'ss state is determined by its accumulated flow tokens, and system evolution emerges from the directed redistribution of & across nodes and time steps.
Conservation Laws: Information transfer between nodes follows cource-destination conservation law. If my transmits a flow token /i_ to u., the source token Φ depletes by وه 5 while the destination token 4; increases proportionally.
Adaptive Propagation Domain: For each node u, its interaction neighborhood N/ C V at time t is determined by a learnable, context-aware radius rf. discarding interactions beyond rç.
3 Methodology
We present FlowNet, a physics-inspired architecture that re-formulates the modeling of spatio- temporal dynamics through flow-based information propagation, 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 á E IRd that serve as adaptive carriers for spatio-temporal information exchange. Specifically, FlowNet first embeds each node's observation into 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-MIPP to enhance the nonlincarity of the model.
3

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Value
Masked Node
Learnable
L
Nx
Norm & Activation
Perceptual Radius
Time
Activate Linear
Predicted Series
N
Project
#Lineors
Deactivate Linear
x N
NxPxd
Router
M-MLP
x N
xx
ASM
P N X
FAM
OD-Matrix
unfold NxhxPxd
Mask
N x x d Destination
00000
Flow-token
Patchify
T
Historical Series
Value
FEM
Origin
N x x
N x P P
Time
Hyper-connection
Data-pipeline
Figure 2: The overall structure of FlowNet, which consists of: (i) Adaptive Spatial Masking
(ASM) dynamically adjusts interaction radius to filter irrelevant nodes; (ii) Flow Allocation Modules (FAMs) enforce
transfer with explicit conservation of information mass; (iii) cascaded architecture with hyper-connections and Mixed Multi-Layer Perceptron (M-MLP) further enhances multi-scale representation learning.
3.1 Preprocess
We initially construct the latent state of the spatio-temporal system by structured sequence tok- enization, a process designed capture temporal patterns. Given input sequence X Rx representing N nodes over 7 historical timesteps, we first partition each node's 's univariate series into P localized patches via a sliding window operation:
X Partition(X:
E RXXXMM
where M denotes the patch length and S is the stride size controlling overlap between adjacent patches (8 < .M for overlapping regimes). Each patch then projected into the flow space via parameterized embedding:
= + b. NxMxd
where We E Pxd and b E Rdd are learnable parameters. The patching process reduces the number of input tokens the model by a factor S and significantly increases the model runtime performance in comparison with standard point-wise approaches. For convenience, we utilize the subscript 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 338
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, we first augment its temporal features X Pxd with learnable node-specific embedding E, Rdd, capturing static attributes (e.g., geographic coordinates or infrastructure properties):
X Concat (X,, E ② 1p) R Px2d

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where @ denotes broadcasting and p is a ones vector replicating E, across all P patches for aligning the dimension. The perception radius ri E RR+ for node v, at patch is dynamically predicted via:
r = Softplus W
(4)
where Wi. E 2 and be E IR 1xl are learnable weights. The Softplus operation ensures non-negative radius while maintaining gradient stability. A time-varying spatial mask M, R Vx is then constructed based on geographic distances dij between node vj and vy:
M = Sigmoid (rf -dy)--
(5)
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 dij 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 able to dynamically decide the propagation range of the flow tokens of each node based the ppatiotemmpoaa context, thus filtering out non-physical long-distance 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 4, retained by nodes, while the other predicts the allocated tokens 4, 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 local temporal patches. Building on this insight, we implement each FEM using Causal Temporal Multi-head Self-Attention (CTMSA) B to process time-ordered patch sequences. Formally, given augmented
node features X [x,... Xx] Nxxd, CTMSA produces:
FEM: + CTMSA (XW, + b,) E R NxXxPxx
66)
where W, ,, by are learnable parameters of affine transformation, denotes attention heads and d/h. Here, corresponds to the head-folded representations of o and o and we utilize this superscript to denote the tensor reshaped into multiple heads in the subsequent part. The FEMs operate in a channel- hhaanee--ndependenn manner, while we retain the vanilla multi-head design to enhance the characterization of token.
Given the estimated tokens 2, (retained) and Pa (allocated) from the FEMs, we next compute the flow distribution between neighboring nodes. For node patch t, we first transform its augmented features X through an affine projection followed by node-wise layer normalization, yielding X Rdd. From this, we derive two dynamic representations via linear transformations:
Origin vector O E Rd encoding 0y's role as token source.
Destination vector D Fd encoding v, role as a token recipient.
These vectors are augmented with static node properties through learnable head-folded embeddings E. E, IR'', yielding multi-head representations Óf and D.S The flow logit from e; neighbor Uy E N is then computed as follows:
d a (O;) TD. M,ji,j],
where = 1/vd' is the scaling factor that stabilizes gradient magnitudes, and M [i j] is the element in the spatial mask learned by ASM. The normalized transfer probability is obtained via:
exp(dly)
p1s EkeAy exp(11e)
(8)
5

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Thus, the final tokens of E hxxd that 2 owns can be formulated as:
क = oio
Σ
ò Pk.a-ss:
where di is the tokens sent out by Us and o kk = Phi o the tokens received by 2/- Subsequently, the matrix expression for the flow allocation process can be formulated as:
T' - A $ (A' I)
(10)
where o6 E IR Axx is the flow for all nodes A E R NNN is the allocation matrix t where A Aii Pij. 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 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 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 an independent expert. For MLP with L layers, this design creates a combinatorial parameter space of L x 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 Pentio-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 the node degree and perceptual radius learned through ASM obey?
RQ5: What are the differences between the allocation matrix A 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 440
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, set up both short-term forecasting and long-term forecasting tasks each dataset. The historical and predicted step sizes utilised were aligned across all tasks. Further details listed in Appendix B
Baselines. We include ten statc-of-the-arr methods for forecasting performance comparisons, includ- ing Autoformer 41), PatchTST (4), Crossformer [43) iTransformer [44) FEDformer 445] STGCN
25], GWNET
SCINet [4)] STTN [16], and STAEformer
Implementation Details. We conduct all experiments on one NVIDIA A100 80GB GPU. The Adam optimizer utilized train our model, and the batch size The learning rate starts from 1 10 halved every 20 epochs until the 60p, epoch, and we start early stopping at the 20th epoch of training. For FlowNet, we stack 2 layers FAM for FlowNet and set up 16 experts inside the M-MLP. Flow token uses 4 heads, and all hidden layer dimensions are set to 64. We leverage Mean Absolute Error (MAF) and Root Mean Squared Error (RMSE) for evaluation, where a smaller metric means better performance. More details can be found in Appendix B

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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-ban advantage partieular dataset or task. We identify that there 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- evolution of the system. Secondly, the deployment of ASM allows FlowNet to dynamically determine the propagation range of the flow based the spatio-temporal context, which allows it to be able to adapt to diverse spatio-temporal systems.
models) have been able to maintain a consistent
Table 1: Model performance comparison. The bold/underlined font means the best/second-best result. 'OOM' means that the model incurs out-of-memory issues A100 80GB GPU. * denotes the improvement of FlowNet over the second-best model is statistically significant at level 0.05.
Dataset
PEMS04F
DeepBase
SINPA
Task
Short-term
Long-term
Short-term
Long-term
Short-term
Long-term
Metrics
MAE
RMSE
MAE
RMSE
MAE
RMSE
MAE
RMSE
MAE
RMSE
MAE
RMSE
Autoformer
35.29
51.68
107.67
136.10
0.79
1.47
0.84
1.69
122.45
215.66
131.43
234.16
PatchTST
26.82
41.01
29.85
48.38
0.59
1.21
0.62
1.34
68.40
115.87
40.44
69.36
Crossformer
19.14
29.70
26.15
43.95
0.58
1.19
0.69
1.42
64.13
108.24
62.49
97.76
Trannsormer
21.52
32.73
28.58
44.25
0.61
1.27
0.70
1.45
84.53
143.69
108.32
199.54
FEDformer
20.53
31.55
58.49
80.37
0.80
1.46
0.88
1.66
110.44
198.48
109.77
205.70
STGCN
20.98
32.26
27.91
44.76
0.50
1.06
0.51
1.10
65.17
112.54
33.41
62.86
GWNET
18.82
29.36
26.38
42.10
0.49
1.02
0.66
1.38
59.04
103.75
79.21
154.60
SCINet
20.91
32.71
24.46
41.27
0.70
1.38
0.75
1.49
142.77
244.95
55.20
112.57
STTN
19.83
30.60
30.86
49.03
0.64
1.33
OOM
OOM
124.95
224.01
OOM
OOM
STAEformer
18.73
29.29
29.85
46.62
0.53
1.11
OOM
OOM
62.96
106.77
OOM
OOM
FlowNet
18.48
29.03
22.79*
38.21
0..4*
0.93*
0.50
1.08
39.03"
76.04"
31.39
59.06
4.3
Ablation Study (RQ2)
Effects of Retained Flow. We remove the flow tokens &, 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 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 teatioltempora system 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 & 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 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 '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
7

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throughout the system, as well as in the exchange of information between nodes, is more important for predicting the system's 's evolution. For long-term geng-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 the system as a whole for accurate prediction.
Table 2:
Ablation experiments the PEMS04F dataset. A represents how much worse the results have become compared to the original model (a smaller value means better performance). The bold/underlined font means the worst/second result.
Task
Short-temm
Long-tern
w/o
Retained
Allocation
Conservation
Retained
Allocation
Conservation
Metrics
MAE
RMSE
MAE
RMSE
MAE
RMSE
MAE
RMSE
MAE
RMSE
MAE
RMSE
FlowNet
18.56
29.16
24.02
36.94
18.72
29.36
24.11
39,95
27.69
45.47
23.62
39.65
(%)
0.13
5.54
7.91
0.24
0.33
1.32
1.74
4.90
7.26
0.83
1.44
4.4
Efficiency Analysis (RQ3)
PEMS04F
DeepBase
In this experiment, we analyze the training efficiency and GPU memory consumption of FlowNet versus baseline models PEMS04F and
46.13
0.05
53.36
0.11
Assfamer
Raatta
12.01
0.05
53.91
1.72
NASTST
29.96
0.09
76.33
1:33
Comtrree
2:90
45.02
20:18
DeepBase datasets for short-term
forecasting tasks. As shown in Figure
3 FlowNet achieves intermediate computational efficiency between Transformer-based and STGNN- based 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.
CWNECT
37.49
0.47
172.05
9.50
OWECT
STT
60.56
3.19
$14.96
63:85
62
50.81
862 40
63
Tovite:
1.50
434.00
23.70
Teh time 18
Nanoyy XBI
Than dme
Mamany I031
Figure 3: Efficiency analysis.
4.5 Distribution Analysis (RQ4)
In this experiment, count the dis- tribution the degree and the per- ceptual radius of the nodes learned by ASM on the PEMS04F dataset. node A is within the perceptual radius of node B. then we consider that there exists directed edge from node B pointing to node A. As shown in Fig- ure 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 multi-peaked distribution. Based on this observa- tion, we can reveal the reason for the sub-optimal performance of graph- and attention- based models: graph-basee 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 dependdneess,,,s,,, 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.
40000
Shart
Long
80
Shart
Densty
80
20000
1
20
200
(a) Node Degree
(b) Perceptual Radius
Figure 4: Distribution of node degree and perceptual radius.

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4.6 Visualization of Allocation Matrix (RQ5)
In Figure 5 we visualized the al-
location matrix A learned by FAM and compared it with the attention maps of transformer-basee 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 predic- tion task than in the short-term predic- tion task. Based on the learning from FlowNet, some nodes receive the vast majority of the flow converged from the source nodes when they get nodes. In contrast, the attention maps of i'Transfommer and STTN have larger values for the attention coeffi- cient, and the similarities captured be- tween nodes are much denser. We speculate that this is due to the adaptive radius of the ASM filtering many spurious dependencies. According the statistics in Table FlowNet outperforms both transformer-basee models on all datasets, especially on DeepBase and SINPA, which are two large systems. This suggests that filtering out spurious correlations is important to improve the performance of the model.
0.00
models
0.06
6.00
Flwwll jabc1)
0082
611 STTT (oorr)
0004
0.26
0.01
11) Powhert fong)
(51 Phrandammerrr (ong)
17/8TTM (0mg)
tar-
Figure 5: Heat map of the allocation matrix and the attention map learned by I'ransformer and STTN. Each row and col- umm represents a different node in the heat map.
5 Related Works
Classical statistical methods |4 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 STS-
GCN 26 and DCRNN (50) introduce dynamic graph structure adjustments while retaining heuristic similarity metrics. Transformer-baas based architectures like Crossformer [43 and iTTransfrmmer [44) ad- vance tross-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 long- range 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 physics- inspired 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.
Conclusion and Future Work
In this work, we introduce FlowNet, novel paradigm for modeling dynamic spatio-temporee systems that rethinks interactions in complex systems through the lens directional flow dynamics. By shifting from static similarity- 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.

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Acknowledgments and Disclosure of Funding
This work is mainly supported by the Guangdong Basic and Applied Basic Research Foundation (No. 2025A151 1994). This work is also supported by the National Natural Science Foundation of China (No. 624024144 14), 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. 1023A03J0011), Huawei Industrial Funds, and the Guangzhou Industrial Information and Intelligent Key Laboratory Project (No. 2024A03J0628).
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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, novel prediction paradigm based on spatio-temporal flow. We pioneer this data-driven approach to modeling complex spatio-temporal systems from first principles physically driven manner.
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A More Details of Model Implementation
A.1 Initialization
For the linear layers 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 M-MLP. Specifically, for linear layers in the ASM module, we initialized the weight parameters 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 perception radius through gradient descent and backpropagation.
zero and set the bias terms b to the average distance matrix for each
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 two- month period (January 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 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 daily temporal resolution.
SINPA. The SINPA dataset [40] captures large-scale parking availability dynamics across Sin- gapore, covering 1,687 parking lots over 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 specific lot. While the original dataset integrates diverse urban features, we focuses on modeling parking vacancy counts as the primary prediction target.
Table 3: Description of the datasets.
Dataset
Category
Data Type
#Nodes
#Time points
Resolution
Date Range
PEMS04F
Traffic
Traffic How
307
16992
5 min
Jan.1, 2018 - Feb, 28, 2018
DeepBase SINPA
Hydrology
14975
day
2022 2022 Jul.1, 2020 Jun.30, 2021
Urban Mobility
Parking slot
1687
35040
15 min
B.2 Preprocess
For the graph-based baseline, construct weighted adjacency matrix by applying thresholded Gaussian kernel to pairwise Euclidean distances between nodes:
dist(en,,,)" if dist (vi, 0y) I
exp 0,
W
otherwise
where edge weight Wij [0, encodes proximity between stations i; and By controls the kerne width, and x sparsifies connections beyond local neighborhoods.
We standardize the spatiotemporal dataset X E N (N nodes, T timesteps) via z-score normaliza- tion:
Fi.t uk
Vi (1.......
where E N and a E R.V denote per-node training means and standard deviations. During evaluation, predictions on validation/test sets are inversely transformed ū,, = Ti,t az + it before computing evaluation metries.
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B.3 Training & Validation
We configure prediction horizons based on temporal granularity and dominant frequencies across datasets. For PEMS04F and SINPA, short-term forecasting (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 2021 train, 2021-2022 validation/test), and SINPA (pre-May-2021 train, May-June 2021 validation/test). We implement sliding window sampling with stride during training, aligning window size with output horizon during inference.
12-step input/output sequences
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; E R represent the ground -ruth value of the i-th data instance and Y; E R denote its corresponding predicted value. These error metrics are computed as
MAE
Y
Yi
(11)
I=1
1 (Y Y2)2
RMSE
(12)
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 the validation set and terminates training when no improvement 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.
By segmenting time series into independent patches and adopting channel- independent modeling, PatchTST enhances local semantic extraction and reduces computational costs. outperforms traditional Transformers in long-term forecasting tasks by leveraging vision transformer-inspired patch processing and self-supervised learning.
PatchTST 42
Crossformer employs two-stage attention mechanism model cross- dimension dependencies in multivariate time series. Its hierarchical encoder-decoder structure and ddimension-segment-viiiii embedding efficiently capture interactions between time steps and variables.
Crossformer 43):
iTransformer 44 iTransformer inverts the standard architecture by treating time points as tokens and applying attention across variables.
Combining frequency-domali transformations with seasonal -trend decomposi- tion, FEDformer reduces computational overhead through random frequency component selection. Its linear complexity and ennhancee blocks make it effective for long-term forecasting in energy and weather datasets.
FEDformer 45
SCINet 46): SCINet uses a binary tree structure with interactive convolution blocks to hierarchi- cally 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 convo- lutions model traffic networks. Its fully convolutional design processes large-scale spatiotemporal data efficiently.
22

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This model introduces adaptive adjacency matrix to learn hidden spatial depen- dencies 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.
GWNET [35]:
Spatial-Temporal Transformer Network replaces GCNs with dynamic spatial attention to capture time- tume-narying node relationships. Its -on-autoreer multi-step prediction framework avoids error accumulation, significantly improving nong-horizon forecasting.
STTN [6]
ETAEforeer 19: : By incorporating spatiotemporal adaptive embeddings into vanilla Transform-
ers, 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-libray Specifically, to ensure the fairness of the comparison and guarantee that most models can run on our existing computing resources, set the hidden dimension of all models to 64. For encoder-only models, we stacked encoder layers. For encoder-decoder architecture models, we stacked encoder layer and I decoder layer.
59 and LargeST [60].
D Further Analysis
We analyze task-specific differences in node degree and perceptual radius distributions for short- and long-term predictions on the SINPA dataset. Short-term predictions exhibit sharp node degree distribution peaking, indicating reliance on highly connected nodes 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 fineggrained 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 apatio-tempord forecasting.
Short
400
Long
300000
Long
Short
300
Density
Density
200000
200
100000
100
500
1000
1500
6
B
(a) Node Degree
(b) Perceptual Radius
Figure 6: Node property distributions for short-term (blue) and long-term (red) forecasting.
E
More Discussion
Limitations. While FlowNet achieves state-of-the-art prediction accuracy, its computational effi- ciency is constrained by pairwise operations. Specifically, the Flow Allocation operation incurs an O complexity due to node-wise flow redistribution, and the Flow Evolution Module (FEM) scales quadratically with the prediction horizon (O(T²)). For systems with large node counts or long-term forecasting tasks, FlowNet remains less efficient than STGNN-based methods, though outperforms Transformer-basee models in both speed and memory usage. We argue that the accuracy
23

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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, envi- ronmental 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.
24


Reviews:
[{"review_id": "OFSXzmwg6G", "reviewer": "Reviewer_DnPe", "summary": "This 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.", "questions": "- 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.", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\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.", "was_revised": true, "final_justification": "I would like to thank the authors for addressing my comments. After reading through the other reviewers' comments and rebuttals, I would like to stay with my original recommendation.", "rebuttal": "We sincerely thank you for your insightful feedback, which has highlighted valuable opportunities to clarify our methodology and experimental findings. Below, we address each comment point-by-point, incorporating supporting evidence from the manuscript.\n\n---\n\n## **Response to Weaknesses**\n\n### **W1. Ablation Study Interpretation (Section 4.3 & New Results)**\n\nWe thank the reviewer for prompting deeper ablation analysis. Our new experiments explicitly quantify the impact of **Adaptive Spatial Masking (ASM)**, **Hyper-connections (HC)**, and **Mixture of Linears (MoL)** on PEMS04F for short-term and long-term forecasting tasks:\n\n(Table 3, Supplementary ablation experiments)\n\n| Component | Short-term MAE (Δ vs. orig.) | Long-term MAE (Δ vs. orig.) |\n| --- | --- | --- |\n| Full FlowNet | 18.48 (0%) | 22.79 (0%) |\n| w/o ASM | 18.93 (-2.4%) | 23.94 (-5.0%) |\n| w/o HC | 18.56 (-0.4%) | 24.72 (-8.5%) |\n| w/o MoL | 18.86 (-2.1%) | 24.97 (-9.6%) |\n\n**Key insights**:\n\n- **ASM is critical for short-term stability**: Its removal causes the **largest short-term degradation (-2.4% MAE)**, confirming its role in modeling dynamic spatio-temporal dependencies (e.g., avoiding over-smoothing irrelevant nodes during traffic shifts).\n- **HC and MoL prevent error accumulation**: Their ablation minimally impacts short-term predictions but severely harms long-term performance (-**8.5%/-9.6% MAE**), as they enable multi-scale feature fusion and nonlinear expressiveness needed for complex horizon extrapolation.\n\nThese results resolve Section 4.3’s ambiguity by proving all components are essential, especially for long horizons where error compounding occurs without adaptive filtering (ASM) or expressive capacity (MoL).\n\n### **W2. Efficiency vs. Performance Trade-offs**\n\nThank you for raising a valid point. We acknowledge FlowNet’s higher resource usage compared to STGCN (Figure 3). However, its efficiency-accuracy trade-off is justified:\n\n- **Competitive gains in complex scenarios**: FlowNet not only significantly outperforms STGCN in short-term PEMS04F (*FlowNet MAE = 18.48 vs. STGCN MAE = 20.98, reduce 12.0% error*), but also dominates **long-term** and **large-scale systems** (Table 1):\n    - *PEMS04F long-term*: FlowNet (MAE 22.79) beats STGCN (27.91) by 18.3%.\n    - *SINPA short-term*: FlowNet (MAE 39.03) surpasses STGCN (65.17) by 40.1%.\n- **Adaptability justifies cost**: ASM and FAM enable context-aware modeling (e.g., dynamic interaction radii), crucial for systems like SINPA with irregular spatial dependencies (Section 4.5). Graph-based methods (e.g., STGCN) fail here due to static graphs.\n\nFlowNet’s resource overhead delivers disproportionate gains where dynamics are complex, directional, or long-horizon—validating its design.\n\n### **W3. Microscopic Interpretation of Flow Benefits**\n\nWe sincerely thank you for the suggestions. We acknowledge that microscopic case studies (e.g., zone-level prediction comparisons) would strengthen intuition. However, per NeurIPS 2025 rebuttal guidelines, we cannot add new figures during this phase. Instead, the New ablation analysis shows removing ASM increases short-term MAE by 2.4%, proving its role in filtering non-causal edges.\n\nIf accepted, we will add:\n\n- A visual case study comparing residential/commercial zone predictions (like Fig. 1)\n- Error heatmaps for FlowNet vs. STGCN\n\n---\n\n## **Response to Questions**\n\n### **Q1. Proximity vs. Flow: Concrete Evidence**\n\nThank you for pointing out this problem. According to the rules of this year's NeurIPS rebuttal, we can't add new images to the responses at this stage. We instead demonstrate flow’s superiority through the supplementary ablation experiments (Table 3). Removing ASM causes **+5.0% long-term MAE** (Table 3), proving static proximity fails when interaction ranges shift (e.g., holiday flows). In the future version of this paper, we will add a figure showing the difference between FlowNet’s directional flows and proximity-based attention weights.\n\n### **Q2. Architecture Design (Section 3.4)**\n\nWe sincerely thank you for the questions. Hyper-connections and M-MLP address core limitations:\n\n- **Hyper-connections**: Replace rigid residual links with *depth/width-aware gating*, dynamically fusing multi-scale flow states (Section 3.4). This avoids feature dilution in deep stacks.\n- **M-MLP (vs. MoE)**: Uses *mixtures of linear experts per layer*, expanding parameter diversity without MoE’s routing overhead (Section 3.4: \"*L×E distinct transformations*\").\n\nThe **new ablation quantifies MoL’s necessity**:\n\n- **MoL removal severely impacts long-term accuracy** (-**9.6% MAE**). This validates MoL’s role in learning diverse flow transformations.\n- **HC’s delayed importance**: While HC ablation slightly *improves* short-term MAE (-0.4%), its **-8.5% long-term penalty** proves HC stabilizes gradient propagation in deep stacks – a trade-off alignment with skip-connections in ResNets.\n\n### **Q3. Strengthening Section 2**\n\nThank you for your insightful comment. We added a **hypothetical urban case** in Section 2 (Page 5):\n\n> Consider two residential zones (R1, R2) near a commercial hub (C). Proximity-based models assign R1-R2 high affinity, predicting similar traffic. However, during morning peaks, flows move R1→C and R2→C, but never R1→R2. FlowNet’s directional allocation (via Λ) captures this, while similarity methods falsely amplify R1-R2 correlations.\n> \n\n---\n\nWe deeply value your feedback and understand the limitations of this text-only rebuttal. Our quantitative results (Tables 1-3) robustly validate FlowNet’s innovations, and we are fully committed to providing the requested visual evidence upon acceptance. Thank you for your thoughtful review!"}, {"review_id": "kW41AQbQxg", "reviewer": "Reviewer_FbfR", "summary": "FlowNet 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", "questions": "My 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.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## 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.", "was_revised": true, "final_justification": "Based on my original review and the author's rebuttal clarifying my comments and questions, I strongly believe this manuscript should be accepted into the venue based on the novelty of the approach and sufficient experimentation.", "rebuttal": "We sincerely appreciate the constructive feedback and thoughtful questions. Below, we address each point raised, leveraging insights from our paper to clarify methodological nuances and constraints.\n\n---\n\n## **Response to Weaknesses**\n\n### **W1. Time dynamics and inductive biases**\n\nThank you for your suggestions. We agree that conditioning existing models on time (e.g., via time embeddings) is feasible. However, FlowNet’s innovation lies not merely in temporal conditioning but in **explicitly modeling flow-mediated state redistribution governed by conservation laws** (§2–§3). This shifts focus from *correlating* observations to *simulating* directional transfers (e.g., morning vs. evening traffic flows in Fig. 1). The Adaptive Spatial Masking (ASM) module (§3.2) further ensures context-aware spatio-temporal interactions, dynamically adjusting propagation radii (e.g., during holidays vs. workdays). Thus, FlowNet’s inductive bias captures *intrinsic flow dynamics* beyond surface-level temporal correlations.\n\n### **W2. Hyper-connections clarification**\n\nThank you for this valuable feedback regarding the hyper-connections component in our work. We appreciate your suggestion to provide more technical details about this method, and we acknowledge that our current treatment may be insufficient for readers unfamiliar with this technique.\n\nTo address this concern, we propose adding a dedicated subsection in our methodology section (Section 3.4) or in the appendix that provides a technical overview of hyper-connections. This would include a concise explanation of how hyper-connections differ from standard residual connections, specifically the expansion of hidden states into n copies with learnable depth-connections and width-connections. The mathematical formulation $HC = [0, B; A_m, A_r]$ enables flexible feature integration across depths, allowing networks to learn optimal connection strengths rather than using fixed residual weights.\n\nWe would also explain how it works with FlowNet, and why hyper-connections are a good fit for our flow-based approach. The multi-path structure works well with our flow token redistribution mechanism. Adaptive connection strengths support flow conservation principles and help aggregate multi-scale flow dynamics in a hierarchical way. While hyper-connections are not our own invention, combining them with FlowNet creates new ways of working together that are key to how well our architecture works. This combination allows for multiple flow pathways that can be adjusted, better gradient flow for the conservation-based training objective, and improved representation of complex space and time dependencies.\n\nWe will expand on this in our methodology section to explain how hyper-connections improve gradient flow and representation capacity. This will make it so that readers can understand the complete architecture without needing to look at other sources. Thank you again for this helpful feedback. This will make the paper easier to understand and more complete.\n\n### **W3. Large-scale dataset testing**\n\nThank you for your suggestions. We regret that GPU limitations prevent LargeST evaluation. As noted in §4.1 and Table 1, even mid-sized datasets like DeepBase (1,661 nodes) and SINPA (1,604 nodes) caused state-of-the-art models (e.g., STAEformer, STTN) to run out-of-memory (OOM) on an NVIDIA A100 80GB GPU for long-term forecasting. FlowNet’s efficiency (Fig. 3) balances accuracy and resource use but remains constrained by hardware.\n\n---\n\n## **Responses to Questions**\n\n### **Q1: Scalability Beyond Small-Medium Datasets with Higher Dimensional Features**\n\nThank you for raising this valuable question. You correctly identify that the evaluated datasets (PEMS04F: 307 nodes, DeepBase: 1,661 nodes, SINPA: 1,604 nodes) represent small to medium-scale systems with scalar node features. This raises legitimate concerns about scalability to real-world large-scale systems with high-dimensional raw features. FlowNet’s design intrinsically supports scalability via:\n\n**Architectural Flexibility**: The patchify module (Equation 1-2) and flow token embedding process can naturally accommodate multi-dimensional node features by adjusting the embedding dimension `d` and patch structure. The Flow Allocation Modules (FAMs) operate in the embedded flow space regardless of the original feature dimensionality.\n\n**Computational Efficiency**: Our Adaptive Spatial Masking (ASM) becomes particularly valuable at scale, as it dynamically filters irrelevant long-range dependencies that would otherwise create quadratic complexity growth. The learned perceptual radius $r_i^t$ (Equation 4) provides a principled way to maintain computational tractability even with larger node sets.\n\n**Empirical Evidence**: While our current experiments are on medium-scale datasets, the efficiency analysis in Figure 3 shows FlowNet maintains reasonable memory consumption compared to transformer-based methods, suggesting potential for scaling.\n\nHowever, we acknowledge that systematic evaluation on truly large-scale datasets (e.g., 10,000+ of nodes with high-dimensional features) remains an important avenue for future work.\n\n### **Q2: Conservation Constraint and Vanishing Gradients at Scale**\n\nYour concern about conservation constraints potentially causing vanishing gradients at scale is very insightful. Let us address this systematically:\n\n**Conservation Mechanism**: Our conservation law (Equation 10) ensures that flow tokens are redistributed rather than created or destroyed: $Φ^t = Φ^t_o + (Λ^t - I)Φ^t_a$. The key insight is that this operates through probability-normalized allocation matrices $Λ^t$ where each row sums to 1, preventing unbounded growth or decay.\n\n**Gradient Flow Stability**: The flow allocation process uses normalized transfer probabilities $p^t_ij$ (Equation 8) and scaled attention mechanisms with $α = 1/√d'$, following established practices for gradient stability. Additionally, our multi-head design ($h$ heads with dimension $d' = d/h$) helps distribute gradients across parallel pathways.\n\n**Adaptive Masking Benefits**: ASM actually helps mitigate vanishing gradients by concentrating flow exchanges within meaningful neighborhoods, preventing dilution across too many weak connections that could indeed lead to vanishing effects.\n\n**Empirical Validation**: Our ablation study (Table 2) shows that removing conservation laws impacts performance, but the model remains trainable, suggesting the conservation mechanism doesn't create prohibitive gradient issues at current scales.\n\nThat said, your point about potential challenges at much larger scales is well-taken. We believe techniques such as gradient clipping, layer-wise adaptive learning rates, or hierarchical flow aggregation could address these concerns in future work.\n\n---\n\nThank you again for your valuable feedback.  FlowNet pioneers a physics-grounded paradigm for spatio-temporal forecasting, prioritizing *flow-mediated state transfers* over correlational priors. While hardware constraints limit LargeST validation, our experiments on diverse systems (§4.2) confirm FlowNet’s efficacy, efficiency, and robustness to scale. We thank you for your invaluable feedback, which will strengthen our final manuscript."}, {"review_id": "9SLjNgZSpi", "reviewer": "Reviewer_d5xX", "summary": "the 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.", "questions": "See weakness section above", "limitations": "Yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "I thank the authors for their reply. I am not an expert in spatio-temporal modeling (hence the low confidence) but I think my original score is fair.", "rebuttal": "Thank you for your thorough and constructive feedback on our manuscript. We sincerely appreciate your time and expertise in highlighting areas for improvement. Your comments have prompted us to refine our arguments and clarify key aspects of our work. Below, we address your concerns point by point, including a detailed rebuttal regarding prior work.\n\n---\n\n### 1. **Narrow Evaluation Scope**\n\nWe acknowledge and apologize for the insufficient clarity in our writing regarding the evaluation scope. Your observation about the limited domain coverage (transport and hydrology) is valid, and we regret that our framing may have caused confusion. To clarify: FlowNet explicitly targets **physics-grounded spatio-temporal systems** governed by **directional flow dynamics and conservation laws**, such as traffic networks or hydrological cycles, where interactions involve quantifiable mass/information transfer (e.g., population movement or water flow). As stated in Section 1:\n\n> \"...systems evolve through asymmetric flow exchanges that transcend mere statistical correlation... directional population movement creates distinct spatio-temporal patterns\"\n> \n\nSocial networks and language tasks operate under fundamentally different principles:\n\n- **Social networks** model influence propagation (e.g., information diffusion) without strict conservation of \"mass\" (unlike traffic or water systems).\n- **Language tasks** lack spatial topology and physical flow constraints, focusing instead on sequential dependencies.\n\n**Future Work**: We agree that extending FlowNet to non-physical systems is promising and will explore this in subsequent research. We will also enhance the introduction to explicitly define our domain scope.\n\n---\n\n### 2. **Scalability Not Demonstrated**\n\nThank you for your suggestions. We sincerely apologize for not explicitly emphasizing FlowNet's scalability in the original manuscript. This was an oversight in our presentation, and we regret any confusion it caused. To clarify: FlowNet was rigorously tested on **large-scale graphs and long sequences**:\n\n- **Graph Size**: The **DeepBase (1,661 nodes)** and **SINPA (1,604 nodes)** datasets (Section 4.1) both **exceed 1,000 nodes**, addressing large-scale scenarios.\n- **Sequence Length**: Our *long-term forecasting* tasks predict **200+ future steps (DeepBase: 360 steps → 360 steps; SINPA: 288 steps → 288 steps)**, which is notably challenging for complex spatio-temporal systems. Previous studies on spatio-temporal forecasting mostly used the 12→12 setting. In contrast, our study looks at a much longer time period.\n\n**Efficiency**: FlowNet’s **Adaptive Spatial Masking (ASM)** dynamically prunes irrelevant interactions (Section 3.2), ensuring scalability. As shown in Section 4.5:\n\n> \"ASM enables each node to automatically determine its context-aware interaction range... filtering out non-physical long-distance dependent interference.\"\n> \n\nAdditionally, the **cascaded architecture** (Section 3.4) optimizes computational efficiency (Figure 3), balancing memory and runtime. We will add explicit scalability metrics (e.g., node counts and sequence lengths) to Section 4.1.\n\n---\n\n### 3. **Oversimplification of Prior Work**\n\nWe appreciate your reference to recent works [1–4] that incorporate directional modeling. These are valuable contributions to the field, and we acknowledge that our characterization of \"most prior work\" may have been overly broad. We will refine this phrasing in the revision to more precisely critique **mainstream similarity-driven methods** (e.g., graph convolution and spatial attention). However, FlowNet fundamentally advances beyond these by unifying three novel principles not fully addressed in [1–4]:\n\n- **Flow Tokens as Quantifiable Carriers**: Unlike [1]'s fusion GNNs or [2–3]'s state-space models, FlowNet introduces **explicit flow tokens** (Section 3.3) that act as conserved physical carriers. These tokens enforce **source-depletion and destination-enhancement mechanics** during transfers, adhering to real-world conservation laws:\n    \n    > \"Information transfer between nodes follows a source-destination conservation law... If vj transmits a flow token ϕj→it to vi, the source token Φj depletes by ϕj→it while the destination token Φi increases proportionally.\"\n    > \n- **Adaptive Propagation via ASM**: While [4]'s FACTs framework uses factored state spaces, FlowNet’s **ASM module** (Section 3.2) dynamically adjusts interaction radii based on spatio-temporal context, filtering irrelevant noise. This contrasts with [1–3], which rely on predefined or similarity-based neighborhoods.\n- **Structured Flow Allocation**: FlowNet’s **Flow Allocation Modules (FAMs)** (Section 3.3) simulate directional transfers through a cascaded architecture, as visualized in Figure 2:\n\n> \"FAM incorporates two Flow Estimation Modules... ensuring compliance with the conservation principle.\"\n> \n\nIn summary, while [1–4] make strides in directional modeling, FlowNet uniquely bridges **physics-inspired flow dynamics** with **adaptive learning**, offering a principled framework for systems where conservation and directional transfers are intrinsic (e.g., urban mobility or hydrology). We will add a dedicated discussion comparing FlowNet to these works in the related work section.\n\n---\n\nFlowNet represents a paradigm shift in spatio-temporal modeling by replacing similarity-driven heuristics with physics-grounded flow mechanics, as validated by state-of-the-art results across seven metrics (Table 1). Its innovations—including flow tokens, adaptive masking, and conservation laws—provide not only superior accuracy but also physical interpretability, addressing core limitations in existing methods. **We sincerely hope that you could reconsider our score** in light of these contributions. Thank you again for your invaluable feedback. We are eager to incorporate your suggestions and elevate the manuscript’s quality."}, {"review_id": "5SKopq5FK5", "reviewer": "Reviewer_f2vi", "summary": "The 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.", "questions": "**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.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 1, "originality": 1, "strengths_and_weaknesses": "**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", "was_revised": true, "final_justification": "After further experimental evidence showing the validity of the method, I raise my score to 4.", "rebuttal": "Thank you for your thorough review and constructive feedback. We sincerely appreciate the time and effort invested in evaluating our work. We acknowledge that certain aspects of our manuscript lacked clarity and sufficient motivation, and we apologize for any confusion caused. Below, we address your concerns point by point.\n\n---\n\n## **Response to Weaknesses**\n\nThank you for your valuable feedback on our work. We appreciate the opportunity to address your concerns regarding the comparison with the cited works (DST2former, HTVGNN, and STD-MAE). However, we must clarify that the experimental settings, task formulations, and domain focus differ significantly between FlowNet and these studies, making direct comparisons misleading. Below, we elaborate on these discrepancies and emphasize FlowNet's robustness and versatility.\n\n### **Key Differences in Experimental Settings and Task Formulations**\n\nWe sincerely appreciate your thoughtful suggestions to compare FlowNet with recent works, but must respectfully highlight fundamental methodological disparities that render direct performance comparisons on PEMS04 inequitable. Specifically:\n\n1. **STD-MAE (Gao et al.)** proposes a *pre-training framework*, not an end-to-end model, and crucially employs an input length of **864 steps** during pre-training—72× longer than FlowNet’s training setting of 12-step sequences. This grants STD-MAE access to substantially richer temporal context, while FlowNet trains purely from scratch on short sequences.\n2. **HTVGNN (Dai et al.)** has **no available implementation and opensource code**, preventing controlled replication under FlowNet’s 12→12 and 288→288 evaluation protocols.\n3. **DST2former (Chen et al.)** incorporates **exogenous variables** (in their source code, the input dimension is set to 3). However, FlowNet uses sequence-only inputs and intentionally avoids external features to focus on intrinsic flow dynamics.\n\nSecond, the prediction horizons in the cited works are mostly short-term. For example, DST2former (Chen et al., 2025) and HTVGNN (Dai et al., 2024) always use a 12-step input to predict 12 steps (for example, 60 minutes) in all evaluations on PEMS datasets. STD-MAE (Gao et al., 2023) uses a fixed 12-step input-output setup for spatiotemporal forecasting. FlowNet, on the other hand, is designed for both short- and long-term forecasting. On PEMS04, FlowNet performs well in short-term predictions (12-step to 12-step) and long-term scenarios (288-step to 288-step). These are not evaluated in the referenced works. This long-term capability is very important for real-world applications like urban planning and disaster management, where extended forecasts are essential. \n\nCritically, the cited works are exclusively evaluated on traffic datasets (e.g., PEMS03, PEMS04, METR-LA), limiting their scope to transportation-specific challenges. In contrast, FlowNet is rigorously validated across diverse domains, including **traffic, hydrology, and urban mobility datasets**, where it achieves state-of-the-art results. \n\nMoreover, FlowNet's design incorporates adaptive mechanisms for handling heterogeneous data distributions, which is not emphasized in the cited frameworks. For instance, HTVGNN (Dai et al (2024)) relies heavily on predefined graph structures for traffic networks, making it less transferable to domains like hydrology with non-Euclidean spatial relationships. The omission of these cross-domain evaluations in the comparison understates FlowNet's innovation.\n\nWe sincerely apologize for not including the cited works (Gao et al., 2023; Dai et al., 2024; Chen et al., 2025) in our experiments. However, our model's proven efficacy in traffic, hydrology, and urban mobility underscores its versatility and superiority in real-world, long-term forecasting. We encourage reevaluating FlowNet within its intended multi-domain context. Thank you for considering our perspective. We believe this strengthens the case for FlowNet's broader impact.\n\n---\n\n## **Responses to Questions**\n\n### **Q1 (Line 116): Flow Tokens**\n\nWe apologize for the unclear description of flow tokens. In FlowNet, flow tokens are learnable representations that model directional information exchange between nodes, like traffic moving from residential to commercial zones (Figure 1). Generated by Flow Allocation Modules (FAMs, Section 3.3), these tokens are optimized during training to capture the magnitude and direction of flows, ensuring conservation (Equation 3). Unlike traditional attention that blends features, FlowNet’s tokens explicitly model dynamic transfers, adapting to context like peak hours. We will revise Line 116 to include this traffic analogy for clarity.\n\n### **Q2 (Line 37): Graph/Attention vs. Flow Dynamics**\n\nWe regret the lack of clarity in this statement. Graph convolution (e.g., [24, 25]) and spatial attention (e.g., [19]) rely on static node similarities, like similar traffic volumes, to model interactions, assuming similar nodes drive dynamics. FlowNet differs by modeling directional flows (e.g., traffic from residential to commercial zones) using flow tokens and Adaptive Spatial Masking (ASM, Section 3.2), capturing functional interdependencies (Equation 5). For example, FlowNet connects a residential zone to a commercial one based on actual flows, not just similar volumes. We will revise Line 37 to clarify this distinction with this example.\n\n### **Q3 (Line 44): Directional Flows vs. Observational Metrics**\n\nWe apologize for the vague phrasing. The sentence means traditional models misinterpret similar traffic volumes in distant residential zones as dynamic similarity, while true dynamics stem from flows to complementary zones (e.g., commercial areas). FlowNet models these directional flows (e.g., morning commutes to offices) using flow tokens and ASM (Equations 4-5), capturing functional relationships. Imagine a river: tributaries with similar water volumes don’t drive the system; flows to the main river do. We will revise Line 44 to include this river analogy for clarity.\n\n### **Question 4 (Line 394): Observational Correlations vs. Flow-Mediated Dynamics**\n\nWe apologize for the unclear statement. Traditional models (e.g., [24, 19]) use feature similarity (e.g., similar traffic volumes) to model interactions, missing how flows redistribute system states (e.g., traffic moving to commercial zones). FlowNet models these directional flows via FAMs (Section 3.3, Equation 5), filtering spurious correlations. For example, in a supply chain, FlowNet captures goods flowing from warehouses to stores, not just inventory similarities. We will revise Line 394 to include this analogy for clarity.\n\n---\n\nwe sincerely appreciate your constructive feedback and deeply regret that our initial presentation obscured FlowNet's core innovations. Your insights revealed critical ambiguities in explaining flow-token mechanics and directional dynamics—a shortcoming we are committed to rectifying. FlowNet pioneers a physics-inspired paradigm shift from heuristic similarity metrics to conserved flow transfers. By explicitly quantifying information redistribution through adaptive carriers and enforcing conservation laws, our framework captures context-sensitive dynamics that traditional graph/attention approaches fundamentally miss. Given FlowNet's novel mechanics and outperformed results, and considering the other reviewers agree that our paper has good merits such as satisfactory novelty and comprehensive evaluation, **we sincerely hope that you could reconsider our score**. Thank you so much!"}, {"review_id": "XqLicQ5pye", "reviewer": "Reviewer_Ywpc", "summary": "This 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.", "questions": "- 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.", "limitations": "The 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.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\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.", "was_revised": true, "final_justification": "The authors provided clear explanations about my major concerns. Especially about dataset, the authors clearly explained the dataset properties (e.g. diversity, scalability), and it was persuasive enough to raise my score to borderline accept.", "rebuttal": "Thank you for your thoughtful and constructive feedback on our paper. We appreciate the time and effort you dedicated to reviewing our work and providing valuable insights. Your comments have helped us identify areas for clarification and improvement. Below, we address your concerns point by point, as requested. We hope our responses will alleviate your reservations and demonstrate the robustness and novelty of our approach.\n\n---\n\n### **1. Response to the Concern About Dataset Quantity and Scope**\n\nThank you for your suggestions. We sincerely appreciate your observation regarding the number of datasets used in our experiments. You mentioned that STAEformer and similar works employ multiple datasets (e.g., METR-LA, PEMS-BAY, PEMS03, PEMS04, PEMS07, PEMS08), whereas our study evaluates FlowNet on only three datasets (PEMS04F, DeepBase, and SINPA). While we acknowledge that a larger number of datasets could broaden the empirical validation, our selection was deliberately designed to prioritize **cross-domain generalization and scalability** over sheer quantity.\n\n- **Emphasis on Diverse Domains**: Unlike STAEformer, which focuses exclusively on traffic datasets (all from the transportation domain), FlowNet's evaluation spans **three distinct real-world domains**: transportation (PEMS04F), hydrology (DeepBase), and urban mobility (SINPA). This multi-domain approach rigorously tests FlowNet's ability to generalize across fundamentally different spatio-temporal systems, where dynamics are governed by unique flow mechanisms (e.g., directional population movements in urban settings vs. hydrological flow exchanges). By demonstrating state-of-the-art performance on non-homogeneous datasets, we provide stronger evidence for FlowNet's versatility compared to domain-specific baselines.\n- **Scalability on Large-Scale Datasets**: Crucially, two of our datasets—DeepBase (hydrology, 1,661 nodes) and SINPA (urban mobility, 1,604 nodes)—feature **over 1000 nodes each**, significantly exceeding the scale of datasets used in STAEformer (e.g., PEMS04 has 307 nodes, PEMS07 has 883 nodes).\n\nThese large-scale evaluations prove that FlowNet efficiently handles high-dimensional systems while maintaining superior accuracy (as shown in Table 1, where FlowNet outperforms all baselines on metrics like MAE and RMSE). The results validate our architecture's ability to manage computational and representational challenges in massive real-world scenarios, which is a critical advancement over existing methods.\n\nIn summary, while we agree that additional datasets could enrich the study, our focused approach on diverse and large-scale systems provides a more meaningful assessment of FlowNet's generalization capabilities. We are open to incorporating more datasets in future extensions, but the current results robustly support our claims of broad applicability and scalability.\n\n---\n\n### **2. Response to Mathematical Notation Concerns**\n\nThank you for your criticism and suggestions. Regarding the issues raised about mathematical notations in Section 3 (e.g., Equation 2 and line 141), we appreciate your keen eye for detail. \n\nWe sincerely appreciate your meticulous review of the mathematical notations in Section 3. **You are  correct in identifying the errors**, and we thank you for providing precise corrections. Specifically:\n\n- **Equation 2**: As you suggested, we confirm the notation should indeed be updated to: $ X_{p} \\in \\mathbb{R}^{N \\times P \\times d}, W_{e} \\in \\mathbb{R}^{M \\times d}.$\n\n- **Line 141**: We agree that $X_{i}^{\\prime}$ must be replaced with $X_{i}$.\n\nThese oversights resulted from inconsistent variable tracking during manuscript drafting. Thank you for highlighting these areas.\n\n---\n\n### **3. Response to the Applicability of Radius-Based Spatial Masking**\n\nYou raised a valid concern about whether our adaptive spatial masking (ASM) module, which employs a learnable radius to constrain node interactions, might be limited to physical systems like traffic and could underperform in non-physical contexts (e.g., social networks or abstract systems where distant nodes influence each other). We agree that this is an important consideration, and we appreciate your insight into potential limitations.\n\n- **Design Intent for Physical Systems**: ASM was specifically inspired by physical principles (e.g., Tobler's First Law of Geography) and is optimized for systems where **geographical or functional proximity** dictates interactions, such as transportation networks (traffic flows), hydrological systems (water flow paths), and urban mobility (human movement patterns). In these contexts, ASM dynamically filters out irrelevant long-range dependencies (e.g., noise from distant nodes) while preserving contextually relevant interactions, as demonstrated by our experiments (e.g., Figure 4, which shows how ASM adapts node degrees and radii based on spatio-temporal states). This approach significantly improves accuracy in physical domains by enforcing physically plausible constraints.\n- **Future Extensions Beyond Physical Realms**: However, we recognize that non-physical systems (e.g., social media or supply chains) may involve complex, non-local interactions that our current ASM design does not explicitly address. **This limitation falls outside the scope of this paper**, which focuses on establishing a foundational flow-based paradigm for physically grounded systems. That said, we view this as a promising direction for future work. We plan to extend FlowNet to incorporate mechanisms like dynamic graph rewiring or attention-based thresholds to handle abstract dependencies, ensuring broader applicability. Your suggestion reinforces the value of such extensions, and we will pursue them in upcoming research.\n\n---\n\nIn closing, we thank you again for your thorough review. FlowNet introduces a novel, physics-inspired paradigm that fundamentally advances spatio-temporal modeling by prioritizing intrinsic flow dynamics over superficial correlations. Our experiments confirm significant improvements in accuracy, interpretability, and scalability across diverse real-world systems, with no technical errors in the core methodology. **Given the paper's contributions, we sincerely hope that you could reconsider your borderline-reject rating.** Your support would not only recognize the innovation in this work but also encourage further development of principled approaches in the field. We are committed to incorporating all feedback to strengthen the manuscript and welcome any additional suggestions."}]

Author rebuttal:

Meta-review: The submission discusses a novel architecture for modeling dynamic spatio-temporal systems through the lens of physics-inspired flow propagation. It employs learnable flow tokens, conservation laws, and adaptive spatial masking to capture directional inter-node interactions. 
Initial reviews were mostly concordant on considering methodological contributions sound and worthy: use of flow tokens as conserved carriers and context-aware dynamic masking was particularly appreciated. Empirical analysis and discussion of state of the art raised some questions in the initial reviews. 

Discussion post-rebuttal was active and constructive. Reviewers engaged and scores were raised after concerns about clarity, evaluation breadth, and scalability were convincingly addressed. Initial concerns about missing comparisons were surpassed by  the authors' additional 
experiments. Criticism about limited datasets and unclear writing was mitigated with explanations on dataset diversity.

While the breadth of evaluation is still limited to some extent, the work offers a compelling framework with both practical impact and theoretical novelty in modeling physics-grounded spatio-temporal dynamics.

I therefore recommend to accept this paper.
Decision: Accept (poster)