FlowNet: Modeling Dynamic Spatio-Temporal Systems via Flow Propagation

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.

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1. Response to the Concern About Dataset Quantity and Scope

Thank 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.

• 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.
• 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).

These 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.

In 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.

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2. Response to Mathematical Notation Concerns

Thank 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.

We 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:

• Equation 2: As you suggested, we confirm the notation should indeed be updated to: $X_{p} \in \mathbb{R}^{N \times P \times d}, W_{e} \in \mathbb{R}^{M \times d}.$

• Line 141: We agree that $X_{i}^{\prime}$ must be replaced with $X_{i}$.

These oversights resulted from inconsistent variable tracking during manuscript drafting. Thank you for highlighting these areas.

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3. Response to the Applicability of Radius-Based Spatial Masking

You 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.

• 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.
• 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.

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In 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.

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.

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Response to Weaknesses

Thank 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.

Key Differences in Experimental Settings and Task Formulations

We 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:

1. 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.
2. HTVGNN (Dai et al.) has no available implementation and opensource code, preventing controlled replication under FlowNet’s 12→12 and 288→288 evaluation protocols.
3. 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.

Second, 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.

Critically, 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.

Moreover, 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.

We 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.

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Responses to Questions

Q1 (Line 116): Flow Tokens

We 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.

Q2 (Line 37): Graph/Attention vs. Flow Dynamics

We 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.

Q3 (Line 44): Directional Flows vs. Observational Metrics

We 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.

Question 4 (Line 394): Observational Correlations vs. Flow-Mediated Dynamics

We 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.

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we 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!

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.

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1. Narrow Evaluation Scope

We 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:

"...systems evolve through asymmetric flow exchanges that transcend mere statistical correlation... directional population movement creates distinct spatio-temporal patterns"

Social networks and language tasks operate under fundamentally different principles:

• Social networks model influence propagation (e.g., information diffusion) without strict conservation of "mass" (unlike traffic or water systems).
• Language tasks lack spatial topology and physical flow constraints, focusing instead on sequential dependencies.

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.

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2. Scalability Not Demonstrated

Thank 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:

• 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.
• 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.

Efficiency: FlowNet’s Adaptive Spatial Masking (ASM) dynamically prunes irrelevant interactions (Section 3.2), ensuring scalability. As shown in Section 4.5:

"ASM enables each node to automatically determine its context-aware interaction range... filtering out non-physical long-distance dependent interference."

Additionally, 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.

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3. Oversimplification of Prior Work

We 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]:

• 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:

  "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."

• 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.

• Structured Flow Allocation: FlowNet’s Flow Allocation Modules (FAMs) (Section 3.3) simulate directional transfers through a cascaded architecture, as visualized in Figure 2:

"FAM incorporates two Flow Estimation Modules... ensuring compliance with the conservation principle."

In 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.

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FlowNet 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.

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.

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Response to Weaknesses

W1. Time dynamics and inductive biases

Thank 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.

W2. Hyper-connections clarification

Thank 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.

To 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.

We 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.

We 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.

W3. Large-scale dataset testing

Thank 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.

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Responses to Questions

Q1: Scalability Beyond Small-Medium Datasets with Higher Dimensional Features

Thank 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:

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.

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.

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.

However, 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.

Q2: Conservation Constraint and Vanishing Gradients at Scale

Your concern about conservation constraints potentially causing vanishing gradients at scale is very insightful. Let us address this systematically:

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.

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.

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.

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.

That 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.

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Thank 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.

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.

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Response to Weaknesses

W1. Ablation Study Interpretation (Section 4.3 & New Results)

We 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:

(Table 3, Supplementary ablation experiments)

Key insights:

• 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).
• 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.

These 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).

W2. Efficiency vs. Performance Trade-offs

Thank 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:

• 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):
  • PEMS04F long-term: FlowNet (MAE 22.79) beats STGCN (27.91) by 18.3%.
  • SINPA short-term: FlowNet (MAE 39.03) surpasses STGCN (65.17) by 40.1%.
• 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.

FlowNet’s resource overhead delivers disproportionate gains where dynamics are complex, directional, or long-horizon—validating its design.

W3. Microscopic Interpretation of Flow Benefits

We 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.

If accepted, we will add:

• A visual case study comparing residential/commercial zone predictions (like Fig. 1)
• Error heatmaps for FlowNet vs. STGCN

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Response to Questions

Q1. Proximity vs. Flow: Concrete Evidence

Thank 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.

Q2. Architecture Design (Section 3.4)

We sincerely thank you for the questions. Hyper-connections and M-MLP address core limitations:

• 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.
• 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").

The new ablation quantifies MoL’s necessity:

• MoL removal severely impacts long-term accuracy (-9.6% MAE). This validates MoL’s role in learning diverse flow transformations.
• 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.

Q3. Strengthening Section 2

Thank you for your insightful comment. We added a hypothetical urban case in Section 2 (Page 5):

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.

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We 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!