Micro-Macro Coupled Koopman Modeling on Graph for Traffic Flow Prediction
Authors: Bairan Xiang
Chenguang Zhao
Huan Yu
Venue: ICLR 2026 Poster (2026)
Keywords: Koopman Operator; Traffic Flow Prediction
Primary area: learning on time series and dynamical systems

Abstract: Traffic systems are inherently multi-scale: microscopic vehicle interactions and macroscopic flow co-evolve nonlinearly. Microscopic models capture local interactions but miss flow evolution; macroscopic models enforce aggregated consistency yet overlook stochastic vehicle-level dynamics. We propose Micro–Macro Coupled Koopman Modeling (MMCKM), which lifts the coupled dynamics to a high-dimensional linear observation space for a unified linear-operator representation. Unlike grid-based discretizations, MMCKM adopts a vehicle-centric dynamic graph that preserves microscopic perturbations while respecting macroscopic conservation laws by discretizing   PDEs onto this graph. At the micro scale, scenario-adaptive Koopman evolvers selected by an Intent Discriminator are designed to model vehicle dynamics. A Koopman control module explicitly formulate how flow state influences individual vehicles, yielding bidirectional couplings. To our knowledge, this is the first work to jointly model vehicle trajectories and traffic flow density using a unified Koopman framework without requiring   historical trajectories. The proposed MMCKM is validated for trajectory prediction on NGSIM and HighD. While MMCKM uses only real-time measurement, it achieves comparable or even higher accuracy than history-dependent baselines.  We further analyze the effect of the operator interval and provide ablations to show the improvement by intent inference, macro-to-micro control, and diffusion. Code and implementation details are included to facilitate reproducibility.

Published as a conference paper at ICLR 2026
MICRO-MACRO COUPLED KOOPMAN MODELING ON GRAPH FOR TRAFFIC FLOW PREDICTION
Bairan Xiang, Chenguang Zhao & Huan Yu Thrust of Intelligent Transportation The Hong Kong University of Science and Technology (Guangzhou) (bxiang1 C7, czhao 2ohho0044p 2 @coonect. Mikkust-gz.ece .n huanyu@hkust gz. edu. en
ABSTRACT
Traffic systems are inherently multi-scale: microscopic vehicle interactions and macroscopie flow co-evolve nonlinearly. Microscopic models capture local inter- actions but miss flow evolution; macroscopic models enforce aggregated consis- tency yet overlook stochastic vehicle-Ievel dynamics. We propose Micro-Macro Coupled Koopman Modeling (MMCKM), which lifts the coupled dynamics a high-dimenssonal linear observation space for a unified linear-operator repre- sentation. Unlike grid-based discretizations, MMCKM adopts vehicle-centric dynamic graph that preserves microscopic perturbations while respecting macro- scopie conservation laws by discretizing PDEs onto this graph. At the miero scale, scenario- seenario-ddaptive Koopman evolvers selected by an Intent Discriminator are de- signed to model vehicle dynamics. mulate how flow state influences individual vehicles, yielding bidirectional cou- plings. To our knowledge, this is the first work jointly model vehicle trajecto- ries and traffic flow density using unified Koopman framework without requiring historical trajectories. The proposed MMCKM is validated for trajectory predic- tion NGSIM and HighD. While MMCKM uses only real-time measurement, it achieves comparable or even higher accuracy than distory-dependent baselines. We further analyze the effect of the operator interval and provide ablations to show the improvement by intent inference, macro-to-micro control, and diffusion. Code and implementation details included facilitate reproducibility.
Koopman control module explicitly for-
INTRODUCTION
Traffic flow modeling is fundamental challenge in intelligent transportation systems, since it re- quires simultaneously understanding intrinsically coupled microscopic vehicle behaviors and macroscopic flow dynamics. Individual vehicle maneuvers aggregate to form traffic patterns, while macroscopic states constrain and influence microscopic driving decisions. The bidirectional cou- pling creates a complex nonlinear system (Wang al. 2024). Existing methods typically adopt either a microscopic or macroscopic perspective, failing to capture the critical cross-scale interac- tions. Microscopic-based approaches model individual vehicle dynamics through causal temporal inference (Mukherjee et al. 2020; Messaoud et al., 2020) or multi-agent interactions on spatial do- main (Rahmani al.] [023] Shi al. [2024] Gao al.) 2025). While these methods capture local behaviors and stochastic events, they struggle to maintain global flow consistency and scale poorly with vehicle numbers. On the other side. macroscopic models use partial differential equations (PDEs) such as the continuity (Hu et al. [202]] Mahjourian al. (022), but they are deficient at responding to indi- vidual vehicle events that critically influence traffic evolution (Haghighi & El Amine Hamri 2024 Rowan et al. 2025).
Shihill-Whthhm--Riee
(LWR) model to ensure conservation laws and flow
Despite recent advances, fundamental gap remains: few existing methods simultaneously bridge microscopic and macroscopic traffic while maintaining computational tractability and physical inter- pretability. Recent efforts bridge micro- macro scales through game-theoretic frameworks/ (Hunng et al. 2020) or kinematic limits(Crsstiani Sahu, 2016) offer theoretical insights, but they either assume homogeneous drivers asymptotic regimes, limiting their real-world applicability. (Lat-

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pr o 2
VK Encoder
Intent
Diseriminator
IntentAware
Predicted
Fusing
VK Decode
VKEvalver
Trajectory
Tiux
NN
Prodicted
(V,, W.)
GK Encoder
GKEvolver
GK Decude
Density
We propose a Micro-Macro Coupled Koopman Modeling (MMCKM) framework that lifts microscopic vehicle trajectories and macroscopic flow evolution into linear observation spaces, enabling unified prediction. The traffic environment is formulated directed graph. At the macro- scopie level, vehicle-centric graph discretization captures how microscopic perturbations affeet wave propagation. At the microscopic level, evenario-adaptin Koopman operators and Koopman control incorporate macroscopic flow conditions into vehicle dynamics.
Figure 1:
lanzio & Piccoli, 2010, [ellomo et al. 2014 Wang al. 2024 Fan et al. 2025 The fundamental challenge remains how to model the nonlinear, bidirectional coupling a unified, computationally tractable framework.
We propose a Micro-Macro Coupled Koopman Modeling (MMCKM) framework that leverages Koopman operator theory transform nonlinear multi-scale dynamics into high- -igh--imennional linear observation spaces, enabling unified prediction through linear operators as in Fig. [ Our key insight is that both microscopic vehicle trajectories and macroscopic flow evolution be lifted into respec- tive observation spaces where their dynamics become approximately linear. Besides, the Koopman operator exhibits the Markovian property when observation functions time-invariant (Wu & Noë 2020 Kostic et al.) 2022 Tatazzol et al.) 2024). Therefore, our framework relies solely on current state information without requiring historical tracking or continuous object detection. The bidirec- tional coupling is achieved through the following mechanisms: microscopic vehicle events influence macroscopic flow through a diffusion term added the LWR model, capturing how individual ve- hicle perturbations propagate through traffic flow. Unlike existing approaches that discretize traffic flow on fixed spatial grids of the Euclidean coordinate, we propose a vehicle-centric graph dis- cretization the Lagrangian coordinate that preserves microscopic perturbations while maintain- ing macroscopic conservation laws. For the microscopic dynamics, we design a scenario-adaptive Koopman operator selection mechanism capture vehicle dynamics in different driving scenarios. And macroscopic flow states affect microscopic vehicle dynamics through Koopman control, where flow conditions as external inputs to affect individual vehicle dynamics.
Our framework makes three major contributions:
Vehicle centric PDE on graphs: vehicle graph with skew-symmetric advection and positive semi-definin diffusion, ensur- ing energy-preserving advection and nonnegative diffusion process, with a constructive parameterization for antisymmetry.
We derive an advection diffusion evolution on a dynamic
2. Unified history-free Koopman modeling: We show how macro/micro observables can be evolved by time-invariant Koopman operators, enabling accurate trajectory prediction from a single snapshot; we align Koopman spectra with graph-PDE spectra to improve stability and interpretability.
3. Physics-guides multi-regime micro dynamics: lightweight intent discriminator selects among parameter-boume Koopman evolvers; Koopman-control path injects macro flow with ISS-style bounds on control/output, reducing long-horizon drift.
Beyond predictive performance, framework provides explicit insights through learned edge weights that quantify interaction intensities between vehicles. These weights evolve dynamically to reflect changing driving conditions, offering interpretable measures for downstream decision- making modules in autonomous driving systems, a capability unique to our vehicle-centric graph formulation.
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2
BACKGROUND KNOWLEDGE
2.1 KOOPMAN OPERATOR THEORY & KOOPMAN CONTROL THEORY
The Koopman operator provides a powerful framework for analyzing nonlinear dynamical systems through linearization in an observation space. Considering discrete-time nonlinear dynamical system:
e++ = f(x)), E R
(1)
By Koopman operator theory, there exists dimenie--d Hilbert space of observable func- tions:
g { = R
where system dynamics become linear. The Koopman operator K. acts on observables as:
Kgg =
(3)
where K. is linear (though anfinite-dimensionn operator. In the lifted observation = g [ói (zt). o1 (r.) the evolution becomes:
2t+1 Katt
The original state can recovered through reconstruction functions v
Finite-dimensional approximation via DMD: While Koopman operator requires infinite dimen- sions, Dynamical Mode Decomposition (DMD) enables finite-dimensional approximation suitable for practical applications(Brunte) et al. 2016). DMD identifies a finite set of modes that capture the dominant dynamics, with each mode characterized by an eigenvalue X, whose real part determines growth/decay rates and imaginary part captures oscillation (Avila & Mezić 2020).
Neural network parameterization: Recent advances leverage neural network's universal approx- imation capabilities to learn both observable functions á Iº" + R2d and reconstruction functions & : Rd + R (Lusch et al. 2018). The key insight is to learn representation where the Koopman operator becomes approximately diagonal:
3 &h (KK())))
where K diag(Ai, The diagonalization enables efficient evolution:
= = e
(6)
providing both computational efficiency and interpretability through modal decomposition.
Extension to controlled system: For systems with external inputs:
It+1 = fxx, u.)
Koopman control theory introduces an actuation operator B that lifts control inputs into the obser- vation space(Proctor et al. 2018 Strässer et al. 2023):
Zt4 KCa, Bu,
This formulation maintains linearity in the observation space while accommodating external influ- ences. Crucially, the framework imposes minimal constraints on the control structure, allowing us to model abstract influences, such as macroscopic traffic flow effects on individual vehicles, as control inputs. This flexibility is particularly valuable for multi-scale systems where cross-scale interactions lack explicit mathematical forms but significantly impact dynamies.
The Markovian property of the Koopman evolution, when observation functions are time-invariant, enables prediction using only current state information eliminating the need for historical tracking that burdens conventional sequence-based methods.
2.2 MACROSCOPIC TRAFFIC FLOW MODELS
Macroscopic models treat traffic as a continuum fluid, with the LWR model serving as the funda- mental first-order theory. The traditional LWR model describes traffic density evolution through conservation equation describing the fluid propagation dynamics:
Op Øt
+
(()) =
(9)
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where p(z, represents traffic density and Q(p) pv(p) is the flux function with v(p) denoting the density-dependee velocity and wave propagation speed field direction which can be viewed as an advection process.
Traditional PDE discretization methods for traffic flow work on the Euclidean coordinate and divide roads into fixed spatial cells. This approach fundamentally limits their ability to capture vehicle- level perturbations. Stochastic behaviors are averaged within each cell, which eliminates the high- frequency dynamics that is crucial for understanding traffic flow. Our work addresses this limitation by introducing a Lagrangian discretization on vehicle-centric graphs.
PROBLEM FORMULATION
We address the multi-scale traffic flow modeling problem where both individual vehicle dynamics and macroscopic traffic density evolution are jointly considered. Unlike traditional models that focus solely vehicle behaviors or flow modeling, between microscopic vehicle dynamics and macroscopic flow patterns.
framework captures the bidirectional coupling
We assume that the ego vehicle can only get measurements of surrounding vehicles within a detec- tion range mmax- At each time step I, we represent the traffic system as a dynamical weighted directed graph 9.- Ev, W/). where:
Nodes V4 represent vehicles. For vehicle i, its state of includes position pt IR22 velocity "/ E IR22 lane ID 1/ and vehicle size 8 E (0, 1} with s² = being large vehicles whose length is longer than 6.5 m and s² 1 being small vehicles. For the ego vehicle, we use the notation as x = [pt, et, 44, si].
Edges & describe spatial interaction between vehicles. We construct using k-nearest neigh- bors (k-NN) based on Euclidean distance balance computational efficiency with inter- action coverage. We use incidence matrix B, to represent the interaction between them. For two vehicles and j. we set Br,jj = 1 if there is an edge from Vy to e; interaction between them and Big = otherwise.
Edge weights W; quantifies vehicle interaction intensity and traffic flow propagation effi- ciency. To reflect the propagation of traffic flow and the effect of vehicle interaction on traffic flow, use two edge weights and learn them from node features.
Usually, the graph topology is described by graph Laplacian operator L TTBB = D-A, where D is the degree matrix and A is the adjacency matrix. The graph Laplacian has a conjugate edge Laplacian L. B W B which has the same non-zero eigenvalues L. This formulation enables us to express traffic dynamics through graph operators that preserve physical laws.
We adopt Koopman- based approach lift the nonlinear coupling micro-macro system to higher- dimensional linear observation space. A critical advantage of Koopman- based approach is its Markovian property, which enables requiring historical trajectories. The model predicts the ego vehicle future trajectory position and traffic density evolution over T'y prediction.
predict using only current state information G. without
METHODOLOGY
In this paper, we model the cross-scale traffic dynamics. The framework is illustrated in Fig
4.1 TRAFFIC FLOW EVOLUTION ON GRAPH
Traditional spatial discretization fundamentally cannot capture how individual vehicle perturbations propagate through traffic. We pioneer Lagrangian approach that discretizes PDEs directly onto vehicles as graph nodes. This is not merely a change of coordinates-in fundamentally preserves information that spatial methods inherently lose. We give the detailed proof Appendix A.I which is one of the core innovation in this paper. The evolution on graph is:
aaady p + diffft
Crady
BB Wady B, J.dill
B Wrifff B
(10)
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Nodes: Vehicle (State Local Density)
Physics Gruph:
Edges:
Interaction Strength (Advection and Diffusion Weights)
Diffusion:
Flow by density gradient
@diff = w Bp, _dfff D wwdlffB Advection: Flow carried by velocity Qadv -wady Bp, L,av = B
(wadv
wrifft
(wady w
Resulting ODE: p caddy p + _dlff
(x),
Velocity of
(x)
V2
(f
V?
(x7
Diffiusion Flux Qdiff
Advection Flux gadv
(x)
VF
(xf.pE)
Velocity uf
(a) Vehicle-Centric Lagrangian Graph
(b) Edge Flux Decomposition
Figure 2: Illustration of traffic flow evolution on vehicle-centric Lagrangian Graph. (a) Lagrangian discretization: Unlike traditional Eulerian methods that discretize space into fixed grids, we treat vehicles as dynamic nodes Vi in the graph Gz. The mesh moves with the traffic flow, allowing the model to preserve high-frequency microscopic perturbations. (b) Edge flux decomposition: The advection flux captures velocity-induce innuut- transport via skew-symmetric Cody, while the diffusion flux captures density-gradient interactions via PSD operator L.
where p = [PI,.... .... P.v] a vector with Pj being the density at the position of vehicle i, N is the num- ber of vehicle, Cadly is the advection operator which is antisymmetric matrix describing the traffic flow, and 7,ditf is the diffusion operator which is a positive semi-definite (PSD) matrix representing the perturbation from surrounding vehicles. We two edge weight matrices Wadv and Wrdfff to denote the advection and diffusion coefficients respectively. This formulation is fundamentally dif- ferent from previous work that discretize space on the Euclidean coordinate and calculate density as average traffic state within cells. Instead, we discretize on the Lagrangian coordinate and get density values around each vehicles. Therefore, how each vehicle affects flow propagation is explicitly for- mulated. Specifically, we use edge weights Wadv and Wrdiff to explicitly encode vehicle-tee cchicle-toveer tes-tt-veeiie -ehicie interaction strengths, providing interpretable measures that are unavailable grid-based methods.
According to graph spectrum thoery, if T.diff and Caady are commute, i.e., dift Cradv Cady J.diff they can diagonalized by the same eigenvectors U:
i. diff : T, U = Diag(n.. :NN),
(1I)
Eadv := C Diag(iC .jEx),
where ja = -1 is the imaginary unit, 1) are the eigenvalues of I.diff and jú are the eigenvalues of Cadv, n and m are the number of eigenvalue. Since I.diff positive semi-definite, all eigenvalues are all real number. The advection term Crady is antisymmetric, so the eigenvalues are all conjugate pure imaginary or O. We further project the density through and get dynamics in the projected space as:
p = Cad Up U diff diff Up (Diag(n) - jDiag(E))
(12)
where P = Upp. The solution for the ODE equation 12 is:
Diag(n) -Digg(i) p(0),
(13)
In traffic flow, I, diff and Cudv are usually not commute, leading them generally not simultaneously diagonalizable in finite dimensions, however, Koopman lifting enables us to approximate their joint evolution in higher-dimensional linear space, where eigen-decooppositinn is no longer strictly re- quired.
4.2 MACRO DYNAMICS WITH VEHICLE INTERACTION
The dynamics of f in equation 12 share the same structure as the linear dynamics in the observation space in Koopman theory, which inspires to lift the macro dynamics to linear space.
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At each time-step, the edge weights W consists of two matrices Wadv and Widfft, which are usually unknown directly from measurements. To reflect how vehicle interactions affect flow dynamics, we use two GNNs get the edge weights W {Wrdiff Waavy. }. The two GNN takes (V,E) inputs, and outputs the diffusion and advection weights respectively.
To reflect their different physical properties, design specific structure for diffusion operator and advection operator to ensure L,diff PSD and Casly antisymmetric. For diffusion operator, we initialize the diffusion edge undirected and adopt Softplus activation for W keep PSD for diffusion oper- ator. While in advection operator, we reconstruct the graph into directed graph with edges aligned with the direction of speed field, after obtaining the weights Wiady via advection network, we add reverse edge of equal weight for each original edge to ensure the antisymmetric. Therefore the cor- responding diffusion operator L.ifff symmetric and the advection operator Cady is antisymmetric.
Inspired by the linear dynamics in the projected space P. we lift the graph features to an observation space Ze by the Koopman Operator theory.
Encoder: Zz = oz()),,
(14)
Evolver Zi+1 = Kz
(15)
Decoder Pt41 (2((1+1)
(16)
where oz is GNN to lift the graph features to the observation space Z, v' is MLP to decode Z to density, and Ki are a learnable matrix that represents the linear dynamics in the observation space.
We design the loss function for the encoder and decoder as:
Aaacro
cnoode
= (( - K Z (G))l
I = 11pe+1 - Pe+1
(17)
where f is the ground truth density as label when training.
In practical traffic flow, the diffusion operator L dif and advection operator Cadv generally do not commute. We therefore do not assume exact commutation, instead, we penalize the commutator:
LAAD
|| gradv caddy I diff
(18)
where the subscript F is Frobenius norm. This design reduces basis rotation and improves operator- splitting stability under the Lie-Trotter scheme:
A se e AtL Acc
(19)
this regularizer is fully differentiable and requires no eigen-decomposition thus being numerically stable and further motivates aligning the Koopman operator equation
To ensure the consistence between Koopman operator K with diffusion operator Li and advection operator Cady, we set = d log(K) (K) (principal matrix logarithm) and align real 0 with A(L and imaginary 0 with (Cadv), formally:
Cypec min [ ([(((()))))))))))))))))))))))))) IIA.11 + [[Im(((0) + Ilc)
(20)
IT is permutation operator, we compute log log(K) via a numerically stable real Schur form with small Tikhonov regularization on near unit eigenvalues. When K is not diagonalisable, the Schur-log remains well-defined and differentiable almost everywhere, avoiding branch ambiguities. This ex- plicitly couples the Koopman dynamics to the learned graph-PDE operators, consistent with the splitting used in training.
4.3 MICRO DYNAMICS WITH FLOW PROPAGATION
Similar to macroscopic traffic dynamics, vehicle behaviors are also highly nonlinear. Besides, when drivers make decisions, they often consider not only the motion state of themselves, but also sur- rounding traffic. Therefore, the macroscopic flow propagation also have a direct impact on mi- croscopic dynamics. To avoid the analysis and computation complexity, measurement of vehicle state of to a higher linear observation space. To reflect how flow propa- gation affects microscopic vehicle behaviors, we adopt the Koopman Control Theory and design
also lift the original

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CrossAttention block to take the role of Actuation Operator that projects macroscopic influence into the vehicle observation space:
Encoder 2t = (2((5)
(21)
Evolver: Ze+1 = K22 + Brur, Ur = CA Zi)
(22)
Decoder: Pi+1 = (2++1)
(23)
where CA is CrossAttention block fusing 2, from miero and Ze from macro, Φz.b' are MLP, K; and B, are trainable matrix, pi_, is predicted trajectory. We choose a CrossAttention module rather than simple linear projection for ur because traffic influence is highly context-dependent: vehicles in different positions or modes contribute unequally. Attention naturally captures this heterogeneity, aligning with the actuation operator role. To ensure the design of Koopman control is input-state sta- ble (ISS). we need to constrain the output of CrossAttention u, as bounded and the spectrum radius <(Kz) which ensures that errors decay geometrically with rate k k(K.) and that external influ- ences remain bounded. This provides a formal guarantee that MMCKM will not suffer unbounded error growth over iterative Koopman applications. The detailed proof is provided in Appendix [A.2]
In practical driving process, the microscopic dynamics always present varying features in different driving scenarios: free flow, car-following, lane changing, merging. and emergency maneuvers. Driver intent is commonly modeled as discrete variable and can switch abruptly, implying that a single operator must account for different dynamical regimes. However, directly increasing the dimension of Koopman operator matrix K2 is computationally prohibitive when we calculate the eigenvalue decomposition. Furthermore, different modes may present distinct Koopman spectra and control response. For example, in free flow, drivers tend to keep constant maximum speed. The spectrum radius k: k(K,)) )) is approximate one. The imaginary part of the eigenvalues small since the oscillation in trajectory present low frequency variance. Besides, since there is few vehicles, its interaction with surrounding vehicles is weak, and the injects from Cross Attention is small. However, in the lane-changing scenario. vehicles have longitudinal and lateral interaction, leading the coupled oscillation at both real and imaginary eigenvalues of Koopman operator.
To capture various driving scenarios with a low computation cost, construct family of Koop- man operators consists of multiple 2 x 2 complex-valued blocks and diagonal real-valued blocks with distinct initialization schemes such that each operator presents a distinct driving mode. The variation among these operators is achieved by (i) imposing different bounds on the spectral radius of K2 reflecting stability margin, (ii) tuning the coefficients of the complex block control terms 0 to adjust oscillation frequency, and (iii) constraining the maximum actuation operator Bmax for actua- tion strength. The detailed settings and their physical interpretations are provided in Appendix [A.2 [A.2 To determine which operator is most consistent with current traffic environment, we design an Intent Discriminator, which implemented as a -eeepeersss (MoE) that evaluates the current vehi- ele state oa and the graph-based observation Z, to accordingly select the most plausible Koopman operator from the candidate set. We implement this Intent Discriminator by a MLP through super- vised learning, training label are generated in data preprocessing calibrated by acceleration and lane variance. In this way, the high computation burden of a single, over wer-generalizee Koopman operator is alleviated. Instead, the system leverages structured ensemble of specialized operators, with the Intent Discriminator serving as the gating mechanism that adaptively aligns operator selection with the underlying driving intent.
For the loss function design, the loss functions for the micro encode and decode are calculated by:
Eimero
(#( (x))
Lmmero
||
1f
(24)
Finally, after training, we only obtain Koopman encoder (GNN and MLP), Koopman operator (ma- trix) and decoder (MLP) without any eigendecomposition. The computation cost focuses on GNN forward passes and linear Koopman operator iteration.
EXPERIMENTS
5.1 DATASETS
We use trajectory prediction and traffic flow prediction to validate the proposed method. We utilize two highway datasets for our experiments: NGSIM and HighDD. We use NGSIM US-101 highway
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Table I: RMSE NGSIM Dataset. Trajectory reported RMSE. The operator interval
in our model is set to 0. and 1.0s
With historical data
Without historical data
Prediction
Ours
Ours
BAT
MS-STGCN
Vit-Traj
CV
Horizon (sec)
1.0sec
0.1sec
0.27
0.42
0.39
0.64
0.54
0.33
2
0.90
1.00
0.95
1.48
0.98
0.92
3
1.43
1.66
1.58
2.63
1.57
1.63
4
2.76
2.44
2.22
4.33
2.26
3.17
5
3.80
3.05
2.89
5.62
2.93
4.65
Table Comparison of accuracy Different Operator Interval on the Highl) dataset
Interval
0.04s
0.1s
0.2s
0.4s(*)
Is
ADE 2.84
2.06
1.88
1.65
2.90
subset, which captures vehicle trajectories sampled at 10 Hz. The HighDD datsaset is collected in German highways with frequency at 25 Hz.
5.2 EXPERIMENT SETTINGS
We establish perception space centered the ego vehicle, covering three lanes (the ego vehicle lane and two adjacent lanes) over a longitudinal range of 90 m ahead and 60 m behind. The k-NN al- gorithm is set identify the nearest vehicles within the perception space build edge connections. We use kernel density estimation to get density from vehicle positions. We use Gaussian Kernel and set the kernel bandwidth 25 m. We set the dimension of the observation space Z and both as 128. For the Intent Discriminator, we set five driving modes: free flow, car-following, lane changing, merging, and emergency. Intent labels derived directly from raw trajectories using simple. de- terministic rules based on longitudinal acceleration, relative headway, and lateral displacement. We assign free flow when vehicles move with stable high speed and minimal interaction; car-following when longitudinal motion is governed by leading vehicle with reduced headway; lane changing when sustained lateral motion accompanies a lane transition; merging when lateral entry is initiated from on-ramp; and emergency when abrupt braking or rapid acceleration indicates evasive maneu- vers. These labels are fully reproducible and require no learned classifier or manual annotation.
We compare against widely-cited recent baselines that representative of history-dependent pre- dictors: BAT (Liao et al. 2024), MS-STGCN (Tang et al.) 2023), Vit-Traj (Cheng et al., 2025). Because distory-dependent methods assume access 3 8 of trajectories while ours is strietly history-free, our comparison focuses on identical prediction horizons and the same current-state in- puts, thereby take CV (Mercat al. 2019) papers, we adopt each method's 's native setting and report our model at matching horizons to avoid conflating sampling effects with modeling capacity. We view these results as conservative for our approach, since removing historical inputs typically disfavors accuracy.
history-free reference. Where intervals differ across
To our knowledge, no prior work evaluates vehicle-centric graph discretization of LWR with diffu- sion on highway data with publicly-standardized density labels. We therefore treat KDE-estimated density as operational ground truth and report absolute errors to KDE as an internal consistency metric. The default bandwidth of KDE kernel is 25 meters. KDE provides consistent operational definition, though we acknowledge its limitations and encourage future benchmarks with sensor- derived density labels. Through ablation study, we identify the efficiency of diffusion operator. Cross-paper macro density SOTA comparison are left for future benchmarks.
5.3 RESULTS
Table I presents prediction accuracy on the NGSIM dataset. We see that the proposed method has lower prediction than the SOTA CV method for all prediction horizons. Remarkably, our approach, as a completely history-free model, achieves performance comparable to ofate-of-the-an methods that require historical trajectory data during the past 3-8 seconds. Traditional sequence-
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based methods like BAT and MS-STGCN impose substantial computational overhead through tra- jectory tracking and storage, limiting real-time applicabilityy. Our Markovian approach eliminates these requirements and also maintains competitive accuracy.
From Table I we observe that the operator interval plays a central role in the error accumulation pattern. Our method exhibits an approximately linear growth of error with respect to the number of iterative steps, which is governed by the Koopman sampling interval. The 0.1s operator achieves excellent short-term accuracy (RMSE=0.33 @ Is) approaching history-dependent models. Never- theless, its error accumulates more rapidly over longer horizons because each prediction requires repeated applications of the operator. For example, forecasting seconds ahead involves 50 itera- tions. In contrast, the 1.0 s operator begins with a higher short-term error but demonstrates superior long-term prediction accuracy. For the prediction over 5 seconds, only five iterations are required. This controlled growth in enror stands in clear contrast to the exponential error amplification often observed in recurrent architectures. These results suggest that, over extended horizons, our approach can surpass existing algorithms as the dependence initial historical data diminishes.
To further examine the effect of operator intervals, we use the HighD dataset, which has higher sampling frequency and provides a more suitable data for evaluating how interval settings affect prediction. We evaluate prediction accuracy for operator intervals at 0.04s, 0.1s, 0.2s, 0.4s, and 1.0s. We compare the Average Displacement Error (ADE) in Table. 2 There is a trade-off between the ability to capture high-frequency and numerical stability: small intervals preserve high-frequency dynamics but suffer numerical instability, while large intervals ensure stability but sacrifice dynamic fidelity. An excessively small interval (e.g., 0.04s) forces each operator to represent minimal dy- namic changes, causing eigenvalues to cluster near unity and creating numerically ill-conditioned system highly sensitive to noise. This configuration requires numerous iterations for prediction, am- plifying computational errors. On the other side, an overly large interval (e.g., 1.0s) cannot capture fegh-frequenyy dynamics because within a single second, significant portions of driving maneuvers occur compared to complete steering actions typically span 5-6 seconds. The optimal interval of 0.4s achieves the best ADE performance, balancing dynamic representation with numerical stability.
5.4 ABLATION STUDY
Component-wise Analysis To validate our bidirectional coupling mechanism, we systematically ab- late the Intent Discriminator and CrossAttention modules. Table 3 presents RMSE comparisons for four variants: the complete model (MMCKM), Intent Discriminator ablation (MMCKM-D, Koop- man control ablation (MMCKM-C), and both components removed (MMCKM-IC).
The results demonstrate complementary roles in multi-scale coupling: The Intent Discriminator pri- marily enhances short-term predictions (29% improvement at 1s), effectively selecting appropriate dynamics for immediate vehicle behaviors. However, its effectiveness diminishes over longer hori- zons as the latent macroscopic state Z, evolves independently, accumulating errors that degrade intent classification accuracy. Maintaining accurate Intent Discrimination would require simultane- ous state updates for all surrounding vehicles leading computationally prohibitive requirement that would negate our method's efficiency advantages.
The Koopman control module proves crucial for long-term stability, reducing error by 37% at 5 seconds compared MMCKM-C. By injecting macroscopic flow information control inputs, maintains the bidirectional coupling essential for accurate long-horizon prediction. Removing this module reduces the system single-operator framework, eliminating the critical macro-to-miero influence that constrains vehicle trajectories within physically plausible flow patterns.
Validation of Diffusion Term Comparing the complete advection-diffusion model (I.C) with advection-only variant (C) on NGSIM. From Table the dramatic degradation confirms our the- oretical framework: the diffusion operator d captures how microscopic vehicle perturbations propagate through traffic flow, a phenomenon completely absent in traditional LWR models. With- out this term, the model reverts to deterministic flow evolution, unable to represent the stochastic perturbations that characterize real traffic. This result validates fundamental contribution: the first successful incorporation of microscopic stochasticity into macroscopic PDE models through vehicle-centric discretization.

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Table 3: Ablation Study of Different Components
HighD, Operator Interval 0.2s
Model
18
2s
38
4s
58
MMCKM
0.29
0.60
1.21
1.72
2.73
MMCKM-1
0.74
1.39
1.96
2.90
3.81
MMCKM-C
0.41
1.01
1.89
2.50
3.46
MMCKM-IC
0.80
1.74
2.54
3.48
4.62
Table 4: Comparison of Removal Diffusion Term on NGSIM
Model
1s
3s
4s
5s
LC
3.2%
4.1%
5.6%
7.4%
9.5%
6.1%
7.2%
10.7%
12.0%
14.1 %
Table 5: Comparison of Different KDE Bandwidth on NGSIM
Bandwidth
1s
2s
3s
4s
5s
10 6.4%
7.7%
11.6%
12.8%
14.0%
20
4.7%
5.9%
7.3%
9.6%
11.1%
25
3.2%
4.1%
5.6%
7.4%
9.5%
30
4.0%
5.2%
7.0%
9.2%
10.4%
40
4.8%
6,3%
8.0%
10.5%
11.7%
50
5.5%
6.6%
8.5%
11.1%
12.6%
Analysis of KDE Bandwidth Sensitivity The KDE bandwidth implicitly determines the spatial frequency content of the density field, and thus affects the supervision signal for learning dif- fusion and advection operators. We compare the performance macro module with bandwidth h {10 20 25 30 m, 40 m, 50 m and report the macro-level RMSE Table. The band- width at 25 meters achieves the best accuracy, while both extremely small and excessively large bandwidths lead to a lower prediction accuracy. By comparing the advection-only variant (C) model in Table. []aadd the complete model (LCC) with different kernel length values in Table. we see that the proposed model achieves more accurate under wide range of bandwidth values, from 20 to 50 m. We also note that with a extreme small bandwidth of 10 m. LC has worse prediction than the model. The possible reason explained follows. When the KDE bandwidth is reduced to 10 meters, the density labels become dominated by high-frequency noise, causing arbitrary ve- hicle perturbation to variance in diffusion term. In other words, according to Eq. 32] small KDE bandwidth brings noise on the density prediction. To fit this noisy, the learned Wain amplifies the noise rather than capturing genuine perturbation propagation. With the perturbation of Wan: commutator of LJAD and spectral alignment of Lypee is disrupted, leading the macro evolver K z to wrong direction. As a result, LC with bandwidth 10 performs even worse than C. This result reveals an important insight: diffusion improves macro prediction only when the density supervision carries physically meaningful gradients; otherwise, diffusion becomes a harmful channel that injects noise into operator learning.
CONCLUSION
We introduce Micro-Macro Coupled Koopman Modeling for traffic prediction, unifying microscopic vehicle dynamics and macroscopic flow evolution within single Koopman-basee architecture. On macro side, we discretize ddveecion-diffusion traffic flow PDE onto vehicle-centric dynamic graph and construct diffusion operator and advection operator ensuring the physical property graphs. On micro side, we design scenario-adaptive Koopman evolvers selected by Intent Discriminator and utilize Koopman control to inject macro-flow influence into vehicle dynamics. Experiments and ablation studies validate the roles of operator interval, intent gating. KDE kernel sensitivity and macro-to to- ro-micro control. Future work will leverage learned edge weights for interpretable interaction measures vehicle planning and control. Another direction is to expand the framework into urban scenarios under heterogeneous graph structure.
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7 ETHICS STATEMENT
This research does not involve human subjects, personally identifiable information, or sensitive data. All experiments were conducted on publicly available datasets (HighD, NGSIM), which are widely used in the transportation research community. No personally identifiable information is included in these datasets. The potential societal impact of this work lies in its application to intelligent transportation systems, where improved modeling and prediction could enhance traffic safety and efficiency. the same time, we acknowledge possible risks, such as misuse surveillance or decision-makin systems that may raise fairness concerns if applied without proper consideration. We encourage responsible and transparent deployment of the proposed methods.
8 REPRODUCIBILITY STATEMENT
We are committed to ensuring the reproducibility of our work. To this end, we provide detailed descriptions of our model architecture, training procedures, and evaluation metries in the main pa- per and appendix. The datasets used in this study (HighD, NGSIM) are publicly available. We will release our code, including data preprocessing pipelines and training scripts, an anonymous repository upon publication. Hyperparameters and implementation details are also provided in the supplementary materials to facilitate replication of our results.
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Jean Mercat, Nicole El Zoghby, Guillaume Sandou, Dominique Beauvois, and Guillermo Pita Gil. Kinematic single vehicle trajectory prediction baselines and applications with the ngsim dataset. arXiv preprint arXiv: 1472, 2019.
Kaouther Messaoud, Itheri Yahiaoui, Anne Verroust-Blondet, and Fawzi Nashashibi. Attention based vehicle trajectory prediction. IEEE Transactions on Intelligent Vehicles, 6(1):175-185, 2020.
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Joshua L Proctor, Steven L Brunton, and J Nathan Kutz. Generalizing koopman theory allow for inputs and control. SIAM Journal on Applied Dynamical Systems, 17(1):909-995 2018.
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A APPENDIX
A.1 TRAFFIC FLOW EVOLUTION ON GRAPH
Starting from the 2D LWR equation with diffusion on multi-lane highway in equation density is described in a Eulerian coordinate I y, and decompose the flux into advection and diffusion components:
9 where
dp 01
V Q 0
(25)
To avoid averaging vehicle perturbation within cell segmented discretization, instead of discretizing on a fixed Eulerian grid, we adopt Lagrangian perspective in dynamical directed weighted graph
G. = = W,) where vehicles form nodes. At time t, we have nodes V. = {ur,... .... UN } rep- resenting N vehicles perceived within in range Tmax containing node positions space, while density at nodes (t(t) representing local traffic density around vehicle i. The vehicles connected using k-NN to form edges &, based Euclidean distance. Each edge eij from node i
(zi, yt) in 2D
to defines a direction vector d ||x,
x, -لاl and decomposed diffusion weight wiiff and advection
weight wadv be determined based on vehicle relative states thj f(v,, U2). These edges form a non-orthogonal, redundant directional template for approximate gradients. Unlike regular grid with two orthogonal directions (rr and y), we have k directions per node that sample the 2D space irregularly and totally M edges.
In continuous space, a difference is essentially the functional difference between adjacent points, while the gradient is the spatial derivative of the function:
Af(x) f(x + Az) f(x), Vf(x) f(x + Az) f(x)
(26)
Ar
While on graph, there is no regular grid Ar and density p is discretized, but exists the adjacent relationship connected by eij. Assume two adjacent nodes and j carry a scale field p; and pj re- spectively, the whole density on discretized nodes on graph is collected by vector = Ip,,... PN] where N is the vehicle number which depicts the density scalar at the position of each vehicle, the difference alongside edge Eiy equals:
(Vap)c Pj Pi
(27)
the relationship can be denoted as incident matrix B:
(Bp)es, Pi + Pj = Pj - Pi Thereby, the gradient graph can be written
(28)
V&p= = Bp
(29)
Similarly, the divergence on the graph computes computes the net flux on node across all incident edges:
Q d T
(30)
We take the total flux Q as a summarization of advection flux Qadv + V Qdiff and diffusion flux where advection flux Qadv pv transported by velocity field v and diffusion flux Quill DDV follows Fick's 's law for density gradients, D is the diffusion coefficient.
Diffusion Term According to Fick's Law, the diffusion flux Qdift = - DV can be discretized on graph according equation 29) and consider diffusion coefficient D is anisotropy but related to adjacent dynamical node features, we can derive:
Q dill -Wdiff Bp,
Widiff = pdiff (dii sj
(31)
Finally, we calculate the divergence according to equation 30
diff According to graph Laplacian definition L = BWBT, TT, this equation can be described through diffusion operator Lam which is not a graph Laplacian but called edge Laplacian:
diff div Q diff = B difff Bp = Lamp
(32)
ddiff = Tampp
Laa B Wiill B
(33)
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In diffusion term which describes the stochastic perturbation propagates along density gradient, it's an entropy production process, leading the dissipation entropy nonnegative:
d d त. _ - p' p- p (B" p=(B'T W p)p (Bp) T W (Bp) 2
J. [((p)
(34)
define y - Bp, this equation turns
y Wrditty y 2 0
(35)
thereby, Wrdiff must be positive seimi-definite (PSD).
Advection Term In continuous space, Q p(x, t) v(x, t), which is a vector field, represents density transported by speed field. On graph, there is no more continuous speed field v(x), but discretized nodes and edges. In this setting, the direction of speed field v(z) is not solely 2D x and y coordinates, but formed by M non-orthogonal, redundant direction vector d,y ->,,~,
continuous condition, the flux via section is:
adv f pV pv- dA
(36)
where n the normal vector of section A. While in graph, there is no more section vertical to speed field but connected with edge channels. We account the flux passing the edge between node and
The flux on edge is bidirectional Qi (Q Qaar)) and the flux on edge for one direction equals:
Q = Ps (v: diy) dAss
(37)
1+ Pÿ (v3 dji)= dAji
define:
Oij == (v, d:y)) dAg. age == - (v, diz) dAj
(38)
we decompose P and Pj to average and difference:
1
1
A: Đ+ 2 5(: pj), Pj -= pi), p 5 Pi))
(39)
then, the advection flux on edge is converted to:
1
Q ady
ady = axjps - ajiPj = 2 (pr - pj)(ar + +aji) + p(ais az))
(40)
the first term can be rewritten graph Wadv Bp and Wadv j (aij + aji).
According to equation 25 we calculate the divergence on node:
pudv
Qady = div
B ndv
(41)
and the discrete conservation law requires the total density preserved:
T pady = T Q =0
(42)
since B1 = 0, this is guaranteed if Qniv depends only on the difference Bp and the second term p(a.y afs) vanished.
Finally, the equation 41 can written as:
judy B Wadv Bp
(43)
use a Laplacian operator Codv to denote:
gody
Cayy
Cudv
B
Wrady B
(44)
Furthermore, according to physical law, we require advection is dissipative:
because it's non-
dt (1llB) = p = p ==
(45)
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this requires Cady to be antisymmetrie matrix C -Cady, To ensure Cady, we develop special parameterization method built from original graph.
We first reconstruct the graph into directed graph with edges aligned with the direction of speed field, and let Alme {0, 13 f x denote the line-graph adjacency, where Aline (e, [e, iff edges and e' share node. Define a symmetric locality mask:
M
2 (Aliic lime AlineT I)
(46)
and introduce an unconstrained parameter matrix P E RAxxM, We parameterize the advection weights and operator by:
W addv M (P-P)) PT), Cav := B Wadv B
(47)
where o is the Hadamard product and B is incident matrix. Because M is symmetric and P-P" PT skew-symmetric, we have:
(Wadv)T= = _0 (PT P) = -Waadv
(48)
hence, we can get:
CadvT (B Wadd B) = B TT (Wadr) -C
(49)
therefore Cady is antisymmetric, ensuring non non-dissipative advection term.
In summary, equation 25 can be discretized graph:
- p L difl
P
(50)
A.2 KOOPMAN OPERATOR DESIGN
In equation 22 even if K2 is stable, if control is infinite and no constrain Bz, 2, is also diver- gent. Furthermore, practice, the augmented Koopman operator [K B3] of vehicle dynamics is constraint by physical limitations such as limited steering angle, throttle and acceleration. Thereby, we need to ensure u, is input-state stable (ISS), equals to:
3 const c 2 1, E (0,1)
|20ll + cB.
cB=
(51)
s.t. ||||||
S cA
sup
||u||| cc11|2201|
U max
1- -osssrr-1
To ensure that the observation z is always bounded, we require that the output of Crosstttention we bounded and |A| 1. For the first requirement, add Sigmoid operator before the final output of the Cross module.
For the second requirement, it equals ensure the spectrum radius <(Ka) 1. In our design, the Koopman operator K₂ (A) is a trainable approximately diagonal matrix with real and complex eigenvalues. We use Ne 2 x complex blocks and Nr 1 real blocks construct (A).. where Ne and Nr is a hyperparameter related to Koopman operator dimension. To ensure k(K,) < 1, we need simultaneously promise the mode of each complex and real blocks strictly less than For each complex block, we construct it with two learnable parameter, radius R and rotation 0 to parameterize the complex conjugate eigenvalue pairs À = Re (2:00, apply to 2 x 2 blocks:
cos (i
sin cos(0)
K_ = R
(52)
further constrain each radius R less than 1 by applying R Kmxx sigmoid(1). where 7 is the eigenvalue advection operator Cadv which governs the traffic flow propagation, while 0menn and Bm represents the vibration frequency as diffusion operator I diff In real block, we apply same constraint on = fémax x sigmoid(1) to ensure stability. Furthermore, through this design, we can directly record max(R) to quickly estimate spectrum radius of each Koopman operator.
Even if we ensure u, is bounded by Sigmoid operator, while training, B, is possible to divergence to big enough to break the physical constraint, to avoid it, constrain by Bmax Tanh Bz).
15

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Table 6: The Hyperparameter of Each Mode
Mode
Fimax
Bmux
Badd
m
free flow
0.95
0.20
0.01
0.00
car-following
0.85
0.60
0.02
0.00
lane changing
0.90
0.75
0.08
0.25
merging
0.88
0.80
0.05
-0.15
emergency 0.70
0.40
0.01
0.00
These design is convenient for us to introduce prior knowledge into different Koopman operator because only need to set different Kmaxx 6 6meun and Bma for different driving mode. In free flow mode, persist speed and fixed direction is reflected by the slowly decaying rate, equals (Kmax + 1), small perturbation from stochastic vehicle behavior (0sn, Omean - 0), and trivial external influence (Bmax + O), the difference of car-following mainly comes from front vehicle influence which can be reflected by larger Bmax > 0. While in lane-changing and merging, the external influence is obvious (Bmax - 1) and there is explicit vibration period (fmean # 0), here we lane changing initialized be larger than 0 and merging initialized to be smaller than zero is to avoid lane changing and merging to be trained to same mode quickly. Finally, for emergency mode, vehicle accelerates quickly, represented by Kyna
while suffered from larger external influence
Bmax > 0.
We list all parameter setting Table
A.3
RUNTIME EFFICIENCY ANALYSIS
Our Koopman- based formulation provides a clear inference- advantage over conventional spa- tiotemporal models. Unlike recurrent or attention-based architectures that must repeatedly unroll temporal dynamics or compute global pairwise attention, our approach decouples temporal lution into single linear Koopman operator propagations, allowing the full multi-step future to be generated in one pass. Consequently, the inference complexity message passing stage with (EEd) cost, where E' = k N is the number of edges, plus Koop- man evolution whose cost depends solely on the latent dimension of complexity O(Td²) totally O(kNd + T Compared to the complexity of spatiotemporal GNN of (NN + kNd)) and spatiotemporal Transformer OT d²d), the computation complexity is greatly reduced.
dominated by sparse graph
Experientially, on an NVIDIA 4090, our model achieves average inference time 0.20 for 10 batches with a batchsize of 128, which already satisfies the real-time deployment.
A.4 USE OF LLMS STATEMENT
We used large language models (LLMs) during the process of preparing this manuscript. Specif- ically, LLMs were employed for language polishing, grammar checking, and improving clarity of expression in certain sections of the paper. All technical content, derivations, experiments, and sults were designed, implemented, and verified by the authors. The authors take full responsibility for the correctness and integrity of the scientific contributions presented in this work.
16


Reviews:
[{"review_id": "OTMzfmmRqN", "reviewer": "Reviewer_dPBn", "summary": "The paper introduces Micro-Macro Coupled Koopman Modeling (MMCKM), a framework for unified traffic flow prediction that combines microscopic vehicle interactions and macroscopic flow dynamics by using Koopman operator theory. This combines microscopic dynamics, where every single vehicle is simulated, to macroscopic dynamics where the overall traffic flow is evolved. The method predicts the future state from the current time step, without relying on historical trajectories, by leveraging a high-dimensional linear observation space, and performs better than history-dependent baselines on experiments on NGSIM and HighD datasets.", "questions": "- Is the model applicable to urban or more complex scenarios than structured highways?\n- Can the authors comment on the runtime efficiency of the method? Is it able to run at real-time?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "The paper tackles an important problem, to bridge the gap between microscopic and macroscopic dynamics in traffic modeling, both of which have advantages and drawbacks. For instance the microscopic scale allows for simulating individual vehicle dynamics while the macroscopic scale enforces high-level flow dynamics such as conservation laws. The authors use an innovative Koopman-based unified model to jointly learn both scales, and leveraging a graph discretization of the PDE, as opposed to standard grid discretization.  Empirical validation on two real-world datasets (NGSIM & HighD) support the approach, achieving competitive or superior results without relying on historical trajectories. The ablation studies validate the impact of each module. The paper is well-written overall and well-motivated.", "weaknesses": "- The paper is quite technical and math-heavy, which can make it hard for readers not familiar with the area to follow the core concepts. Some sections such as 4.1/4.2 could benefit from more intuitive explanations or visual illustrations.\n- Experiments are limited to highway datasets with estimated (not sensor) density labels, so it’s unclear how well the method would generalize.\n- There are no direct comparisons with other hybrid or Koopman-based approaches, so it makes it harder to tell how much of the improvement comes from the coupling design itself.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OV4JuJjdOD", "reviewer": "Reviewer_EX1y", "summary": "The paper introduces Micro–Macro Coupled Koopman Modeling (MMCKM), a unified Koopman-based framework that jointly models microscopic vehicle interactions and macroscopic traffic flow evolution. \n\nUnlike conventional microscopic or macroscopic approaches that treat these scales separately, MMCKM lifts the coupled dynamics into a shared high-dimensional linear observation space. On the macroscopic side, the method discretizes advection–diffusion traffic flow PDEs onto a vehicle-centric dynamic graph, preserving physical flow consistency without grid constraints. On the microscopic side, scenario-adaptive Koopman evolvers, guided by an Intent Discriminator, model diverse vehicle behaviors, while a Koopman control module captures the bidirectional influence between flow states and vehicle dynamics. MMCKM adopts a vehicle-centric dynamic graph that preserves microscopic perturbations while respecting macroscopic conservation laws by discretizing PDEs onto this graph. Evaluated on NGSIM and HighD datasets, MMCKM achieves comparable trajectory prediction accuracy to state-of-the-art history-dependent models despite using only real-time inputs. Ablation studies confirm the contributions of intent inference, macro-to-micro control, and operator interval design. The authors also outline future work on interpretable interaction analysis and extending the framework to urban traffic scenarios.", "questions": "-", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The introduced method seems to be very original;\n- The article is well-written; it has a good structure, and I like the fact that it includes an ablation study and a reproducibility statement, and the use of LLMs statement\n- The results of the experiments are good\n- The method has the potential to be significant, taking into account that it does not require historical trajectories", "weaknesses": "- The method still does not outperform the best methods utilizing historical data, but the fact that the results are comparable is noticeable\n- There are some typos and minor writing issues, e.g., due to the lack of space,: \n  - l. 49: \"frameworksHuang\" -> \"frameworks Huang\" \n  - l. 50: \"limitsCristiani\" -> \"limits Cristiani\"\n  - l. 128: \"applicationsBrunton\" -> \"applications Brunton\"\n  - l. 133: \"R^nLusch\" -> \"R^n Lusch\"\n  - l. 144: \"spaceProctor\" -> \"space Proctor\"\n  - l. 370: \"3 8 s\" -> \"3-8 seconds\"", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "f7u4DNGeH5", "reviewer": "Reviewer_EnKM", "summary": "The paper proposes a Micro–Macro Coupled Koopman Model (MM-Koop) for unified modeling and prediction of multi-agent traffic flow dynamics.\nBy embedding both micro-level nonlinear interactions and macro-level fluid statistics into a shared Koopman space, the method enables a cross-scale linearized representation of complex traffic systems.\n\nThe framework consists of:\n\n1. A micro-level encoder mapping individual agents’ states to Koopman-observable embeddings;\n\n2. A macro-level aggregator capturing density, velocity, and energy fields;\n\n3. A cross-scale coupling operator that models bidirectional feedback between individual and collective dynamics.\n\nExperiments on the NGSIM (highway) and ETH/UCY (pedestrian) datasets demonstrate superior multi-step prediction accuracy and interpretability compared to LSTM, GraphKoopman, and other baselines.", "questions": "1. Is the Koopman operator learned using neural networks or estimated through regression? How is numerical stability ensured?  \n2. Does the cross-scale coupling introduce instability into the operator spectrum?  \n3. Could the model benefit from graph-based message passing to represent heterogeneous agent interactions?  \n4. Would the framework generalize to domains without explicit macro-aggregators, e.g., aerial swarms?  \n5. How do the authors interpret the sparsity of Koopman spectra under high-density conditions? Is it an artifact of strong linearization?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Presents a \\textbf{novel cross-scale Koopman operator framework} that linearly couples micro-level agent dynamics and macro-level flow statistics.  \n- Provides a \\textbf{mathematically rigorous formulation} with well-defined block operator structure and spectral regularization for stability.  \n- Offers strong \\textbf{interpretability and physical insight}, explaining emergent traffic phenomena (e.g., congestion waves, flow bifurcation) via Koopman spectral modes.  \n- Demonstrates \\textbf{solid predictive performance} compared to LSTM and GraphKoopman baselines.  \n- The theoretical structure is \\textbf{generalizable} to broader multi-agent dynamical systems such as robotic swarms or fluid-based networks.", "weaknesses": "- Experiments rely primarily on controlled simulations; no real-world deployment or sensor-based evaluation is shown.  \n- The paper’s \\textbf{learning component is limited}: Koopman matrices are estimated rather than learned via gradient-based training, reducing its alignment with mainstream ML innovation.  \n- Lacks \\textbf{comprehensive ablation studies} to assess the contribution of each coupling term ($K_m, K_M, K_{mh}, K_{hm}$).  \n- The related work section omits recent Koopman-based learning frameworks (e.g., NeuralEDMD, DeepKoopman++, Contrastive Koopman Learning).  \n- While theoretically strong, the overall \\textbf{ML novelty and scalability} remain somewhat constrained.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ty4Ec729P0", "reviewer": "Reviewer_qGhn", "summary": "This paper proposes **MMCKM (Micro–Macro Coupled Koopman Modeling)**, a unified framework for traffic flow prediction that links microscopic vehicle trajectories and macroscopic flow dynamics. Both scales are lifted into **Koopman spaces**, where their evolution is governed by linear operators aligned through **spectral and physical constraints**.\n\nAt the macro level, advection–diffusion PDEs are discretized on a **vehicle-centric dynamic graph**, preserving physical properties such as antisymmetric advection and positive semi-definite diffusion. At the micro level, multiple Koopman operators represent different driving intents, chosen via an **Intent Discriminator**, and macro information is injected through a **Koopman control mechanism**.\n\nThe model predicts from a single snapshot (history-free) and achieves strong results on **NGSIM** and **HighD** datasets, with ablation studies validating the importance of macro–micro coupling, intent modeling, and diffusion components.", "questions": "1. How sensitive are the macro-level results to KDE bandwidth? Please provide results under different bandwidths (e.g., 10 m, 25 m, 50 m).\n2. Could you describe the process of intent labeling in more detail? What thresholds or criteria were used, and how accurate is the Intent Discriminator?\n3. What is the computational complexity and runtime per prediction step? How does it scale with the number of vehicles?\n4. Could the model generalize to more complex network structures (urban intersections), and what modifications might be required?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. **Innovative modeling perspective** — integrates micro-level trajectories and macro-level PDE dynamics within a unified Koopman framework.\n2. **Physically consistent design** — antisymmetric advection, PSD diffusion, and spectral constraints ensure stability and interpretability.\n3. **Modular and efficient structure** — multiple small Koopman operators (for different intents) improve computational efficiency and interpretability.\n4. **Strong practical motivation** — “history-free” prediction is appealing for real-world deployment with intermittent data.\n5. **Clear ablation and physical insight** — experiments reveal the distinct contributions of each model component.", "weaknesses": "1. **KDE-based macro labels** are not ground-truth densities; performance depends on kernel parameters. Sensitivity analysis is missing.\n2. **Computational scalability** not analyzed — unclear runtime behavior as vehicle count grows. And there is a lack of specific runtime comparison with other baseline models\n3. **Intent Discriminator details** (labeling rules, accuracy, and noise sensitivity) are insufficiently described.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)