STDDN: A Physics-Guided Deep Learning Framework for Crowd Simulation
Authors: Zijin Liu
Xu Geng
Wenshuai Xu
Xiang Zhao
Yan Xia
You Song
Venue: ICLR 2026 Poster (2026)
Keywords: spatiotemporal-learning;physics-informed;neural ODE;crowd simulation;
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)

Abstract: Accurate crowd simulation is crucial for public safety management, emergency evacuation planning, and intelligent transportation systems. However, existing methods, which typically model crowds as a collection of independent individual trajectories, are limited in their ability to capture macroscopic physical laws. This microscopic approach often leads to error accumulation and compromises simulation stability. Furthermore, deep learning-driven methods tend to suffer from low inference efficiency and high computational overhead, making them impractical for large-scale, efficient simulations. To address these challenges, we propose the Spatio-Temporal Decoupled Differential Equation Network (STDDN), a novel framework that guides microscopic trajectory prediction with macroscopic physics. We innovatively introduce the continuity equation from fluid dynamics as a strong physical constraint. A Neural Ordinary Differential Equation (Neural ODE) is employed to model the macroscopic density evolution driven by individual movements, thereby physically regularizing the microscopic trajectory prediction model. We design a density-velocity coupled dynamic graph learning module to formulate the derivative of the density field within the Neural ODE, effectively mitigating error accumulation. We also propose a differentiable density mapping module to eliminate discontinuous gradients caused by discretization and introduce a cross-grid detection module to accurately model the impact of individual cross-grid movements on local density changes. The proposed STDDN method has demonstrated significantly superior simulation performance compared to state-of-the-art methods on long-term tasks across four real-world datasets, as well as a major reduction in inference latency.

Published as a conference paper at ICLR 2026
STDDN: A PHYSICS-GUIDED DEEP LEARNING FRAMEWORK FOR CROWD SIMULATION
Zijin Liu', Xu Geng', Wenshuai Xu², Xiang Zhao?, Yan Xia', You Song² School of Computer Science and Engineering, Beihang University, China School of Software, Beihang University, China
"Beijing Digital Native Digital City Research Center, Beijing, China (liuzijjhs jin, xu, by2321118, songyou @buaa. edu. cn, [gengxu, xiayan)@bdnre..omm org..
ABSTRACT
Accurate crowd simulation is crucial for public safety management, emergency evacuation planning, and intelligent transportation systems. However, existing methods, which typically model crowds trajectories, are limited in their ability to capture macroscopic physical laws. This microscopic approach often leads to error accumulation and compromises simu- lation stability. Furthermore, deep learning-drivee low inference efficiency and high computational overhead, making them imprac- tical for large-scale, efficient simulations. To address these challenges, we pro- pose the Temporll Decoupled Differential Equation Network (STDIN), novel framework that guides microscopic trajectory prediction with macroscopic physics. We innovatively introduce the continuity equation from fluid dynamics as strong physical constraint. A Neural Ordinary Differential Equation (Neu- ral ODE) is employed to model the macroscopic density evolution driven by in- dividual movements, thereby physically regularizing the microscopic trajectory prediction model. We design density-velocity coupled dynamic graph learning module to formulate the derivative of the density field within the Neural ODE, effectively mitigating error accumulation. We also propose a differentiable den- sity mapping module to climinate discontinuous gradients caused by discretization and introduce a cross-grid detection module to accurately model the impact of in- dividual cross-grid movements local density changes. The proposed STDDN method has demonstrated significantly superior simulation performance compared state-of-the-ar methods on long-term tasks across four real-world datasets, well as major reduction in inference latency.
collection of independent individual
methods tend to suffer from
INTRODUCTION
Crowd simulation Karamouzas & Overmars 2011 Feng et al. 2016 Mathew al., 2019 Yang et al. 2020 Rasouli 2021) has become cornerstone technology in domains such as public safety management and intelligent transportation systems. holds both significant research value and practical importance for real-world problems, such as crowd evacuation in densely populated areas and passenger flow organization in transportation hubs. Accurate simulation of crowd dynamics not only aids in optimizing spatial design and management strategies but also offers critical decision support for emergency responses to unforeseen events (Yang al. 2020).
Existing mainstream approaches to crowd simulation generally fall into three categories: physics- based methods, data-driven deep learning approaches, and physics-gue -usiiee-guiedd deep learning methods (Rasouli] 2021] Zhang et al. 2022, (Shi et al. 2023] Chen et al. 2024 /Zhou et al. 2024). Physics- based methods simulate local behaviors like avoidance and following by characterizing physical interactions among individuals (Helbing & Molnar 1995 Sarmady al. 2010). These methods are grounded in solid theoretical foundations and offer strong physical interpretability. However, they typically built upon simplified linear mechanies assumptions, making it difficult to capture the nonlinear and stochastic characteristics inherent crowd behaviors. As a result, their performance drops significantly in high-density highly interactive environments.

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With the advancement of deep learning, data-driven methods (e.g., recurrent neural networks CChen et al. 2018), graph neural networks (Xu et al. 2022), and diffusion models (Ho et al.) 2020 have achieved remarkable progress in trajectory prediction by learning motion patterns from scale trajectory datasets. For example, STGCNN (Mohamed et al. (020)) learns trajectory evolution from a diffusion mechanism to estimate the probability distribution of future trajectories. However, the absence of physical constraints in these methods often leads unrealistic behaviors that violate fundamental physical laws, such as unnatural crowd congestion or collisions that ignore avoidance principles.
historical data, while MID (Gu al. 2022)
To bridge the gap between physical consistency and expressive modeling capacity, physics- guided deep learning approaches have emerged. These methods aim to integrate physical models into neu- ral network architectures or training processes as inductive biases (e.g.. PCS (Zhang et al., 2022). SPDiff (Chen et al., 2024)). While improving physical consistency, most focus on microscopic in- teractions and struggle to capture macroscopic density evolution, leading to error accumulation and suboptimal results. Additionally, SPDiff's diffusion-based paradigm suffers from slow inference and high computational cost, limiting its scalability.
Based on these insights, we propose novel Spatio- Temporal Decoupled Differential Equation Net- work (STDDN) for crowd simulation. By incorporating the continuity equation from fluid dynamics as structural constraint, STDDN guides the trajectory prediction process from global perspec- tive of density evolution. In contrast to existing physics-guided methods that primarily rely on individual-level modeling, STDDN treats the crowd as continuous medium and reformulates tra- jectory evolution as a transport process in the density field. It uses neural networks to learn the coupling between the density and velocity fields, enabling macroscopic modeling of collective dy- namics. Meanwhile, STDDN retains the strong representation capability of data-driven methods, enhancing both physical consistency and global interpretability while supporting end-to-end train- ing. Experiments on multiple benchmark datasets demonstrate that the proposed method achieves higher prediction accuracy and greater inference efficiency in long-term simulation tasks, showing strong generalization ability and engineering applicability.
Key contributions of this paper include:
A unified macro-micro coupled modeling framework. We incorporate the continuity equa- tion as a differentiable physical constraint, achieving a tight, end-to-end integration dif- ferential equations with deep neural networks for the task of crowd trajectory prediction. This design significantly enhances the physical consistency and global stability of the sim- ulation.
Physically interpretable design of a dynamic graph network. A cross-timestep dynamic graph neural network is constructed by using current velocity as incoming edges and fu- ture velocity as outgoing edges, enabling explicit modeling of density flux over time and improving the interpretability of the simulation process.
Two differentiable structures for enhanced physical consistency. We propose differen- tiable density mapping module based on radial basis functions and a continuous cross-grid detection module. These designs mitigate gradient discontinuities caused by the discretiza- tion process and ensure both mass conservation and smooth backpropagation.
Superior accuracy and efficiency. Experimental results on four real-world datasets demon- strate that STDDN outperforms existing state-of-the-art methods in long-tern crowd sim- ulation accuracy, while also achieving a substantial increase in inference speed.
2 RELATED WORK
Crowd simulation. As discussed the previous section, there are three main approaches in crowd
simulation research: physics-based methods, data-driven deep learning methods, and physics-guided
deep learning approaches. Physics-based methods were among the earliest frameworks studied (Di-
etrich & Köster 2014 Helbing & Molnar 1995 Pelechano et al. 2007 Corbetta & Toschi 2023)
For example, Social Force Model (SFM) HHeliigg & Molnar 1995) pioneered dynamic modeling of sparse crowds by formulating resultant force system between pedestrians, the environment, and their goals. However, its underlying linear force assumptions exhibit clear limitations in high-density scenarios. To account for speed differences among pedestrians, fine-grid cellular automaton model
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(Sarmady et al. 2010) was proposed to enable more realistic crowd simulations. With advancements in data acquisition and modeling techniques, data-driven deep learning methods have gained wide adoption. PECNet employs Conditional Variational Autoencoder (CVAE) (Ivanovic & Pavone 2019) to generate multiple possible trajectories and samples a plausible endpoint based on observed data. STGCNN (Mohamed al. 2020) leverages graph neural networks to model interactions with surrounding pedestrians at each time step. MID (Gu et al., (022) directly models the distribution of future trajectories using diffusion model and designs Transformer-based architecture cap- ture temporal dependencies in historical data. Although these methods achieve good performance in trajectory prediction, their lack of physical constraints limits generalization under extreme or un- foreseen scenarios, often resulting in physically implausible behaviors. In recent years, researchers have begun exploring physics- guiiddd deep learning methods for crowd simulation (Kochkov et al.) 2021). PCS (Zhang et al. 2022) incorporates the SFM as a prior and jointly trains two branches: one data-driven and one physics- through an interaction mechanism to enable information sharing. NSP (Yue et al. 2022) explicitly embeds physical parameters into the neural network to
enhance interpretability. SPDiff (Chen et al.] (004) integrates the SFM with a diffusion model, building physics-aware denoising network that has shown strong performance in crowd simulation tasks. However, since the SFM focuses solely on individual-level microscopic interactions, these methods suffer from error accumulation over time, especially in iterative simulations. Additionally, because crowd simulation often relies on autoregressive prediction combined with diffusion-bar based generation, inference becomes computationally intensive. To address these limitations, we propose a novel framework that integrates macroscopic physical laws, enabling simulation to be completed in single forward pass. This significantly improves both efficiency and stability.
Physics informed models. In recent years, the integration of physical principles into neural net- works has gained considerable attention in spatiotemporal prediction (Kochkov et al., 2021 Karni- adakis et al. 2021; Fang et al., [02T]] Luo et al. 2023] Chen et al., [023] Zou Guol 2024). For traffic flow forecasting, STDEN (Ji et al. (022) proposed a deep learning framework based on the continuity equation, embedding traffic flow dynamics into neural networks to bridge the gap between data-driven and physics- based models. ST-PEF (Ji et al., 2020) introduced spatiotemporal potential energy fields to simulate traffic flow, and its subsequent work, ST-PEF+ (Wang et al., 2022). fur- ther incorporated the field theory of human movement. AirPhyNet (Hettige et al. (2024) modeled particle diffusion and convection processes as differential equation networks, while Air-DualODE Tian al. 2024) integrated data-driven and phyeedd-e methods, achieving advanced per- formance in air quality prediction. In trajectory prediction, neural differential equations have been widely applied to model agent dynamics. TrajODE (Liang al.) 2021) employ Neural ODEs to model continuous-time dynamics in trajectory data, Social ODE (Wen et al. 2022) utilized neural ordinary differential equations to characterize continuous trajectory evolution, NSDE (Park et al., 2024) proposed neural stochastic differential equations to enhance cross-domain generalization, and Social LODE (ee et al., 2024) introduced latent ordinary differential equations to overcome the lim- itations of RNNs in long-sequence modeling. Inspired by these advances, we integrate continuity equations with crowd simulation by employing graph structures to connect historical trajectories with future predictions. This approach effectively decouples the spatial and temporal dimensions of trajectory forecasting, addressing key challenges in the field.
3 PRELIMINARIES
3.1 PROBLEM FORMULATION
We consider a group of M pedestrians moving in an environment with S static obstacles. At each time step I, the state of the crowd is denoted as Q' = vr',"',a a', d, h² E,E, where p', v², and a' E MR2 represent the positions, velocities, and accelerations of all pedestrians, respectively; d E Mx2 denotes their destinations; h' = (p' -h:t, y aa h-t) represents the recent motion history over window of h frames; and E Rxx contains the positions of static obstacles in the environment. Crowd simulation is formulated as a spatiotemporal prediction problem, where the goal is to model the future evolution of individual trajectories over time and space. The model fo initializes from the initial state and generates the next moment's state by entering the current state, i.e.,
Q++1 So(Q', F.)
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which is continuously iterated until all individuals in the crowd reach their respective destinations, completing the simulation process. To ensure physical consistency, we predict acceleration a at each time step, and update the position and velocity using the standard integration rules: vf of + At and p'+1 p² v² At. The Appendix A.1 provides the domains and descriptions of all symbols used in the paper.
3.2 CONTINUITY EQUATION
The continuity equation (Lienhard & Lienhard 2008 Grady al., 2010) one of the fundamental conservation equations in fluid mechanics, describing the conservation of mass during flow. Its differential form expressed as:
0p Ot
+ (pv) =0
where p represents the fluid's density field, denoting the mass density at spatial position (2r, y) and time 4; represents the fluid's velocity field at spatial position (æ, y) and time I; V (pv) denotes the divergence of the mass flux In spatiotemporal modeling, the continuity equation naturally enables decoupling of time and space: the temporal evolution is governed by the time derivative of density, while the spatial transport is guided by the velocity field.
4 METHOD
4.1 THE MODEL FRAMEWORK
We propose the Spatio-Temporal Decoupled Differential Network (STDDN), physically consis- tent framework designed to unify the modeling of the coupling between microscopic motion and macroscopic evolution. The core idea of STDDN is to leverage the continuity equation (Eq. 2 from fluid dynamics as a strong physical prior. To this end, we employ a Neural ODE to model the evolution of the macroscopic density field driven by individual movements. To compute the time derivative required for this evolution, we innovatively embed a microscopic trajectory prediction network within whose predictions drive the dynamic changes in the density field. This design enables macroscopic physical laws to impose physical regularization on the microscopic trajectory prediction in an end-to-end manner, thereby significantly enhancing the simulation's 's stability and physical consistency. The overall framework is illustrated in Figure
ODESolver(p (e FFg. P, 10:00 :))
FolDi,
3p
Macroscopic
p
dt
Gomel (.))))
GNN
NE
GNN
DVCG
DVCG
DVCG
DDM
CGD
Microscopic
fo
So
( P.d))
(p..v.))
Figure 1: Overview of the STDDN framework, consisting of microscopic trajectory prediction network (bottom left), neural ODE-based macroscopic density evolution module (top left), and the DVCG module (right), which connects microscopic trajectories with macroscopic density and velocity fields. The DVCG module includes three components-
CGD, and NE- combindd in dynamic Graph Neural Network (GNN) to model density evolution. These components build the inflow term G. ( f, p¹) and outflow term Goue (2, t, p²), which are combined via Eq 4 to form the derivatives of the density field. Colored arrows show velocity vectors at different time steps.
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We design Density- Coupled Graph Learning (DVCG) module as the core evolution fune- tion of the neural ODE. This module takes as input the density and velocity fields at adjacent time frames, and uses dynamic graph neural network to predict the temporal rate of change of the density field, therehy driving its continuous evolution over time. Specifically, the DVCG module consists of three key submodules: Differentiable Density Mapping (DDM) module, a Continuous Grid Detection (CGD) module, and a Grid Node Embedding (NE) mechanism. These components jointly estimate the temporal density derivatives while maintaining physical consistency and smooth gradient flow during training.
Furthermore, STDDN enables joint training by allowing macroscopic density predictions to guide and refine the microscopic trajectory prediction module. During the inference stage, the model utilizes the trained single-step trajectory prediction network to perform autoregressive inference ac- cording to Eq. 1 generating future trajectories step by step, thus enabling high-precision simulation of crowd dynamics. It is important note that during inference, we only require the trained model fe and do not rely on the ODE part.
4.2 DENSITY-VELOCCH COUPLED GRAPH LEARNING MODULE
Inspired by Chapman 2015), we discretize the spatial region into regular grids and use them to construct graph structure. Leveraging the trajectory information of individual pedestrians, propose an innovative density-velocity coupled dynamic graph learning module (DVCG), designed to efficiently compute the temporal derivative of the macroscopic density field. Specifically, the velocities of all individuals at the current time step form the set of incoming edges for the graph nodes, while the predicted velocities the next time step constitute the outgoing edges. The flux each grid node is determined jointly by the pedestrian velocities and the current node density. This design transforms the trajectory prediction task into spatiotemporal density transport optimization problem, offering stronger physical interpretabilityy. In discrete local spatiotemporal field, the density evolution o of spatial grid node and its connected nodes j and which are linked by the velocity vectors Vjz and Vik, at time can be expressed as:
Op: dt
(mensyllvyallo + bye) (V)6ー>1
Σ
Σ
(me+1waal t+1
O)lee O)l (vlc:) 0) + bak)
The above flux operations can compactly represented in matrix form as:
Op
di Fo(0, G.. G..( ('),p') Gove ('),p"))
Ci, (") (m, A W o V A' о B)
(4)
(o, p't C (1,m, A ++ W VV +V B)²)
Here, Fg denotes the time derivative of the density p. & represents all learnable parameters, includ- ing the weight matrix W, the bias matrix B, and 0. The terms vv'' 8) and o² (p': refer to the predicted velocity and density the next time step, respectively, as generated by neural net- work fa. The operator @ denotes element-wing wise (Hadamard) multiplication. The cross- -gid masks my and m are provided by the CGD module. The matrices A' and A++1 represent the dynamic connectivity matrices at time and & + 1, respectively, and the terms Wyj and Wikk are the learnable parameters W. while by: and bu are the learnable parameters in B. Theoretically. fo can be any existing microscopic prediction model. However, to ensure a fair comparison, we adopt the same network architecture for fo as in Chen et al. (2024). (More details about the neural network are provided in the Appendix(A.2] Since individual velocities vary over time, the constructed graph ex- hibits dynamic evolution characteristics. Starting from the initial density po, we use an ODE solver to compute the future density sequence over the time steps, denoted as p4:T;
p Jir ODESolver(p) Fg. [0: 7])
(5)
More details about the ODE solver are provided in the Appendix A.3
4.3 DIFFERENTIABLE DENSITY MAPPING MODULE
The Differentiable Density Mapping (DDM) module maps continuous individual positions onto a discrete spatial grid to produce density values. Traditional hard assignment methods cause discon-
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tinuous gradients at grid boundaries, hindering model optimization. To address this, adopt probability- soft assignment strategy. Specifically, we first compute the coordinates of each grid center ç, forming the matrix D [Cp, C c CN-il E RIXN For any predicted position pt. we calculate its squared Euclidean distances to all grid centers:
d(p') = [|lp' - coll", Ipp' - c ... lll' - -N-11l]
These distances are then converted into a probability distribution using a -emperature-contrlledd softmax function:
exp(-3 1p -,,1²) دコ=
q: (pp')
(6)
N- exp(-2 3 1pp' e2l²)
where 3 is temperature parameter that controls the smoothness of the distribution.
Finally, by summing the position distributions of all K individuals, obtain continuous and differentiable density representation:
K
((p')
4.4 CONTINUOUS CROSS-GRID DETECTION MODULE
crowd movement simulation, net flux arises only from trajectories crossing grid boundaries. Tra- ditional discrete methods for statistical detection cause gradient discontinuities, hindering end-to- end training. address this, we first obtain a continuous spatial distribution via differentiable position-to- grid probability mapping, then quantify the extent of grid crossing by computing the divergence between probability distributions at consecutive time steps:
I ((llr")a(e 1 [KL(a(p)), M) + (C)(
where K.C denotes the Kullback-Leible divergence, and M (g(p') + 1))²+ represents the mean distribution. The divergence values are then transformed into differentiable cross-grid masks using a temperature-controlles sigmoid activation function:
m aa((T r)) [0.01, 0.99)
where a-)) is sigmoid activation, a a scaling factor and r is a threshold parameter, This formula- tion maintains gradient continuity while preserving physical constraints.
4.5 NODE EMBEDDING
The granularity of grid partitioning affects how strongly the continuity equation constrains trajectory prediction: finer grids capture more detailed cross-cell movements but incur significant memory overhead, as traditional weight matrices scale with ( To address this, we propose lightweight node embedding representation. Each grid node is associated with learnable embedding and bias vectors: w [wo:, W', ... wN-1 E Nxx, = [b. bi.... -N=
[ E R The weight and
bias matrices are then dynamically constructed via outer products: W wwT B = bb? This design reduces the storage complexity to O dd), significantly improving memory efficiency while preserving modeling capacity.
4.6 TRAINING
By coupling the trajectory prediction network with the differential equation, the training objective jointly supervises both velocity and density through combined loss function:
Ljoint = INN lODE = X ||v voll + >ellp poll
(10)
where X₁ and ' balancing coefficients that control the relative importance of velocity prediction accuracy and density evolution consistency, respectively. The full training procedure provided as pseudocode in the Appendix A.4

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5 EXPERIMENTS
5.1 EXPERIMENTAL SETUP
Datasets. We evaluate our framework on four open-source trajectory datasets: GC (Zhngg et al.) 2022), UCY (Lerner et al. 2007) which includes three sub-scenes (ZARAI, ZARA2, and UCY), and ETH PPellegrini et al. 2009), which includes two sub-scenes (ETH and HOTEL). These datasets span diverse scene types, spatial scales, time durations, and pedestrian densities, providing a com- prehensive benchmark for assessing the simulation capabilities of our framework. Following the approach of SPDiff (Chen et al. 2024), we select dense and long-duration segments for GC and UCY (300s and 216s respectively), focusing regions with over 200 pedestrians per minute. For ETH and HOTEL, following the approach of STGCNN (Mohamed et al., ( we adopt the original training/test splits. Further preprocessing details, including coordinate transformation and temporal interpolation, are provided in the Appendix A.5
Baselines. We divide the baseline methods into physics-based data-driven, and physics-guided methods. Within the physics-based methods, we choose the widely used Social Force Model (SFM) (Helbing & Molnar 1995) and Cellular Automaton (CA) (Sarmady et al. 2010) for comparison. Within the data-driven methods, we select three representative approaches recently published, in- cluding STGCNN (Mohamed al. 2020) which utilizes graph convolutional neural networks to compute a spatiotemporal embedding. PFCNet (Mangalam et al. (020)) which uses VAE to sam- ple multi-modal endpoints and MID (Gu et al.) 2022) which is based on the diffusion framework to model indeterminacy. For physics-guided methods, we select PCS ZZang et al. (022). whose backbone graph networks, NSP (Yue et al. 2022), based on sequence prediction models combined with CVAE, SPDiff (Chen et al. 2024) which employs a diffusion based model for next time step prediction model.
Metrics. We employ four evaluation metrics to comprehensively assess the model's performance and inference efficiency. For trajectory accuracy, Mean Absolute Error (MAE) and Optimal Trans- port (OT) (Villani (02T) distance are used to measure point-to-point error and trajectory shape similarity (Sanchez-Goozalezz al., 2020; Gretton et al., 20022), respectively. These are widely used statistical metrics in physical process modeling. For inference efficiency, we adopt number of pa- rameters (#Pars) and single-frame inference latency (Latency, measured in ms). Detailed definitions and implementation specifics are provided in the Appendix A.6 along with comparative results for additional metrics in the Appendix A.7
5.2 OVERALL PERFORMANCE
To substantiate the effectiveness of our proposed framework, experiments were executed across four large-scale crowd simulation datasets, followed by systematic comparison against existing crowd simulation methods. The empirical findings (presented in Table [ and Table 2 more results can be found in the Appendix A.7 unequivocally demonstrate STDDN's substantial advantages over the second-best method, SPDiff, across all evaluation settings. Specifically: for the GC dataset, inference duration was reduced by 50%, concurrently yielding MAE and OT metric enhancements of 2.6% and 2.46%; on the UCY dataset, inference time saw 90% reduction, with MAE and OT metrics boosting by an impressive 5.39% and 10.01 %; on the ETH dataset, inference time decreased by 50%, alongside accuracy gains of 6.0% and 19.81% %; and for the HOTEL dataset, inference time experienced a 75% reduction, while MAE and OT metrics consistently improved by 12.66% and 12.21%. Specifically, our observations are as follows: Firstly, in comparing pure data-driven meth- ods with purely physical model approaches, STDDN achieves a superior balance within the network by more effectively integrating the two paradigms. not only assimilates the objective laws in- herent in physical models but also captures general pedestrian movement regularities. Secondly, while both SPDiff and MID utilize diffusion models to condition future trajectories on historical data, STDDN attains superior performance merely by constraining its model parameters via the continuity equation. Furthermore, single-frame frame simulation with diffusion models necessitates mul- tiple forward passes, whereas STDDN requires only a single pass, leading to significantly reduced inference time and a considerable efficiency advantage over diffusion models. Lastly, despite the significant variations in pedestrian speeds observed in the ETH and HOTEL datasets, method
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Table I: Overall performance comparison on GC and UCY datasets. The bold and underlined font show the best and the second best result respectively. Performance averaged over runs.
GC
UCY
Model
MAE]
OT
#Pars}
Latency
MAE1
OT
#Pars
Latency4
CA
2.7080
5,4990
8.3360
79.4200
SFM
1.2590
2.1140
2.5390
6,5710
STGCNN
8.1608
15,8372
7.56K
8.0539
7.5121
18.7721
7.56K
8.0539
PECNet
2.0669
4.3054
2.10M
51.3781
3.7694
16.1412
1.23M
49.5681
MID
8,4257
35.1797
3.52M
245.7762
8.2915
47.8711
2.47M
276.8964
PCS
1.0320
1.5963
0,97M
30.4784
2.3134
6,2336
0.62M
21.2818
NSP
0.9884
1.4893
1.93M
60.7866
2.4006
6,3795
2.14M
31.3471
SPDiff 0.9116
1.3925
1.14M
206.9909
1.8760
4.0564
0.22M
471.0496
Ours 0.8875
1.3582
0.17M
86.8537
1.7747
3.6503
0.07M
44.6565
consistently achieved better performance, thereby further validating the efficacy of the continuity equation constraint. Visualization of generated trajectories are provided in the Appendix AA.9
Table 2: Overall performance comparison on ETH and HOTEL datasets. The hold and underlined font show the best and the second best result respectively. Performance averaged over runs.
ETH
HOTEL
Model
MAET
OT
#Pars
Latency
MAEJ
OT
#Pars
Lateney
CA
0.7211
2.5318
0.5328
0.3102
SFM
0.6933
1.1903
0.4977
0.2371
STGCNN
3,8711
15,3766
7.56K
8.0539
5.6711
7.3851
7.56K
8.0539
PECNet
0.6533
0.9712
2.10M
48.3381
0.4592
0.2380
1.93M
31.8613
MID
4.1823
16.8301
3.79M
73.6639
4.3867
23.7882
2.86M
60.6521
PCS
0.6573
1.0121
0.62M
33.2728
0.4278
0.2068
0.88M
13.5936
NSP
0.6343
0.9372
1.57M
40.4512
0.3972
O. 1919
1.86M
21.3488
SPDiff
0.5527
0.8706
0.18M
81.4144
0.3380
0.1646
0.16M
68.5705
Ours
0.5185
0.6918
0.20M
30.5723
0.2952
0.1445
0.05M
17.4986
5.3
ACCUMULATED ERROR ANALYSIS THE SIMULATION
To further investigate the enror accumulation behavior of our method in long-term prediction, analyze the temporal evolution of MAE and OT metrics on the GC and UCY datasets, and compare the results with two crowd simulation baselines: SPDiff and PCS. As shown in Figure both our method and SPDiff exhibit a "rise-then-fall" trend over time. However, our method consistently achieves the lowest overall error, indicating that it is the least affected by error accumulation during long-term prediction. This performance advantage is primarily attributed to the incorporation of macroscopic physical constraints into our model, which guide it toward globally optimal solutions and effectively suppress error propagation over time. To more comprehensively evaluate our model's long-term density prediction capability, we provide additional analysis of ane-accumulated density prediction errors in the supplementary materials.accummllatie of density prediction errors in the Appendix A.10
5.4 ABLATION STUDY
To further investigate the contributions of key components in STDDN, we conduct an ablation study on the GC and UCY datasets. Specifically, we evaluate the following variants: w/o ODE: The neural ODE constraint is removed, and the model is trained solely via purely autoregressive scheme for trajectory prediction. w/o Cross-net: The Continuous Grid-Crost Crossing Detection (CGD) module is removed. Since approach is flux-based, the absence of this module significantly weakens
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the physical enforcement of the continuity equation. w/o NN loss: The trajectory autoregressive training objective discarded; the model is optimized only using the Sensity-field-Be ODE loss. w/o NE: Learnall node embeddings (Node Embedding) and bias terms removed. Trans: The dynamically constructed adjacency structure-deriee from predicted trajectories- replaced with static attention-based learned weight matrix. Dopri5 RK4: The default Euler solver is replaced with higher-order ODE solvers, namely Dormand-Pring 5 (Dopri5) and Runge- Kutta (RK4), respectively. Discrete NN: The neural ODE solver is replaced by an explicit first-order discrete update implemented via residual connection: p²+1 p + At F& ( t. p²), which corresponds to the forward Euler discretization of the continuity equation: de Fo ( I, p').
neural
d
Table 3: Ablation study on different parts of model design.
The experimental results are pre- sented in Table leading to the fol- lowing conclusions: First, the signif- icant performance drop in w/o ODE demonstrates that incorporating the continuity equation as a physical con- straint effectively mitigates the accu- mulation of errors inherent in purely autoregressive models during long- horizon crowd simulation, thereby validating the necessity of the differ- ential equation framework. Second, substantial degradation observed in w/o Cross-net indicates that accu- rately modeling the flux generated as agents cross grid boundaries is cru-
GC
UCY
MAE
OT
MAE
OT
w/o ODE
1,3784
2,4956
2.4867
6.0586
w/o Cross-net net
0.9784
1,4732
1.8926
4.9532
w/o NN loss
1.2387
2.3466
1.9327
4.2514
w/o NE
0.8921
1.3881
1.7917
3.7131
Trans
0.8901
1.3611
1.7833
3.7055
Dopri5
1.1315
1.7422
1.9654
5,2318
RK4
1,2311
1.8625
2.0381
5,4581
Discrete NN
0.8875
1.3582
1.7747
3,6503
Ours
0.8875
1.3582
1.7747
3.6503
cial for enforcing mass conservation, further confirming the effectiveness of the continuity con- straint. Third, the results of w/o NN loss reveal that model driven solely by physical laws struggles to capture the complex spatiotemporal dynamics of real pedestrian motion, highlighting the syner- gistic advantage of combining data-driven learning with physical priors. Moreover, comparisons dynamic graph structure, together with
between w/o NE and Trans show that the trajectory-drive learnable node embeddings, more effectively encodes spatial relationships and interaction seman- ties than static attention mechanisms. Notably, although Dopri5 and RK4 offer higher theoretical numerical accuracy, they lead to performance degradation solvers introduce intermediate interpolated states that do not align with the actual observation time steps, increasing computational overhead and raising the risk of overfitting. In contrast, the Euler solver naturally aligns with autoregressive-styye discrete-time modeling, achieving an optimal trade-off between computational efficiency and simulation accuracy. Finally, Discrete NN achieves performance comparable to our full method, indicating that our approach fundamentally models inter-framm crowd flow transitions on a discrete spatiotemporal grid, rather than a smooth, continu- dynamical process.
our task. This because higher-order
5.5 SENSITIVITY STUDY
We conducted a sensitivity analysis on the key hyperparameters of our framework, including grid size. ODE time steps т. loss balancing coefficients A and X2, and node embedding dimensions. The MAE results of GC dataset are shown in Figure 3 (More results of sensitivity analysis on dif- ferent datasets can be found in the Appendix (A.II. The results show that variations in grid size, ODE time steps, and embedding dimensions all have a certain impact on the final prediction per- formance, which further demonstrates the effeetiveness of the continuity equation as a physical constraint within the model. Additionally, while different values of the loss balancing coefficients lead to varying results, setting the two terms with comparable weights enables a better trade-off between data-driven learning and physical modeling, thereby achieving improved predictive accu- racy and highlighting the value of incorporating physical constraints into autoregressive prediction models.

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052
MAVE
1.3
PCS
05
KSP
0.0
Texchan:
Grall Sler
ODE: Sinee Sige
ICY
DCC
02-01
SIDON
0.00
230
X57
200
46
68
20
Nede Inbhodia Disucaloo
Figure 2: Accumulate error of the simulation, using MAE and OT as metrics.
Figure 3: Sensitivity analysis the GC dataset for grid size, ODE time steps т, loss balancing coefficients (A1 and X2), and node embedding dimension, using MAE as the evaluation met- ric.
6 CONCLUSION
This paper proposes STDDN, novel crowd simulation framework that integrates macroscopic phys- ical constraints with deep learning. By leveraging the continuity equation, STDDN systematically models the coupled evolution of crowd density and velocity fields, effectively mitigating error ac- cumulation and instability commonly seen in microscopic approaches. Through the design of dif- ferentiable density mapping module, continuous cross-grid detection module, and grid node em- bedding mechanism, STDDN enables high-precision, end-to-end simulation. Experimental results demonstrate that our method significantly outperforms existing mainstream methods on long-term trajectory simulation tasks across multiple real-world datasets, achieving not only superior perfor- mance but also a substantial reduction in inference time. This method offers a physics-consistedt and interpretable deep learning paradigm for crowd modeling, showing strong potential for practical applications.
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A
APPENDIX
A.I NOTATIONS
An overview the notations used in this paper. along with their domains and descriptions, is pro-
vided in Table
Table 4: List of major symbols and descriptions.
Sym.
Descriptions
K
R RN
Number of pedestrian
Number of nodes
p²
Density of nodes at time
me
R
cross cross-grid -rid masks of K pedestrianat time
A
{0, JJNNN
Dynamic adjacency matrix
W
RAXN
Learnable weight matrix Learnable bias matrix Velocity tensor matrix Velocity magnitude matrix
RNXN
Nxx22
|
RNXN
Gin
RIXN
The inflow vector of all nodes The outflow vector of all nodes All-ones row vector
Gout
RIXN
{1]IXN
A.2 NEXT TIMEFRAME TRAJECTORY PREDICTION MODEL
To ensure the fairness of experimental comparisons, we adopt the same denoising network archi- tecture used in the sub-optimal baseline SPDiff, with all diffusion step embeddings fixed to O. The overall network architecture, as illustrated in the Figure 4 consists of three key representation learning components: encoding of historical trajectories, modeling of interactions with surrounding pedestrians, and attraction modeling toward the destination.
The network employs the Equivariant Graph Convolution Layer (EGCL) module for message pass- ing. Its computation logic is as follows: At the layer, the node embedding h' position embedding p', velocity embedding v' and acceleration embedding a' are used as inputs compute the updated embeddings 1/+1, p1+1, 111 11 and |+1 The update process consists the following steps:
my = @h h1y llp! - 11)),
(11)
o 1+1 da (h()))
Σ
dig (p) (mis)
(12)
SEN(1)
J/ = + 0
1+1 .1
1+1 = 1/1 + vtt1
(13)
m4 = Σ
Mijs 1111 = on (hi. m.)
(14)
JEN())
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Po(output)
f m
f ped
Crowd Interaction Module
hist
MLP
EGCL 8
History Module
LSTM
yk
EGCL EGCL
c~N(0,1)
K
Linear
Traning
Yo
at+1
Ct
Figure 4: The detailed of next trajectory prediction model.
A.3 ODESOLVER
In our experiments, we adopt the Euler method the numerical solver for ODEs. The solver is configured with relative tolerance (rtol) of le-4 and absolute tolerance (atol) of le-3. The crowd trajectory and density data are sampled from fixed-frame-rad videos, resulting discretized time steps with consistent intervals. The single-step update mechanism of the Fuler method allows pre- cise alignment with each frame's 's timestamp, eliminating the need for multiple sub-steps within single time interval and avoiding inconsistencies with the actual temporal resolution. Since the Eu- ler method requires only one forward computation per frame, it offers low computational complexity and minimal memory overhead, making it well-suited for long-duration, multi-scent and large-scale crowd simulation tasks. Moreover, neural ODEs are easily implemented using the torchdiffeq pack- age.
A.4 TRAINING ALGORITHM PSEUDOCODE
We provide the pseudocode for the training algorithm, shown in Algorithm T
Algorithm Train
1:
Trajectory prediction model fo
2: while not converged do
3: Draw (p"rr 0. alrr) from dataset
4: Compute initial crowd density n by (q. and (q.
5: for E [0: 7] do
6:
Predict next trajectory p**1 by fe
7:
Execute continuous cross-grid detection module by E8.(8) and (9.
8: Compute density net flux by Eq.(4)
9: Execute the single-step ODE solver in Eq.(5)
10: end for
Compute joint loss Ijoins by (q.
12:
end while
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Table 5: The basic statistics the datasets.
Statisties
GC
UCY
ETH
HOTEL
ZARAI
ZARA2
Average duration of trajectory(s)
11.02
13.25
6.10
6.72
13.92
19.06
Average pedestrian per minute (min
203
132
49
35
44
Pedestrian density (m 3
0.094
0.058
0.016
0.012
0.149
0.241
Average speed (m s-1)
1.155
1.072
2.293
1.038
1.071
0.793
Std of the average speed (m 8 1)
0.565
0.646
0.807
0.695
0.134
0.349
simulate time steps
725
651
5776
9006
4481
5231
A.5 DATASET DETAILS
To evaluate the effectiveness of our proposed model, we conduct experiments on six real-world pedestrian trajectory datasets constructed from surveillance video recordings. These datasets differ in scene complexity, crowd density, and temporal resolution. The key statistics are summarized in Table 5 Detailed descriptions are as follows:
GC. The GC dataset contains 12680 annotated trajectories within an image coordinate system cov- ering approximately 30x35 meters. We select 300-sccond subset within a 20x20 meter area that includes rich pedestrian interactions. The original sampling rate is 1.25 Hz (Ár 0.8s). To en- hance temporal resolution and reduce interpolation error, we apply cubic interpolation using SciPy, resulting in a new timestep of 0.08 s. The coordinates are converted from image space to world coordinates using provided mapping.
UCY. The original UCY dataset includes three sub-scenes: ZARAI, ZARA2, and UCY. The ZARAI scene contains 148 pedestrians over 360 seconds and covers a 16x12 meter area. The ZARA2 scene contains 204 pedestrians over 420 seconds and covers a 16x12 meter area. The UCY scene contains 528 pedestrians over 216 seconds and covers a 23x25 meter area. Trajectories in three scenes converted world coordinates and interpolated three times to reduce the timestep from 0.4 s to 0.08 s, improving temporal continuity and simulation accuracy.
ETH. The ETH dataset contains two sub-scenes: ETH and HOTEL. The ETH seconds with 360 pedestrians in an 80x20 meter space, while the HOTEL scene spans 692 seconds with 389 pedestrians
spans 464
50x30 meters. We follow the original train/test splits provided in prior work. Both scenes undergo coordinate transformation and cubic interpolation to unify the timestep 0.08 s. consistent with the GC and UCY datasets.
We divide each dataset into training and testing sets based on time. For the GC dataset, the training- to-testing ratio is set 4:1, while for the UCY, ZARAI and ZARA2 datasets, it is 3:1. For ETH and HOTEL, the original training and validation sets are merged into single training set, and evaluation performed on the original test set.
A.6 METRICS
MAE: The MAE measures the average magnitude of errors between predicted and true positions, and is defined as:
N MAE = Σ pp N
1
pl
(15)
where N is the total number of prediction instances, p denotes the predicted position, is the corre- sponding ground-truth position, and || denotes the matrix norm.
OT: quantifies the transportation cost between two distributions P and Q, which can be interpreted as the distance between the predicted and ground-truth -tunh trajectory distributions:
(
OT(PIIQ) infz
()c,y, vydddy,
XXY
(16)
T(a, P(x)), / (d)
s.t.
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where x(2, y) is the transportation plan approximated from P(x) and Q(y) using the Sinkhorn algorithm, and c(x, y) = || yll The domain X = Y corresponds to the simulation duration, while P and denote the predicted and ground-truth trajectory distributions.
MMD: It mcasures the difference between two distributions by comparing the mean embeddings in a reproducing kernel Hilbert space (RKHS). It is often used assess the similarity between predicted and real trajectory distributions:
MMD
- Exy-ol((v))l
(17)
where P and Q represent the predicted and ground-truth trajectory distributions, and Φ(.) denotes the mapping to the high-dimensional feature space induced by kernel. In this work, we use the Gaussian kernel to compute the empirical MMD.
DTW: It measures the similarity between two trajectory sequences by aligning them non-linearly along the time axis, allowing for time warping. It is defined as:
(X)
min WEW
E |z, 3/1
(18)
(U)E
where X = (.1,,..., Fn) and Y (y1...,, Um) are two trajectory sequences, W the set of all possible alignment paths, and || denotes the Euclidean distance. DTW enables alignment of trajectories with varying speeds, thus better capturing their shape similarity.
FDE: It measures the distance between the predicted final destination and the ground-truth final destination at the end of the prediction horizon. is defined as:
FDE
= N
Σ 1 MEN
P 12,1,,-111
(19)
where Pa is the predicted final position the last prediction timestep Tp. and pt is the corresponding ground-truth final position.
#Colli: We count two pedestrians with a distance less than 0.5m as a collision and take the sum- mation of collisions in all frames as Collision. But considering that pedestrians could walk with their friends, to whom they won't keep a large social distance, We take the pair of pedestrians that have a collision in more than 2 seconds as friends and do not count the collisions between them into Collision.
DEA: To assess whether the model generates unrealistically dense local clusters, we define the local density of pedestrian at time as:
df Σ U(II p5l12 R)
(20)
j=1,f
.iee., the number of neighbors within radius R meter. Using the mean local density from ground-truth trajectories, μGT, as threshold, we compute:
N T
1
DEA N
I(d MGT)
(21)
t=1
#Pars: This metric measures the total number of learnalee parameters (weights and biases) in the model, directly reflecting the model's static size and storage complexity. A lower parameter count indicates more lightweight model, making it easier deploy. The unit is typically in thousands (K) or millions (M).
Latency: This metric measures the actual time required for the model to complete single-frame prediction on a specific hardware platform. It is a hardware-dependent metric that directly reflects the real running speed of the model. All latency tests in this paper were conducted NVIDIA RTX 4090, with the unit in milliseconds (ms).
GFLOPs: This metric measures the theoretical total number of floating-poin operations required for the model to generate the next frame from its current state. is a hardware- wtwpwwaar--deeeddeeent metric that fairly reflects the complexity of the model's algorithm itself.
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This metric measures the number of simulation frames (Frames Per Second) that the model can generate per second in continuous autoregressive mode. It directly evaluates the model's 's throughput and performance in waal-world applications, serving as the gold standard for assessing whether the simulation system can meet real-time requirements.
FPS:
A.7 ADDITIONAL, EXPERIMENTAL RESULTT
To gain deeper insights into the strengths of our method, we extend the evaluation beyond standard metrics and analyze its performance in terms of trajectory shape fidelity, inference efficiency, and physical plausibility across six benchmark datasets: GC, UCY, ETH, HOTEL, ZARAI and ZARA2.
We first report on key trajectory quality metries: Maximum Mean Discrepancy (MMD) and Dynamic Time Warping (DTW). MMD is employed to assess the macroscopic distributional consis- tency between the predicted and ground-truth trajectories, while DTW excels at measuring the mi- croscopic shape similarity between two trajectories and is robust to non-linear temporal variations. These two metrics serve as powerful complement the main text, offering a more holistic view of the prediction fidelity. Subsequently, we conduct an in-depth comparison from the perspective of model efficiency, reporting two core metrics: the num- ber of floating-poin operations for single forward pass (GFLOPs) and the continuous simulation frame rate (FPS). As a hardware-agnostic metric, GFLOPs reflects the theoretical computational complexity of the model's algorithm itself. In contrast, FPS directly measures the model's practical throughput and real-tin ttmme performance on specific hardware.
the distance-based MAE and OT metries in
As shown in Table 6 and Table 7 our method not only performs exceptionally the trajectory quality metrics but also demonstrates a commanding advantage in runtime efficiency. These results further validate the superiority of our approach and its significant potential for practical applications.
Additionally, we evaluate the physical realism of our method using three metrics Final Displace- ment Error (FDE), collision count (#Colli), and Density Estimation Accuracy (DEA)-across across six datasets (GC, UCY, ETH, HOTEL, ZARAI and ZARA2); the results are reported in Tables 8| and TO respectively. Across all datasets and metries, our method consistently achieves the best performance, demonstrating superior physical plausibility, fewer inter-agent collisions, and more accurate crowd density estimation compared to all basclines.
Table 6: Additional performance comparison on GC and UCY datasets. The bold and underlined font show the best and the second best result respectively. Performance averaged over 5 runs.
GC
UCY DTW4
Model
MMD4
DTW4
GFLOPs|
FPSt
MMDJ
GFLOPs|
FPSt
CA
0.0620
2.0220
SFM
0,0150
0.1209
STGCNN
0.5296
5.1438
0.0002
124.163
0.5149
5.1695
0.0002
124.163
PECNet
0.0397
0.7431
3,4704
18.9913
0.1504
2.0986
2.7134
19.0032
MID
0.3737
4.2773
0.4537
3.7826
0.4384
4.7109
0.3871
4.7312
PCS
0.0126
0.4378
0.8242
32.8106
0.1070
0.9887
0.2442
46,9936
NSP
0.0106
0.3329
0.2176
17.3792
0.1199
0.9965
0.2121
32,8713
SPDiff
0.0092
0.3332
0.1522
4.8315
0.0671
0.7541
0. 0.1261
2.1247
Ours
0.0083
0.3441
0.1994
22.1406
0.0627
0.6905
0.0472
28.3033
A.8 COMPUTATIONAL COST
We provide a comparative analysis of our method against major baseline approaches crowd sim- ulation, focusing on three key metrics: training time, GPU memory consumption, and batch size, as summarized in the Table II Our method indeed incurs higher training costs due to: the node embedding matrix, which remains memory-intensiv even under small batch sizes and the temporal coupling between consecutive frames, which requires storing intermediate tensors for flux compu- tation and backpropagation through the ODE solver. However, this design is intentional: it
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Table Additional performance comparison on ETH and HOTEL datasets. The bold and underlined font show the best and the second best result respectively. Performance averaged over runs.
ETH
HOTEL
Model
MMD4
DTW4 GFLOPs]
FPSt
MMDJ
DTW4
GFLOPs).
FPSt
CA
0,4329
0.3548
SFM
0.3788
0.2301
STGCNN
3,9678
1.5622
0.0002
124.163
0.7418
3.7811
0.0002
124.163
PECNet
0.3573
0.3571
3,4704
19.9936
0.2219
0.2415
3.1927
30.3817
MID
3.7518
1.9512
0.5618
11.7528
0.6957
3.4641
0.4914
17.3411
PCS
0.3577
0.3101
0.3378
30.0575
0.1902
0.2201
0.2098
76.8331
NSP
0.3365
0.2843
0.2863
18.3782
0.1766
0.1883
0.1452
43.1812
SPDiff 0.2221
0.2275
0.1753
12,2830
0.1090
0.1444
0.0586
14.5837
Ours
0.1672
0.2221
0.2250
32.7108
0.1113
0.1464
0.0186
57.148
Table 8: Additional performance comparison on GC and UCY datasets. The bold and underlined font show the best and the second best result respectively, Performance averaged over 5 runs.
GC
UCY
Model
FDE
#Colli
DEAL
FDE
#Colli
DEA
CA
4.937
638
0.0290
13.9788
4602
0.0266
SFM
2.1031
1368
0.0280
3.4871
554
0.0267
STGCNN
15,3913
>9999
0.1867
10.4219
>9999
0.3123
PECNet
3,8124
1593
0.0612
5,9714
1474
0.0296
MID
13.8824
9999
0.1323
13.9821
>9999
0.2134
PCS
1.9742
558
0.0271
3.0814
474
0.0329
NSP
1.4124
1006
0.0265
3,4781
438
0.0332
SPDiff
1.3842
944
0.0248
3.0714
738
0.0467
Ours
1.3628
492
0.0231
2.9413
432
0.0337
plicitly enforces spatiotemporal continuity constraints, enabling high-fidelity, physically consistent predictions at inference time. As demonstrated our experiments, our approach achieves superior trajectory plausibility and density conservation while maintaining competitive inference speed. In essence, we trade a moderate increase in training overhead for stronger physical inductive biases and improved generation quality.
A.9
CASE STUDY
To provide intuitive demonstration of our method's performance and complement the quanti- tative comparisons, we visualized trajectory prediction results of our proposed method and physics- guided crowd simulation methods on four datasets, shown in Figures 5 It can be observed that all methods fit the ground-truth trajectories well in the short term. However, for longer-term pre- dictions, other methods tend to deviate from the paths, while STDDN remains the closest to the ground truth. Moreover, in the leftmost example, it can be seen that part of PCS's predicted trajectory overlaps with obstacles, which is clearly an unreasonable prediction.
A. ACCUMULATED DENSITY PREDICTION ERROR ANALYSIS
To comprehensively evaluate the density prediction performance of the proposed method over long time frames, we analyzed the trend of the average absolute enror (MAE) between predicted and true densities over time on the GC and UCY datasets. We compared the results with two representa- tive methods in crowd simulation, PCS and SPDiff. The experimental results shown in Figure 6 and Figure From Figure 6 and 7 it can be seen that our proposed method consistently main- tains the lowest overall density prediction error. In contrast, while the SPDiff method is capable of
18

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Published as a conference paper at ICLR 2026
Table 9: Additional performance comparison on ETH and HOTEL datasets. The bold and underlined
font show the best and the second best result respectively. Performance averaged over runs.
ETH
HOTEL
Model
FDEI
#Colli
DEAL
FDEL
#Colli
DEA
CA
1.2343
738
0.0184
0.9641
714
0.0326
SFM
1.1489
682
0.0184
0.8731
630
0.0328
STGCNN
5.9631
2410
0.1323
8.7814
2031
0.1322
PECNet
0.9412
780
0.0156
0.8104
1508
0.0320
MID
6.0431
3814
0.1321
7.9873
1873
0.1211
PCS
0.9514
692
0.0240
0.7641
634
0.0332
NSP
0.9312
628
0.0243
0.6971
591
0.0341
SPDiff 0.8536
700
0.0218
0.5931
608
0,0345
Ours
0.8391
596
0.0251
0.5841
544
0.0339
Table 10: Additional performance comparison on ZARAI and ZARA2 datasets. The bold and underlined font show the best and the second best result respectively. Performance averaged over runs.
ZARAI
ZARA2
Model
FDE
MAE
#Colli
DEAL
FDE
MAEJ
#Colli
DEAJ
CA
2.0341
1.2814
262
0.0178
1.9432
1.0311
4031
0.0093
SFM
1.9857
1.1446
262
0.0171
2.9841
1.7120
3146
0.0087
STGCNN
6,9871
3.7361
2085
0.1341
8.7638
4.1324
7113
0.2141
PECNet
1.5314
0.9848
244
0.0162
3.9436
2.0405
4656
0.0192
MID
7.9981
4.0312
3013
0.1492
9.9743
5,4327
6016
0.1934
PCS
1.3289
0.8393
236
0.0157
2.8377
1.1837
3686
0.0072
NSP
1.4312
0.8241
226
0.0147
2.1842
1.1717
3686
0.0083
SPDiff 1.1244
0.7450
246
0.0204
1. 1.9811
0.9327
3021
0.0091
Ours
1.1132
0.7338
220
0.0150
1.8321
0.9000
2920
0.0086
predicting density to some extent, its performance does not significantly outperform other methods. Our method, however, takes the density prediction problem into full consideration and effectively embeds physical laws into the prediction model. Therefore, even without achieving the highest pre- cision density prediction, our method significantly improves the effectiveness long- -ur-diuation crowd simulation.
A.11 SENSITIVITY STUDY
We conducted a sensitivity analysis the key hyperparameters in the framework across multiple datasets, including grid size, ODE time steps (r), balance coefficients of the loss function, and node embedding dimensions. Figure g 10 shows the specific performance of the model in terms of MAE the UCY, ETH, and HOTEL datasets. By examining Figure 8 and 10 it can be observed that the grid size affects the simulation performance to some extent in all three datasets. Only by se- lecting an appropriate grid size can the simulation performance of the microscopic network reach its optimal level. Specifically, the grid size influences the pedestrian flow rate, and an appropriate grid size optimizes the constraint effect the continuity equation; the ODE step size affects the degree of overfitting in the network during simulation extrapolation, and an optimal step size ensures the best performance in long-term simulations; the balance coefficient in the loss function determines the performance balance between the microscopic trajectory prediction network and the macroscopic density prediction, and reasonable balance can achieve the best microscopic prediction perfor- mance; the grid node embedding dimension affects the number of parameters in the neural ODE, and too many parameters can lead to overfitting, thus limiting the constraint effect the continuity equation on the trajectory prediction network.
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Published as a conference paper at ICLR 2026
Table 11: Computational cost compared to other methods on GC and UCY datasets.
GC
UCY
Model
Training time
Memory
Batch size
Training time
Memory
Batch size
PCS
1h
2G
32
1h
2G
16
SPDiff
5h
7G
32
3h
6G
32
Ours
8h
26G
16
5h
22G
4
GC
UCY
ETH
HOTEL
Figure 5: Visualization of predicted trajectories on the GC, UCY, ETH, and HOTEL datasets. Each column represents different dataset, showcasing various scenarios. Observed trajectories shown in ground truth in STDDN in SPDiff in ---,, NSP in --- and PCS in ----
A. BOUNDARY EFFECTS
In our current implementation, we approximate boundary effects through dynamic population han- dling: at each time step, we only model pedestrians currently visible in the scene. Individuals entering or exiting field view are explicitly detected and their states are initialized or termi- nated accordingly. This strategy has proven effective in practice across standard benchmarks, where entry/cxit events are relatively sparse and well-separated. However, we fully acknowledge the theo- retical limitation: in truly open environments, future entries cannot be predicted priori, and strict mass conservation over a fixed spatial domain is indeed violated. A more principled solution would be extend the continuity equation with explicit source/sink terms:
Op Øt
V (pv) S
222)
where models inflow (new entrants) and models outflow (exits). Such extension would not only better reflect open-system dynamics but also enable prediction of potential entry zones-e.g. via learned or context-aware source maps. We agree this a highly promising direction and plan explore it thoroughly in future work.
A.13 LIMITATIONS
Despite the promising results achieved in this study, several limitations remain, which also point to promising directions for future research. First, in terms of spatial modeling, future work could ex- plore more flexible representations, such as implicit function modeling based continuous spatial
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Published as a conference paper at ICLR 2026
0.008
Density MAB
0.006
0.004
STDDN
SPDiff
0.002
PCS
NSP
0.000
100
200
300
400
500
600
700
Time Frame
Figure 6: Accumulated density prediction error comparison on the GC dataset.
0.008
STDDN
SPDiff
0.006
PCS
Density MAE
NSP
0.002
0.000
100
200
300
400
500
600
700
Time Frame
Figure 7: Accumulated density prediction error comparison the UCY dataset.
coordinates, multi-scale graph structures to alleviate the computational overhead and expressive limitations introduced by grid discretization. Such approaches may better capture complex spatial dependencies that extend beyond grid partitions. Second, regarding the use of the continuity equa- tion as a strong physical constraint-whicr while enhancing consistency and stability, may over- ride certain latent motion patterns-futur research could investigate soft-constraint mechanisms or adaptive strategics that balance data-driven and physics-driven modeling. This would preserve physical consistency while improving the model's 's ability to capture hidden dynamics in the data.
In addition, enhancing computational efficiency in large-scale scenarios is another important direc- tion. for instance through sparsification techniques, model compression, parallel acceleration, to further facilitate the practical deployment the proposed framework in crowd simulation, emer- gency evacuation, and intelligent transportation applications.
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Published as a conference paper at ICLR 2026
2.1
1.9
MAB
1.8
MAE
2.0
1.7
1.9
1.6
1.5
0.1
0.2
0.3
0.4
15
20
Grid
Time Steps
1,900
1.875
16.3
2,8867
1.7942
1.8266
1 8448
1,850
A2
2,6867
1,8488
1.7832
1,8348
1.800
1.775
0.11
0.5
15
20
Node Embedding Dimension
Figure 8: Sensitivity analysis on the UCY dataset for grid size, ODE time steps loss balancing coefficients (Aı and A2), and node embedding dimension, using MAE as the evaluation metric.
0.57
0.56
0.58
MAF
MAE
0.53
0.52
0.54
0.51
0.52
0.50
2.0
3.0
15
20
Grid Size
ODE Time Steps
I16829
0,6829
0.6829
075
0.55
0.5473
0.54
A2
MAB
0.7801
0,5472
0.5243
0.5232
0.53
0.7801
0.5501
0.5311
015198
0.52
0.5
10
20
A1
Node Embedding Dimension
Figure 9: Sensitivity analysis on the ETH dataset for grid size, ODE time steps т, loss balancing coefficients (A1 and X2), and node embedding dimension, using MAE the evaluation metric.
A. 14 USAGE OF LLMS
All aspects of this work, including the conceptualization, methodology, experiments, and analysis, were independently carried out by the author. The final manuscript was refined with the assistance of a large language model for language polishing to improve clarity and coherence. However, the intellectual content and scientific contributions remain solely the work of the author.
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Published as a conference paper at ICLR 2026
0.36
0.34
0.36
MAE
MAE
0.34
0.32
0.30
0.30
0.28
Grid Size
ODE Time Steps
0.35
0,8754
0.8754
0.3754
0.7391
0.3084
0,2952
0.352
0.33
A2
0.32
0.5
0.7691
0.3484
0.3219
0.2922
0.30
0.1
0.5
10
20
A
Embedding Dimension
Figure 10: Sensitivity analysis the HOTEL dataset for grid size, ODE time steps т, loss balancing coefficients (A₁ and A2), and node embedding dimension, using MAE as the evaluation metric.
23


Reviews:
[{"review_id": "NtfddkghRg", "reviewer": "Reviewer_JPbf", "summary": "This paper proposes STDDN (Spatio-Temporal Decoupled Differential Equation Network), a novel physics-guided deep learning framework for crowd simulation.\nSTDDN explicitly combines microscopic trajectory prediction with macroscopic density evolution by embedding the continuity equation from fluid dynamics into a Neural ODE structure.\nThe model separates local trajectory dynamics from global density fields, enabling physical consistency and stable long-term simulations.\nExperiments on four real-world crowd datasets (GC, UCY, ETH, HOTEL) show that STDDN outperforms prior physics-guided baselines such as SPDiff and PCS in both accuracy and inference speed.", "questions": "Please see the weakness section", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "1.\tGood motivation on coupling of micro- and macro-level dynamics.\n\nThe paper’s main contribution is conceptually sound. By using the continuity equation as a bridge between trajectory prediction and density evolution, STDDN unifies local motion modeling with global flow consistency.\n\n2.\tPhysically meaningful ODE formulation.\n\nThe introduction of a Neural ODE to simulate density evolution is well justified. It provides continuous-time reasoning while enforcing conservation principles, addressing a key limitation of purely data-driven models that tend to accumulate errors over time.\n\n3.\tStrong empirical performance.\n\nAcross four datasets, STDDN shows consistent gains over all baselines, including both physics-based and deep learning methods. The improvements in both accuracy and latency demonstrate that the proposed framework is practically beneficial.\n\n4.\tInterpretability and efficiency.\n\nThe method retains interpretability through its physically grounded formulation while remaining computationally tractable, which is uncommon in physics-guided models.", "weaknesses": "1.\tLimited experimental diversity.\n\nAlthough the method is tested on multiple datasets, all belong to similar crowd domains. It would strengthen the generality claim to include different physical systems, such as vehicle or swarm simulation.\n\n2.\tAblation breadth.\n\nThe ablation study is informative but it would be useful to show how performance changes under different ODE solvers or with alternative coupling strengths between density and trajectory modules.\n\n3.\tMinor missing citations for ODE-based trajectory forecasting.\n\nThe paper would benefit from acknowledging prior studies that have already explored ODE formulations for trajectory or crowd prediction, such as Social ODE: Multi-agent Trajectory Forecasting with Neural Ordinary Differential Equations (ECCV 2022) and Improving Transferability for Cross-Domain Trajectory Prediction via Neural Stochastic Differential Equation (AAAI 2024).\nThese works share conceptual overlap in embedding physical dynamics into continuous differential frameworks.", "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": "L5TV9TQTNF", "reviewer": "Reviewer_Fbqh", "summary": "This paper proposes STDDN, a novel framework for crowd simulation that addresses the common issues of error accumulation and physical inconsistency in long-term predictions. Its core contribution is the unique integration of a macroscopic physical law—the continuity equation from fluid dynamics—with a microscopic deep learning model for trajectory prediction. By using a Neural ODE to model crowd density evolution, STDDN enforces a strong physical constraint during training. Experiments show that STDDN not only achieves state-of-the-art accuracy but also significantly reduces inference latency compared to leading methods.", "questions": "- The fluid dynamics assumption is a strong prior. Could you clarify the intended scope of your method? In which crowd scenarios (e.g., panic, counterflow) might this assumption become a limitation?\n- Given the model's sensitivity to grid size, can you offer any practical guidelines or a more principled approach for selecting this crucial hyperparameter for new scenes?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper's primary strength is its originality in creating a macro-micro coupled framework. Using the continuity equation to regularize trajectory prediction is a conceptually novel and powerful idea for this field. The quality of the work is good, supported by rigorous and comprehensive experiments that convincingly demonstrate superior performance in both accuracy and efficiency over strong baselines. The paper is also written with exceptional clarity.", "weaknesses": "- **Lack of Direct Physical Metrics**: The paper claims to improve physical realism by avoiding issues like congestion and collisions, but it fails to provide direct quantitative evidence. The evaluation relies on general error metrics (MAE/OT), which are insufficient proxies. The work would be much stronger if it included systematic measurements and comparisons of collision rates, obstacle penetration rates or density extremum analysis to directly support its core claims.\n- **Training Cost**: While the paper rightly emphasizes its fast inference speed, it completely neglects to discuss the training cost. The use of a Neural ODE likely makes the training process computationally expensive and slow. An additional analysis should be included in the paper.\n- **A minor issue**: the table in page 8 has a wrong caption: \"**Figure** 4\".", "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": "0cJlbKr5lK", "reviewer": "Reviewer_n74p", "summary": "This paper proposes STDDN (Spatio-Temporal Decoupled Differential Equation Network), a novel physics-guided deep learning framework for crowd simulation. Unlike prior microscopic or purely data-driven approaches, STDDN introduces a Neural ODE formulation guided by the continuity equation from fluid dynamics, thereby coupling macroscopic density evolution with microscopic trajectory prediction. The model integrates three modules — Differentiable Density Mapping (DDM), Continuous Cross-Grid Detection (CGD), and Node Embedding (NE) — to ensure differentiability and physical consistency. Experiments on four real-world datasets (GC, UCY, ETH, HOTEL) show that STDDN significantly improves both simulation accuracy and inference speed compared with state-of-the-art baselines such as SPDiff and PCS.", "questions": "Can the authors explain why the fluid physics improves results in low-density datasets like ETH and HOTEL, where the fluid assumption barely holds?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1、Novel Integration of Physics and Deep Learning:\nThe paper introduces a principled way to integrate the continuity equation into deep models for crowd simulation. This macro–micro coupling via Neural ODE is both original and physically interpretable.\n2、Methodological Sophistication:\nThe DVCG module cleverly connects density and velocity fields through a graph structure, while the DDM and CGD modules effectively address gradient discontinuity and cross-grid flux detection. These designs are mathematically sound and technically detailed.\n3、Interpretability and Physical Consistency:\nThe approach offers clear interpretability grounded in physics, addressing a key limitation of previous purely data-driven models that violate conservation laws", "weaknesses": "1、The proposed model enforces strict mass conservation through the continuity equation, implying that the total population density within the target spatial domain remains constant over time. However, in realistic datasets and surveillance scenarios, the number of pedestrians in view is not fixed — new individuals may enter the scene, and others may leave. Such open-world dynamics inherently violate the closed-system assumption of the continuity equation. Without explicit treatment of source or sink terms (i.e., inflow/outflow of mass) or adaptive boundary conditions, the model may experience cumulative density drift or numerical instability, particularly when crowd density fluctuates significantly. The authors are encouraged to clarify whether boundary inflows are modeled, or to discuss potential modifications to better handle non-conserved population scenarios.\n\n2、The ablation study provides useful insights, particularly regarding the contributions of the ODE solver and the mass constraint loss. Both components appear meaningful; however, the current experimental setup only uses discrete outputs in the loss computation. As a result, the experiments do not adequately demonstrate the benefit of continuous-time modeling enabled by the ODE formulation. To strengthen this section, I suggest decomposing the “w/o ODE” setting into two variants:\n(1)Purely autoregressive training, as mentioned in the paper (“trained using purely autoregressive methods”).\n(2)Discrete neural network replacement for ODE, where the ODE solver is replaced with a discrete neural module that still leverages the combined loss function including the mass constraint term.\nSuch a refinement would better isolate the contribution of the continuous-time ODE formulation from the general modeling capacity and loss design, making the ablation analysis more convincing.", "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": "dUGAdnMExo", "reviewer": "Reviewer_aHns", "summary": "This work proposes a physics-guided deep learning framework for crowd simulation (STDNN). The authors introduce the continuity equation from fluid dynamics as a strong physical constraint and design a density-velocity coupled dynamic graph learning module. They show that STDNN is significantly superior to simulation performance compared to SOTA methods.", "questions": "1.\tHow is the continuity equation incorporated into the Neural ODE solution process? It requires a more detailed explanation.\n2.\tThe detailed parameters used when solving the Neural ODE in torchdiffeq are not disclosed.\n3.\tFigure 1 contains many typo errors.\n⦁\tFor example, $Gin$($Gout$) should actually be $G_{in}$($G_{out}$).\n⦁\tThe input to Microscopic seems to be $\\pho^0$.\n⦁\tThe DDM and CGD in the figure are also too simple.\n4.\tShould the use of the loss function in Eq 10 be more explicit? Eq 8 does not seem to be included in it.\n5.\tAre there more granular comparative tests, such as what the results were for ETH/HOTEL/ZARA1/ZARA2/UNIV, respectively?\n6.\tShould ADE and FDE also be reported for general trajectories?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1.\tThe authors propose a network of time-space decoupled differential equations combined with the continuity equation, which is helpful for predicting the physical laws of trajectories in the macroscopic world.\n2.\tIn experiments, Tables 1 and 2 clearly illustrate the trajectories and verifies main results from the paper.", "weaknesses": "1.\tThe proposed method uses Neural ODEs to solve $\\rho$, but there are many similar ideas, and the use of Neural ODEs in trajectory prediction is also a very common approach.\n2.\tThe proposed method utilizes constraints based on continuity equations, but the specific implementation of this constraint in the Neural ODE framework requires more detailed explanation.\n3.\tThe authors conducted many experiments, but it seems that it is necessary to split each subset of the dataset and compare with newer baselines and methods. The current baseline only reaches the year 2024. Based on the trajectory dataset used by the authors, it seems that there are a large number of sota in pedestrian trajectory prediction that have not been compared.", "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)