Air Quality Prediction with Physics-Guided Dual Neural ODEs in Open Systems
Authors: Jindong Tian
Yuxuan Liang
Ronghui Xu
Peng Chen
Chenjuan Guo
Aoying Zhou
Lujia Pan
Zhongwen Rao
Bin Yang
Venue: ICLR 2025 Poster (2025)
Keywords: Air Quality Prediction; Physics-guided Deep Learning
Primary area: learning on time series and dynamical systems
TL;DR: A novel approach for physics-guided Neural ODE

Abstract: Air pollution significantly threatens human health and ecosystems, necessitating effective air quality prediction to inform public policy. Traditional approaches are generally categorized into physics-based and data-driven models. Physics-based models usually struggle with high computational demands and closed-system assumptions, while data-driven models may overlook essential physical dynamics, confusing the capturing of spatiotemporal correlations. Although some physics-guided approaches combine the strengths of both models, they often face a mismatch between explicit physical equations and implicit learned representations. To address these challenges, we propose Air-DualODE, a novel physics-guided approach that integrates dual branches of Neural ODEs for air quality prediction. The first branch applies open-system physical equations to capture spatiotemporal dependencies for learning physics dynamics, while the second branch identifies the dependencies not addressed by the first in a fully data-driven way. These dual representations are temporally aligned and fused to enhance prediction accuracy. Our experimental results demonstrate that Air-DualODE achieves state-of-the-art performance in predicting pollutant concentrations across various spatial scales, thereby offering a promising solution for real-world air quality challenges.

Published as a conference paper at ICLR 2025
AIR QUALITY PREDICTION WITH PHYSICS-GUIDED DUAL NEURAL ODES IN OPEN SYSTEMS
Jindong Tian', Yuxuan Liang', Ronghui Xu', Peng Chen', Chenjuan Guo', Aoying Zhou', Lujia Pan', Zhongwen Rao², Bin Yang East China Normal University, 2 Huawei Noah's Ark Lab
HHnng Kong University of Science and Technology (Guangzhou) (jdtian, pchen, rhxu | @stu. ecnu. edu. cn, (ciguo, ayzhou, byang) @dase. ecnu. edu. en, (raozhongwen, pan u jjia @huawei J
com
com
ABSTRACT
Air pollution significantly threatens human health and ecosystems, necessitating effective air quality prediction to inform public policy. Traditional approaches are generally categorized into physics-based and data-driven models. Physics-based models usually struggle with high computational demands and closed-system as- sumptions, while data-driven models may overlook essential physical dynamics, confusing the capturing of spatiotemporal correlations. Although some physics- guided approaches combine the strengths of both models, they often face mis- match between explicit physical equations and implicit learned representations. To address these challenges, propose Air-DualODE, novel physics- guiied approach that integrates dual branches of Neural ODEs for air quality prediction. The first branch applies open-system physical equations capture spatiotemporal dependencies for learning physics dynamics, while the second branch identifies the dependencies not addressed by the first in a fully data-driven way. These dual representations are temporally aligned and fused to enhance prediction accuracy. Our experimental results demonstrate that Air-DualODE achieves state-of-the-art performance in predicting pollutant concentrations across various spatial scales, thereby offering a promising solution for real-world air quality challenges. The code is available at:
INTRODUCTION
Air pollution presents significant threat to both human health and the environment. For individuals, prolonged exposure elevates the risk of respiratory and cardiovascular diseases (Azimi & Rahman 2024]. From an environmental perspective, air pollution contributes to phenomena such as acid rain and ozone depletion, disrupting ecosystems and diminishing biodiversity (Cheng al. [024)) These detrimental effects underscore the critical need for precise air quality forecasting, not only to protect public health but also to inform and shape governmental policies effectively.
Traditional research on air quality prediction largely relies on physics-based and data-driven ap- proaches, each of which has their own inherent limitations. Physics-based models make predictions by solving Ordinary or Partial Differential Equations (ODEs/PDES), which capture the transport of pollutants through diffusion and advection, as illustrated in Fig Although these models achieve high accuracy, they require substantial computational resources, especially when applied on large spatial scales. Conversely, data-driven models excel at uncovering dependencies within data (Cheng et al., 2023) Xu et al.) (024a), yet their lack of integration with physical principles can lead to in- complete or even erroneous representations of spatiotemporal relationships. Therefore, it is crucial to propose a hybrid model that effectively harnesses the strengths of both physics- basee and data- driven approaches. However, the development of such a model presents two primary challenges.
Unrealistic assumptions of physical equations in open air quality system. Existing work em- ploys fundamental continuity equations from physies to model the pollutant diffusion process (Mo- hammadshirazi et al. 2023 Hettige et al. 2024). These equations assume the region of interest is
Cooresponning author

---

Published as a conference paper at ICLR 2025
9
Pollutants diffuse from high concentra- tion areas to low concentration areas.
(a) Diffusson:
(b) Advection: Pollutants spread to other areas under the influence of the wind field.
Figure 1: The heatmaps show PM2s concentrations of individual stations time steps and t+ 1. In this open system, there are sinks that absorbs pollutants and sources that generates new pollutants, along with pollutants exiting and entering the region's 's boundary through diffusion and advection.
a closed system, where the total mass remains constant over time. While such assumptions have proven effective in fields like traffic flow and epidemic modeling (Ji et al. 2022 Gao et al. 2021 they are insufficient for real-world air quality prediction, which operates in an open system where pollutants continuously enter and exit with airflow (e.g.. A, D. E in Fig Meanwhile, new pollu- tants may be introduced by industrial activity and vehicular emissions within the region (e.g.. B, C, D, and E in D while others are absorbed by natural elements like forests and lakes (e.g.. A, Fín Ta). Given these facts, relying on closed-system assumptions fails to adequately model physical phenomena in open systems, and may introduce incorrect inductive biases into models.
Mismatch between explicit physical equations and implicit deep learning representations. Physical equations provide an explicit, interpretable framework for modeling pollutant concentration changes, with each variable tied to a well-defined physical meaning. In contrast, deep learning ap- proaches capture spatiotemporal dependencies through implicit representations, often lacking direct physical interpretation. To harness the strengths of both methods, researchers have begun integrating physical equations into neural networks (Verma et al. 2024). These hybrid approaches design neural architectures around physical principles, ensuring that latent variables align with well-established physical meanings (Ji et al. 2022; Hettige et al. 2024, Nascimento et al. 202I). However, the high-dimensional latent spaces in neural networks often represent nonlinear combinations or ab- stract transformations of physical variables, making it inappropriate to interpret these dimensions as specific physical quantities. Therefore, one of the primary challenge in integrating physics into data-driven models is resolving this fundamental mismatch.
In this paper, we propose Air-DualODE, a Physics-guided Dual Neural ODEs that integrate physics dynamics and data-driven dynamics for open system's Air Pollution Prediction. For the first chal- lenge, we propose discrete Boundary-Aware Diffusiin-AAdvection Equation (BA-DAE) to appro- priately model open air systems. Due to the uncertainties associated with pollutant generation (source) and dissipation (sink) in open air systems, pollutant changes involve significant uncertainty, but both processes are linked to individual stations. To capture these dynamics, BA-DAE introduces a linear correction term, breaking away from the unrealistic closed system assumption and providing more accurate description of physical phenomena in open air systems. Additionally, we incorpo- rate Data- Driven Dynamics to capture spatiotemporal dependencies that BA-DAE ignores, making our approach better suited for real-world open air systems. For example, Data-Driven Dynamics can learn some patterns of pollutants generation, which cannot be described by BA-DAE.
To address the second challenge, we propose dual dynamics to integrate physical equations into data-driven models while retaining the model's ability to learn spatiotemporal correlations from data. Physics Dynamics generates spatiotemporal sequences consistent with physical phenomena by solving the BA-DAE, while Data-Driven Dynamics uses Neural ODE to learn dependencies not captured by physical equations. To place the physical knowledge and data-driven representations in the same space, we project the output of Physics Dynamics into latent spaces. However, these representations cannot be directly fused, as they correspond in time but are not fully matched. To resolve this, we design a dynamics fusion module with decaying temporal alignment to fuse both representations. In conclusion, we summarize our contributions follows:
Considering that air pollution transport is in an open system, we redefine the discrete diffusion- advection equation in explicit spaces and propose the BA-DAE, which makes physical equations more consistent with pollutant transport in open air systems.
We introduce Air-DualODE, a model that integrates Physics Dynamics and Data-Driven Dynam- ics to harness the strengths of both physical knowledge and data-driven insights. To the best of
2

---

Published as a conference paper at ICLR 2025
knowledge, this is the first dual dynamics deep learning approach specifically designed for air quality prediction in open system.
Experiments have shown that Air-DualODE achieves state-of-the-an results predicting pollutant concentrations at both city and national spatial scales.
PRELIMINARIES
2.1
PROBLEM STATEMENT
Air quality prediction relies on historical data from N observation stations, including pollutant con- centrations X1.T E IR T'xN1,, such as PM2s and NO2. It also incorporates auxiliary covariates AIT E R (x like temperature and wind speed. The objective is to forecast future pollu- tant concentrations these stations. Based on the region's 's geographical information, we construct geospatial graph G = EV, where / is a set of stations = =N,, and E is set of edges. Specif- ically, An edge F,; between nodes Vi and V, indicates a potential pathway for pollutant transport (See Appendix [A.4] for more details). The objective of the air quality prediction task is to learn function í().) that can accurately forecast future pollutant concentrations XT+1.T+7 [ This function can be formalized as:
X f(XIT, AIT, G).
2.2 BASIC CONCEPTS
Continuity Equation. The continuity equation, fundamental principle in physies, describes mass transport. It explains that system mass changes over time due to material inflow and outflow, assum- ing no sources like vehicle emissions or sinks like forest absorption.
0X öt
V (XF) 0,
where X is air pollutant concentration. F is the flux of particles which describes the transport of pollutants particles, and V is the divergence operator. When solving the Continuity Equation, is crucial to determine if the environment is an open or closed system because the boundary conditions differ significantly, affecting the equation's 's discrete form.
Irffusion-Advectii Equation. By modifying the flow field F of Eq Advection equations for pollutants can be derived (Hettige et al. 2024). See Appendix A.2 for more details.
both the Diffusion and
0x = k v V(X-
Öt
where l: is diffusion coefficient, Tis wind field, and 2 is a Laplacian operator combining divergence and gradient.
2.3 RELATED WORK
Air Quality Prediction. Air quality prediction key objective in smart city. Existing ap- proaches can be broadly categorized into two types: physics-based models and deep learning mod- els. The former rely on domain knowledge to construct Ordinary and Partial Differential Equations (ODEs/PDEs) that describe air pollutant transport (Li et al. [003 Daly & Zannetti 2007). How- ever, these models face two major limitations. They rely on closed-system assumptions, and their of numerical methods, such finite difference finite element methods (Özişik et al. 2017 Solín 2005). becomes computationally expensive at larger spatial scales. This makes rapid pre- diction difficult for urban planning (Lai et al.) 2023) and policy-making (Liu et al. 2025). Deep learning (Campos et al. 2023 2024 Qiu et al. 2024 Chen et al. 2024 Qiu et al. 2025) methods have recently gained prominence by using historical data and auxiliary covariates, such as weather, to extract high-di sich-imennsioal spatiotemporal representations for pollutant prediction. These models typically employ RNNs or Transformers for temporal representations and GNNs or CNNs for spatial
dependencies (Yu al. 2017 Shang al. 2021 Zhao et al. 2023 Wu al. 2024). Furthermore,
3

---

Published as a conference paper at ICLR 2025
Mir-DualDEE Geospatial Graph
Physics Dynamics (X,4
Dynamics Fusion
X
Temporal Alignemnt
Encoder,
Caefficient Estimator
Pollutant Concentration
Concentration Wind speed Wind direction.
OUE (" Dr I.... T
Data-Driven Dynamics
Stay Sang Siste Bing Strte
Sang Sist Sinay Strte
Auxiliar Covariates
(X,,
27
Lalent variable
GNN Fusion
Encoder,
2D FD.
Output: XT+UT+
P.
Discrete Boundary-Aware Difiusion-Advection Equation eq
DD: Masked Attention-Bassd ODE Function dz
pu
dt
Gaut
GCN
LayerNor
Spatial MSA
Weighted
dx"
tar yeves
of
Sum
MLP
Gade
GON
Projection
Correction Term: A..X
LayerNorn
Spatial masked self -eff-tteetion
The overall framework of Air-DuaIODE consists of Physics Dynamics, Data-Driven Dy- namics, and Dynamies Fusion. FP and FD represent the ODE functions in Physies Dynamies and Data-Driven Dynamics, respectively.
Figure 2:
there models specifically designed for air pollutant prediction tasks (Yu et al. 2023 Liang al. 2023). However, these deep learning approaches may overlook essential physical dynamics, leading to confused spatiotemporal correlations that could violate physical principles.
Physics Guided Deep Learning. Recently, several studies have explored incorporating physical knowledge into deep learning models (Karpatne et al. 2024). The most direct approach involves using physical knowledge as constraints in loss functions (Raissi et al., 2017: Shi et al.) (021)) penalizing deviations from physical laws encourage physically consistent patterns and improve generalizability (Greydanus et al. 20199 Choi et al. (203b)). However, these methods rely on highly accurate physical knowledge; otherwise, they may introduce inductive bias, limiting the model's learning capacity. Another approach is to design hybrid frameworks that integrate scientific knowl- edge into specific modeling components (Bao et al. 2021 Xu et al. 2024b). such as incorporating Choi et al. 2023a Hettige al., 2024). These methods often require deep latent representations align with physical mean-
physical equations into Neural ODEs (Hwang et al. 2021 et al. Choi
ings, forcing models to switch between abstract representations and physical quantities Nascimento
ct al. 2021); Verma et al. (2024).
3
METHODOLOGY
3.1 MODEL OVERVIEW
To integrate physical knowledge into neural networks, and ensure accurate spatiotemporal correla- tions, propose Air-DualODE, which includes BA-DAE and dual dynamics under open air sys- tems. As shown in Fig the framework consists of three main components:
Physies Dynamics To maintain consistency with the neural network's implicit representations. Physics Dynamics directly solves physical equations to obtain spatiotemporal correlations with physical meanings. Specifically, solves the BA-DAE, namely FP, more realistic equation for open air systems, to generate physical simulation results Xf41744 for the next 5 steps. These results are then mapped into the latent space as Z
Data-Driven Dynamics Although BA-DAE captures pollutant changes in an open air system, does not account for some spatiotemporal dependencies, like the effects of temperature and humid-
4

---

Published as a conference paper at ICLR 2025
ity on pollutant transport. To address this, Data- Dynamics is implemented using Neural ODE with spatial masked self-attentiin (Spatial-MSA) in ODE function FD, This branch captures unknown dynamics in latent space and generates latent dynamics representation Zf+1
Dynamics Fusion Although 17-1 and 1 share the same latent space and time span, they yet aligned. To resolve this, Decaying Temporal Contrastive Learning (Decay-TCL) aligns them over time using decaying weights for effective fusion. A GNN then spatially fuses two representations, and the combined result is decoded to generate the prediction XT+1T+T-
3.2
PHYSICAL DYNAMICS OF OPEN SYSTEM'S AIR POLLUTION
3.2.1
DISCRETE DIFFUSION IITUSION-ADVECTIONN EQUATION UNDER CLOSED SYSTEM
The Method of Lines (MOL) can discretize the PDEs (Eq into a grid of location-specific ODEs
Schiesser 2012 lakovlev al. 202T). The discrete equation can be derived as follows:
dX = Law + T.
dt
Tarrr and Tayy are the discretized graph Laplacian operators for diffusion and advection, respectively, approximated using the Chebyshev GNN as referenced in (Defferrard et al., [2016] Hettige et al. 2024). According to CChapman & Chapman (2015). ), it is necessary to define distinct graph structures represent diffusion concentration gradient fields and advection wind fields.
Diffusion Graph is an undirected weighted graph based on G. with weights computed using the in- verse of the distance between nodes, such that Gam [ajl des (Han et al. 2023), where d, represents
the haversine distance between nodes Vi and Vy-
Advection Graph is a directed weighted graph based on G. where the direction and weight of the edges depend on the wind direction A and wind speed A at ' timestamps. After calculating the included angle & in Fig 3] we project the wind vector of V; onto Eay to obtain the wind intensity from V; to Vj. as shown in the Fig The Guav [Rj] is calculated as follows:
|0| dij
ReL.U
cos
when Gij = 1,
Gadv [ij]
when Gij =
In particular, the wind vector's projection V; does not point in the direction of Vj. i.e., i.e. cos < O, this suggests that the pollutants at V; are currently not influenced by advection to reach V,. Con- sequently, at that time, the directed edge in the advection graph will not constructed. Notably, Gaav differs from Gai Gaif remains static over time, representing static graph. In contrast, Gaav changes dynamically over time as the wind speed and direction at each node vary, making it dy- namic graph that varies different timestamps.
is easy to prove that Eq 3 has conservation property (see Appendix A.3). which means the amount of pollutants concentration remaining constant over time:
dt 0.
(5)
3.2.2 DISCRETE: BOUNDARY-AWARE DIFFUSION-ADVECTIIN EQUATION FOR OPEN SYSTEM
Since air pollutant propagation does not occur in closed system, the total pollutant mass cannot remain constant over time. Particularly, pollutants can dissipate beyond boundaries, such as being carried away by wind in Fig[I] Additionally, industrial zones and vehicles contribute to pollution, while urban lakes and rainfall absorb pollutants. Thus, the dynamics of air pollution are far more complex than closed system's 's conservation assumption.
In an open system, changes in the total amount of pollutants can be categorized into dissipation and generation. Pollutant dissipation refers to sinks, including transport out of the boundary through diffusion and advection, as well as absorption by forests or lakes. Pollutant generation refers to sources such industrial emissions. As shown in Fig 4 the dissipation and generation Point E
5

---

Published as a conference paper at ICLR 2025
latitude
Wind Direction Wa
A
Gody Kil
Tongzhou
Val
Dachang
Luttieng
Emisssio
O
Diffusion
Wind Speed A.
Huoxian
Advection
longitude
Absorption
Figure 3: Construction of Gadv [tj].
Figure 4: The sources and sinks station E.
are highly relevant its current concentrations. Therefore, we incorporate two terms into Eg the dissipation term B @ (3 E [-1, and the generation term G o X (o+ E [0, +oo xx1) These terms be simplified into open system correction term: X (B e +00) Nx1). Boundary stations, experiencing more outward diffusion and advection, tend to have greater negative 3r So, this equation is called Discrete Boundary-Aware Diffusion-Advectionn Equation (BA- DAE):
dX a
F F (X+;
o (-k (-k-Lamm) + (1-cl ax) o (Ladv B
(6)
dt
Here, + represents learnable parameters used to approximate the graph Laplacian operators. The gate value a E IR Nx1 is estimated by linear layer (R2 + I) based on each station's 's diffusion and advection over time, representing the weighted influence of these two effects as they vary across time and space. The diffusion coefficient & and the correction term's coefficient 3 are estimated via the Coefficient Estimator, a Recurrent Neural Network. By solving BA-DAE with ODE solver from Chen et al. (2018), pollutant concentrations from T + 1 to T + 7 obtained, and temporal encoder maps these concentrations to the latent space for dynamics fusion, as illustrated in the Physics Dynamics part of Fig
X X441:4+ ODESolve(XT. (PTTTTT 1T.,
Zf = Encoder, (X
(8)
where ZT+1TT+ encodes the spatiotemporal dependencies of pollutant transport with physical knowledge, which data-driven models may not fully capture. This compensates for the limitations of data-driven approaches and enhances the interpretability of the model.
3.3 DATA-DRIVEN NEURAL ODE
The Data-Driven Neural ODE focuses on modeling dynamical systems that go beyond the limita- tions of the BA-DAE, aiming to uncover knowledge that cannot be described by physical equations alone. For instance, spatiotemporal correlations within historical data patterns are not adequately described by the BA-DAE. However, data-driven dynamics can bridge these gaps by learning from historical data. Additionally, relationships such as humidity and pollutant transport can also be ef- fectively captured through this branch. Following Chen et al. (2018), Latent- ODE is employed in the Data-Driven Dynamics branch, as depicted in Fig[2] Specifically, we utilize (X,,)) A)1:T as input, which is mapped to the latent variables ZP via an Encoderp. This serves the initial condition for learning and solving the Neural ODE, resulting in ZT+11T47*
ZP Encoderp (X (XI,T,
(9)
ZP41 = ODESolve(25) (2, ....T................ I++1...++7:000
(10)
where T represents the learnable parameters in ODE function FD To capture spatial correlations across different time steps within the dynamic system, a masked self-attention mechanism is incor- porated into FD, Specifically, the adjacency matrix of G employed since intuitively, no potential transport pathway exists between two stations, their representations should not be correlated. This approach not only improves computational efficiency but also en- ables the model to focus on relevant information, enhancing its effectiveness across various spatial granularities in prediction scenarios.
mask in the Spatial-MSA

---

Published as a conference paper at ICLR 2025
3.4 DYNAMICS FUSION
Decaying Temporal Constrastive Learning. Before combining the physics and data-driven dy- namics, it is essential to align their latent representations, Z++1 and ZP +1:T+++>> resolve inconsistencies and enable effective fusion. Since both dynamics operate synchronously, three key intuitions are considered: latent representations at the same time step are more similar than those at different time steps, 2) adjacent time steps are more similar than distant ones, and 3) representa- tions from the same dynamics are more similar than those from different dynamics. To satisfy the three intuitions, we propose Decaying Temporal Contrastive Learning (Decay-TCH) -CL). It computes decaying weight w((t,, s) based on timestamp differences and dynamics type to regulate the similarity between latent representations. The hyperparameters Å₁ and A2 are used to differentiate the distinct dynamics, and they must satisfy X1 > 2 to fulfill third intuition.
2 2.0( 0 A1 -tt sl)
if t < if s) >T,
wit, 8) = 1 1 mod 7|)
where (-)) represents sigmoid function. Let 1 Iz", 12" z We define the softmax proba- bility of the relative similarities when computing the loss as:
exp 2,, 2₂)} p(t,t') E exp{Sim( (2,, 2,)}
The function Sim(..) denotes cosine similarity. The decaying temporal contrastive loss for 2 at timestamp is defined as:
2r
((t) log p(t,
Σ
w(t,j) logp(t,j).
(11)
Eq 11 ensures the similarity inequality in the Temporal Alignment shown in Fig 2 aligning the physical and data-driven representations in the temporal dimension to enable better fusion. The final loss for Decay-TCL is defined as:
1
2r
Lus
((t).
(12)
2T
GNN Fusion. After temporal alignment, the two latent representations are concatenated as z concat( z Z,P) at each timestamp. Since the influence of stations is distance-dependent, using MLP structures may introduce noises from distant sites, complicating the learning process thus re- ducing accuracy (see Appendix 16). To address this, GNNs are employed based on the geospatial graph G' to Dynamics Fusion: Z ()) = GNNs(Z, (")), where n represents the GNN layers. The De-
coder then transforms the latent representations Z(m) -T+ITT- into predicted pollutant concentrations X 1T We jointly optimize the prediction loss and the loss functions in Eq 12 with hyperpa- rameter 7 to balance them:
T+r Σ
1
C=
|
& + TCst.
(13)
T
4 EXPERIMENT
4.1 EXPERIMENT SETTING
Datasets. We evaluate the performance of our model using two real-world air quality datasets: the Beijing_ dataset and the KnowAif" dataset. The Beijing dataset is evaluated at the city scale, while the KnowAir Air dataset is evaluated at the national scale. Further descriptions about these two datasets can be found in Appendix A.5] In accordance with previous studies (Liang et al. 2023), we focus on PM25 concentration as the target variable, using meteorological factors- including temperature,
httpss//www.beeettt....////e/eo/ 2018/
https://ggtthbb.ccm/////wuaa

---

Published as a conference paper at ICLR 2025
pressure, humidity, wind speed, and direction auxiliary covariates to ensure consistency be- tween the two datasets. Similar to prior work (Liang et al. 2023: Hettige et al. 2024). we employ 3-hour time step and utilize data from the preceding 72 hours (24 steps) predict the subsequent
72 hours. More details regarding the implementation can be found in Appendix A.7 Additionally, following Liang al.] (2023), analyze the errors in predicting sudden changes. Sudden changes are defined as cases where PM2.5 levels exceed 75 ug/m² and fluctuate by more than =0 ug/m² in the next three hours.
Baselines. We evaluate Air-DualODE against baselines of four categories: 1) Classical Methods: Historical Average (HA) and Vector Auto-Regression (VAR). 2) Differential Equation Network-
Based Methods: Latent-ODE (Chen et al.] (018). ODE-RNI RNN (Rubanova et al. 2019) and ODE-
LSTM (Lechner Hasani, 2020). 3) Spatio-temporal Deep Learning Methods: DCRNN (Li al. 2017 STGCN ( Yu et al. 2017), ASTGCN (Guo et al. 2019). GT' (Shang et al. 2021). MTSF- DG (Zhao et al., 2023). PM2.5-GNV (Wang al. (020), and AirFormer (Liang et al. 2023). 4)
Physics-guided Neural Networks: AirPhyNet (Hettige et al. 2024).
Evaluation Metrics. Baselines are evaluated using Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Symmetric Mean Absolute Percentage Error (SMAPE), with smaller values indicating better performance (see in Appendix A.6))
4.2 PERFORMANCE COMPARISON
Overall Performance. Table T compares the performance of Air-DualODE with baselines over three days and sudden changes across two datasets. Air-DuaIODE outperforms all baselines across all metrics for both datasets. These results demonstrate the model's effectiveness in integrating Physics Dynamics and Prtv-Driven Dynamics within a deep learning framework to capture complex spatiotemporal dynamics relationships.
Table I: Overall prediction performance comparison. Bold fonts indicate the best results, while underlined fonts signify the second-best results.
Beijing
KnowAir
Methods
3days RMSE
Sudden Change
3days RMSE
Sudden Change
MAE
SMAPE
MAE
RMSE
SMAPE
MAE
SMAPE
MAE
RMSE
SMAPE
HA
59.78
77.73
0.83
79.66
94.87
0.87
25.27
38.57
0.53
59.37
79.38
0.72
VAR
55.88
75.64
0.82
77.18
98.73
0.86
24.56
37.35
0.51
$7.40
71.84
0.71
LatenODE
43.35
63.75
0.79
69.71
92.67
0.77
19.99
30.85
0.46
42.59
57.90
0.50
ODE-RNN
43.60
63.67
0.79
70.22
93.12
0.78
20.57
31.26
0.45
42.07
57.76
0.51
ODE-LSTM
43.68
64.23
0.78
70.60
93.27
0.77
20.21
30.88
0.45
41.29
56.47
0.49
STGCN
45.99
64.85
0.82
73.65
97.87
0.76
23.64
32.48
0.52
55.29
73.21
0.54
DCRNN
49.61
71.53
0.82
75.31
96.50
0.81
24.02
37.87
0.53
56.88
74.58
0.73
ASTGCN
42.88
65. 24
0.79
72.75
96.68
0.83
19.92
31.39
0.44
42.06
57.50
0.52
GTS
42.96
63.27
0.80
70.36
92.70
0.76
19.52
30.36
0.43
40.87
56.20
0.49
MTSF-DG
43.12
66.06
0.79
72.26
95.32
0.77
19.17
29.
0.43
40.64
55.65
0.50
PM25GNN
45.23
66.43
0.82
72.45
95.34
0.78
19.32
30.12
0.43
40.43
55.49
0.49
Airformer
42.74
63.11
0.78
68.80
91.16
0.75
19.17
30.19
0.43
39.99
55.35
0.49
Aet
42.72
64.58
0.78
70.03
94.60
0.78
21.31
31.77
0.47
43.23
58,79
0.50
Air-DuaOODE
40.32
62.04
0.74
66.40
90.31
0.73
18.64
29.37
0.42
39.79
54.61
0.49
From Table T we can also observe the following: 1) deep learning approaches outperform classical methods such as HA and VAR, demonstrating their superior ability to capture complex spatiotem- poral correlations. 2) Spatiotemporal deep learning models, originally developed for traffic flow forecasting, such as ASTGCN and GTS, exhibit strong adaptability to air quality forecasting, pro- ducing results comparable to SOTA air quality prediction methods like AirFormer and PM25GNN. 3) Approaches based on Neural DE networks also exhibit competitive performance compared to spatiotemporal deep learning methods, as they capture the ODE function in latent space, which is crucial for accurate air quality prediction. Overall, these results suggest the effectiveness of Air- DualODE in achieving precise air quality predictions while leveraging the physics-based domain knowledge of air pollutant transport. More Discussions about sudden change evaluation can be found in Appendix A.17
8

---

Published as a conference paper at ICLR 2025
4.3 ABLATION STUDY
Effect of Dual Dynamics. To validate the effectiveness of Dual Dynamics, we test each branch of Air-DualODE independently across both datasets: a) w/o Physics Dynamics, which solely relies on Data-Driven Dynamics augmented by the spatial-MSA mechanism; b) w/o Data-Driven Dynamies, employing only Physics Dynamics without the encoder. Both configurations exclude Dynamics Fu- sion. As shown in Table the removal of either branch results in diminished performance. Notably, the absence of Data- Driven Dynamics leads to notably poor performance from BA-DAE on both datasets, since physical equations often idealize real-world scenarios, Conversely, the integration of both branches significantly enhances performance, demonstrating that Data-Driven and Physics Dynamics provide complementary insights.
Table 2: Effect of dual dynamics, fusion on latent space, Decay-TCL and Spatial-MSA.
Beijing RMSE
Know RMSE
Methods
MAE
SMAPE
MAE
SMAPE
w/o Physics Dynamics
42.75
63.45
0.75
19.69
30.65
0.45
w/o Data-Driven Dynamics
44.33
65.60
0.82
21.21
33.09
0.56
Explicit Fusion
41.32
63.08
0.76
19.06
30.28
0.43
Cross-Space Fusion
41.97
63.12
0.77
19.27
30.54
0.44
w/o Decay- TCL
42.34
66.56
0.82
19.10
29.79
0.43
w/o Spatial-MSA
40.52
62.49
0.75
18.97
30.01
0.43
Air-DuaIODE
40.32
62.04
0.74
18.64
29.37
0.42
Effeet of Fusion on Latent Space. To further validate Dynamics Fusion in the latent space, we test alternative fusion strategies: a) Explicit Fusion: After solving ODE-Func FD in the latent space using Data-Driven Dynamics, we apply a Decoder to map it to explicit space (i.c., pollu- tant concentration) before fusion. b) Cross-Space Fusion: We combine the explicit-space output of Physics Dynamics with the deep representations of Data-Driven Dynamics. Both methods use GNN for fusion, consistent with Air-DualODE. As shown in Table 2 both methods led to de- cline in performance. Fusion in explicit space experienced information loss during the decoding of deep representations, whereas fusion across different spaces encounter mismatched representations, which exacerbats performance deterioration. This underscores the importance of aligning and fusing Dynamics within the latent space to achieve optimal results.
Effeet of Physical knowledge. To further demonstrate the impact of incorporating physi- cal knowledge, we compare Air-DualODE with three other closed-system variations: a) Diff-c, which only considers the diffusion process, b) Adv-c, which only considers the advection pro- cess, and c) DAE-c, which accounts for both processes. Additionally, we compare these with d) BA-DAE-o, namely Air-DualODE, as pro- posed in Section 3.2.2 for open air systems. 3 shows that integrated physical knowl- edge is crucial, and inadequate modeling leads reduced performance. Consistent with Air- PhyNet, DAE-c outperforms Diff-c and Adv-c, but its inability to model dissipation and generation makes less effective than BA-DAE-o. This underscores the effectiveness of BA-DAE in model- ing open air systems. While Data-Drived Dynamics can compensate for some deficiencies in the physical equations, it still cannot match the effectiveness of direct physical modeling-
Diff-c
DAE
BA-DAE
42
41.5
19.5
MAE
40.5-
18.5
40
(a) Beijing
(b) KnowAir
Figure 5: Effect of Physical Knowledge on MAE.
Effect of Spatial-MSA. To assess the impact of Spatial-MSA in FD .we compare it with w/o MpSt which uses standard self-attention (SA). out SpS is similar to Air Dir-uulODE on the Beijing dataset, while there is significant per- formance gap on the KnowAir dataset. This is because stations in Beijing are geographically close with no major barriers, making MSA almost equivalent SA. In contrast, with KnowAir being a national dataset with distant stations, the performance gap highlights that distant station representations offer little useful information.
that the performance with-
9

---

Published as a conference paper at ICLR 2025
Effect of Decay-TCL. To validate Decay-TCL, test the model w/o Decay-TCL, where Lui is removed from Air- DualODE. Table shows
shows that directly fusing Dual Dynamics in the latent space leads to performance drop. This is because the latent representations of Physics Dynamics, after being mapped through the Encoder, may not align with those Data-Driven Dynamics, resulting in mismatches and inconsistencies during fusion.
4.4 CASE STUDY
We present two case studies to demonstrate the interpretability of the physical equations within the dual dynamics. To highlight its physical significance, we visualize the predicted 2M2.5 concen- trations alongside wind direction data from the Beijing dataset. Fig displays the PM levels at monitoring stations at two consecutive prediction timestamps. The emergence in Fig causes the pollutant concentrations to shift in the direction of the wind, most noticeably in the circled regions. Specifically, PM25 concentrations decrease in the purple-circled regions, while they increase in the black-circled regions. This indicates that Air- DualODD successfully captures certain spatiotemporal correlations between wind direction and pollutant concentrations based on the Advection equation in the BA-DAE.
southeast wind
Beiling 2018-2-9 0:00
Beijing 2018-3-3 9:00
Beijing 2018-2-15
Beijing 9:00
southeast wind
southeast wind
0.05
0.15
0.30
Beijis
Bei
0.15
Linefang
Langtang
.0.20
0.00
Visualization of predicted PM2s concentrations under advection. The heatmap shows PM2s concentration, while arrows represent wind direction.
Figure 6:
Figure Visualization of (,, where negative and pos- itive values separately indicate dissipation and gener- ation. The arrow direction represents wind direction, and the length reflects wind speed at each station.
Next, we visualize the G in BA-DAE under two scenarios on the Beijing dataset (see Fig 7. As described in Section 3.2.2 negative values represent PM2,5 dissipation (blue colorbar), while posi- tive values indicate PM2. generation (red colorbar). We observe two distinct scenarios: In the first scenario (left side of Fig 7.. with wind present, dissipation significantly increases boundary sta- tions in the wind direction (denoted by circles), resulting in larger negative 3,. Additionally, some boundary stations exhibit positive 3r, which is reasonable because the wind these stations carries pollutants from outside the region. In other words, this can be considered as pollutant generation. In the second scenario (right side of Fig with no wind, boundary stations exhibit greater dissipation due to forest absorption and diffusion beyond the boundary, leading to negative 3; at the outskirts of Beijing. Notably, the city center of Beijing exhibits higher 84 at 9 a.m., indicating pollutant genera- tion. This is expected, 9 a.m. coincides with Beijing's morning rush hour, when large number of vehicles are present in the area. This example demonstrates that our model successfully captures the increase in pollutants caused by vehicle emissions. Overall, by integrating physical equations, Air-DuaIODE demonstrates excellent interpretability.
5 CONCLUSION AND FUTURE WORKS
We design Lir-DulODDE to predict air quality at both city and national levels. Our method employs a dual-dynamies approach, leveraging Neural ODEs to model physical (known) and data-driven (unknown) dynamics, which are subsequently aligned and fused in the latent space. By integrating physical equations into deep learning, Air-DualODE allows existing physical knowledge to enhance the model's learning, inference, and ultimately, its prediction accuracy. While the results are promis- ing, predicting sudden changes remains a significant challenge in air quality forecasting. Therefore, Designing specific mechanisms better handle sudden changes is important and promising search direction. Moreover, we plan to further explore Physics-gue -uiided DualODE and extend its application to general spatiotemporal prediction tasks.
10

---

Published as a conference paper at ICLR 2025
ACKNOWLEDGEMENT
This work is partially supported by National Natural Science Foundation of China No.62372179. Yuxuan Liang's work is supported by the National Natural Science Foundation of China No.62402414.
REFERENCES
Mohammad Naim Azimi and Mohammad Mafizur Rahman. Unveiling the health consequences of air pollution in the world's most polluted nations. Scientific Reports, 14(1):9856, 2024.
Tianshu Bao, Xiaowei Jia, Jacob Zwart, Jeffrey Sadler, Alison Appling, Samantha Oliver, and Tay- lor T Johnson. Partial differential equation driven dynamic graph networks for predicting stream water temperature. 202/ IEEE International Conference on Data Mining (ICDM), pp. 11-20. IEEE, 2021.
David Campos, Miao Zhang, Bin Yang. Tung Kieu, Chenjuan Guo, and Christian S Jensen, LightTS: Lightweight time series classification with adaptive ensemble distillation. Proceedings of the ACM on Management of Data, 1(2):1-27, 2023.
David Campos, Bin Yang, Tung Kieu, Miao Zhang, Chenjuan Guo, and Christian S Jensen. QCore: Data-efficient, on-device continual calibration for quantized models. Proceedings of the VLDB Endowment, 17(1 11:21)8-27777 2721, 2024.
Airlie Chapman and Airlie Chapman. Advection graphs. Semi-Autonomons Networks: Effective Control of Networked Systems through Protocols, Design, and Modeling, pp. 3-16, 2015.
Peng Chen, Yingying Zhang, Yunyao Cheng, Yang Shu, Yihang Wang, Qingsong Wen, Bin Yang, and Chenjuan Guo. Pathformer: Multi-scale transformers with adaptive pathways for time series forecasting. arXiv preprint arXiv:2402.00 05956, 2024.
Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances in neural information processing systems, 31, 2018.
Lingyun Cheng, Siwei Lu, and Runchen Shi. Research on environmental pollution caused by acid rain. Highlights in Science, Engineering and Technology, 91:25-29, 2024.
Yunyao Cheng, Peng Chen, Chenjuan Guo, Kai Zhao, Qingsong Wen, Bin Yang, and Christian S Jensen. Weakly guided adaptation for robust time series forecasting. Proceedings of the VLDB Endowment, 17(4):766-779, 2023.
Hwangyong Choi, Jeongwhan Choi, Jechyun Hwang, Kookjin Lee, Dongeun Lee, and Noseong Park. Climate modeling with neural advection-diffuue equation. Knowledge and Information Systems, 6566):2403-2427, 2023a.
Jeongwhan Choi, Seoyoung Hong, Noseong Park, and Sung-Bae Cho. Gread: Graph neural reaction-diffusion networks. In International Conference on Machine Learning, pp. 5722-5747. PMLR, 2023b.
Aaron Daly and Paolo Zannetti. Air pollution modeling-an overview. Ambient air pollution, pp. 15-28, 2007.
Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. Advances neural information processing systems, 29, 2016.
Junyi Gao, Rakshith Sharma, Cheng Qian, Lucas M Glass, Jeffrey Spaeder, Justin Romberg. Jimeng Sun, and Cao Xiao. STAN: spatio-temporal attention network for pandemic prediction using real-world evidence. Journal of the American Medical Informatics Association, 28(4):733-743, 2021.
Samuel Greydanus, Misko Dzamba, and Jason Yosinski. Hamiltonian neural networks. Advances in neural information processing systems, 32, 2019.

---

Published as a conference paper at ICLR 2025
Chenjuan Guo, Christian S. Jensen, and Bin Yang. Towards total traffic awareness. SIGMOD Record, 43(3):18-22 23, 2014.
Shengnan Guo, Youfang Lin, Ning Feng. Chao Song, and Huaiyu Wan. Attention based spatial- temporal graph convolutional networks for traffic flow forecasting. In Proceedings of the AAAI conference on artificial intelligence, volume 33, pp. 922-929, 2019.
Jindong Han, Weijia Zhang, Hao Liu, and Hui Xiong. Machine learning for urban air quality ana- lytics: A survey. arXiv preprint arXiv:2310.09632 2023.
Kethmi Hirushini Hettige, Jiahao Ji, Shili Xiang, Cheng Long, Gao Cong, and Jingyuan Wang- NirPhyNet: Harnessing physics-guide neural networks for air quality prediction. arXiv preprint 2440022402.03784, 2024.
Jeehyun Hwang, Jeongwhan Choi, Hwangyong Choi, Kookjin Lee, Dongeun Lee, and Noseong Park. Climate modeling with neural diffusion equations. In 2021 IEEE International Conference on Data Mining (ICDM), pp. 230-239. IEEE, 2021.
Valerii lakovlev, Markus Heinonen, and Harri Lähdesmäki. Learning continuous-time PDEs from sparse data with graph neural networks. In ICLR, 2021.
Jiahao Ji, Jingyuan Wang, Zhe Jiang, Jiawei Jiang, and Hu Zhang. STDEN: Towards physics-guided
neural networks for traffic flow prediction. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp. 4048-4056, 2022.
Anuj Karpatne, Xiaowei Jia, and Vipin Kumar. Knowledge-guided machine learning: Current trends and future prospects. arXiv preprint arXiv:2403. 159899 2024.
Siqi Lai, Zhao Xu, Weijia Zhang, Hao Liu, and Hui Xiong. Large language models as traffic signal control agents: Capacity and opportunity. CoRR, 2023.
Mathias Lechner and Ramin Hasani. Learning long-term dependencies in irreularly-samplee time series. arXiv preprint arXiv:2006.044/ 2020.
Lianfa Li, Roxana Khalili, Frederick Lurmann, Nathan Pavlovic, Jun Wu, Yan Xu, Yisi Liu, Karl O' O'Sharkey, Beate Ritz, Luke Oman, et al. Physics-informet deep learning to reduce the bias in joint prediction of nitrogen oxides. arXiv preprint arXiv:2338.0744/1 2023.
Yaguang Li, Rose Yu, Cyrus Shahabi, and Yan Liu. Diffusson convolutional recurrent neural net- work: Data-driven traffic forecasting. arXiv preprint /77.v0/70..01126, 2017.
Yuxuan Liang, Yutong Xia, Songyu Ke, Yiwei Wang, Qingsong Wen, Junbo Zhang, Yu Zheng, and Roger Zimmermann. Airformer: Predicting nationwide air quality in china with transformers. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37, pp. 14329-14337, 2023.
Huiping Liu, Cheqing Jin, Bin Yang, and Aoying Zhou. Finding top-k shortest paths with diversity. IEFE Trans. Knowl. Data Eng., 30(3):488-500) 2018
Sicong Liu, Yang Shu, Chenjuan Guo, and Bin Yang. Learning generalizable skills from offline multi-task data for multi-agent cooperation. In The Thirteenth International Conference Learn- ing Representations, 2025.
Ahmad Mohammadshirazi, Aida Nadafian, Amin Karimi Monsefi, Mohammad H Rafiei, and Rajiv Ramnath. Novel physics-based machine-learning models for indoor air quality approximations. arXiv preprint 2308.:2330..14333 2023.
Renato G Nascimento, Matteo Corbetta, Chetan S Kulkarni, and Felipe AC Viana. Hybrid physics- informed neural networks for lithium-ion battery modeling and prognosis. Journal of Power Sources, 513:230526. 2021.
M Necati Özigik, Helcio RB Orlande, Marcelo J Colaço, and Renato M Cotta. Finite difference methods in heat transfer. CRC press, 2017.
12

---

Published as a conference paper at ICLR 2025
Xiangfei Qiu, Jilin Hu, Lekui Zhou, Xingjian Wu, Junyang Du, Buang Zhang, Chenjuan Guo, Aoy- ing Zhou, Christian Jensen, Zhenli Sheng, et al. TFB: Towards comprehensive and fair bench- marking of time series forecasting methods. arXiv preprint arXiv: 2403::2403. 2024.
Xiangfei Qiu, Xiuwen Li, Ruiyang Pang. Zhicheng Pan, Xingjian Wu, Liu Yang. Jilin Hu, Yang Shu, Xuesong Lu, Chengeheng Yang, Chenjuan Guo, Aoying Zhou, Christian S. Jensen, and Bin Yang. EasyTimm: Time series forecasting made easy. In ICDE, 2025.
Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Physics informed deep learn- ing (part i): Data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv: 10561, 2017.
Yulia Rubanova, Ricky TQ Chen, and David K Duvenaud. Latent ordinary differential equations for iregularly-samplld time series. Advances in neural information processing systems, 32, 2019.
William Schiesser. The numerical method of lines: integration of partial differential equations. Elsevier, 2012.
Chao Shang, Jie Chen, and Jinbo Bi. Discrete graph structure learning for forecasting multiple time series. arXiv preprint arXiv:2101 101.06861, 2021.
Qichao Shentu, Beibu Li, Kai Zhao, Yang Shu, Zhongwen Rao, Lujia Pan, Bin Yang, and Chen- juan Guo. Towards general time series anomaly detector with adaptive bottlenecks and dual adversarial decoders. ICIR, 2025.
Rongye Shi, Zhaobin Mo, and Xuan Di. Physics-informes deep learning for traffic state estima- tion: A hybrid paradigm informed by second-order traffic models. In Proceedings of the AAAI Conference Artificial Intelligence, volume 35, pp. 540-547, 2021.
Pavel Solín. Partial differential equations and the finite element method. John Wiley & Sons, 2005.
Gerrit Van den Burg and Christopher KI Williams. An evaluation of change point detection algorithms. arXiv preprint arXiv:2003.06222, 2020.
Yogesh Verma, Markus Heinonen, and Vikas Garg- ClimODE: Climate and weather forecasting with physics-informed neural odes. arXiv preprint arXiv:2404. 10024 2024.
Shuo Wang, Yanran Li, Jiang Zhang, Qingye Meng, Lingwei Meng, and Fei Gao. Pm2.5-GNN: A domain knowledge enhanced graph neural network for pm2. 5 forecasting. In Proceedings of the 28th international conference on advances in geographic information systems, pp. 163-166, 2020.
Simon Witzke, Noel Danz, Katharina Baum, and Bernhard Y Renard. Mobility data improve fore- casting of covid-19 incidence trends using graph neural networks. In epiDAMIK 6.0: The 6th International workshop on Epidemiology meets Data Mining and Knowledge Discovery at KDD 2023, 2023.
Xingjian Wu, Xiangfei Qiu, Zhengyu Li, Yihang Wang, Jilin Hu, Chenjuan Guo, Hui Xiong, and Bin Yang. CATCH: Channel-aware multivariate time series anomaly detection via frequency patching. In ICLR, 2025.
Xinle Wu, Xingjian Wu, Bin Yang, Lekui Zhou, Chenjuan Guo, Xiangfei Qiu, Jilin Hu, Zhenli Sheng, and Christian S Jensen. AutoCTS++: zero-shot joint neural architecture and hyperparam- eter search for correlated time series forecasting. The VLDB Journal, pp. 1-28, 2024.
Ronghui Xu, Hanyin Cheng, Chenjuan Guo, Hongfan Gao, Jilin Hu, Sean Bin Yang, and Bin Yang. MM-Path: Multi-modal, multi moulti-granularigg path representation learning extended preprint arXiv:241 1.8428 2024a.
ceaming-extended version. arXiv
Wanghan Xu, Fenghua Ling, Wenlong Zhang, Tao Han, Hao Chen, Wanli Ouyang, and Lei Bai. Generalizing weather forecast to fine- greegra temporal scales via physics-ai hybrid modeling. arXiv preprint 4r0iv:2405.11 13796, 2024b.
13

---

Published as a conference paper at ICLR 2025
Bing Yu, Haoteng Yin, and Zhanxing Zhu. Spatio-temporal graph convolutional networks: A deep learning framework for traffic forecasting. arXiv preprint arXiv:1709.04855 2017.
Haomin Yu, Jilin Hu, Xinyuan Zhou, Chenjuan Guo, Bin Yang, and Qingyong Li. CGF: A cate- gory guidance based pm .{2.5) sequence forecasting training framework. IEEE Transactions on Knowledge and Data Engineering, 10105::1025511119 2023.
Kai Zhao, Chenjuan Guo, Yunyao Cheng, Peng Han, Miao Zhang, and Bin Yang. Multiple time series forecasting with dynamic graph modeling. Proceedings of the VLDB Endowment, 17(4): 753-765, 2023.
14

---

Published as a conference paper at ICLR 2025
A APPENDIX
A.I NOTATION
Table 3: Summary of Notations used in the paper
Symbol
Description
G
Geospatial Graph
Gailt
Diffusion Graph (static)
Gadv
Advection Graph (dynamic)
N
Number of Nodes
V, Vi
Nodes the and i-th node Set of Edges in G Pollutant Concentration at t timestamp (X, E Rx1)
E
X,
A
Auxiliary covariates at timestamp (Ac E IR (D-)))
F
Flux of Particles
V
Wind Field
Lan
Discrete Laplacian operator on Gair
Ladv
Discrete Laplacian operator on Gadv
A W.
Wind speed at at timestamp
A
Wind direction at at & timestamp
k
Diffusion Coefficient
B
Correction term's 's coefficient in BA-DAE (3 E RNXI)
8
B Bl.... 31, ..., [N]
A.2 THE DERIVATION OF MIFFUSION-ADVECTIO EQUATION
According to the Eq by modifying the flow field F. both the Diffusion and Advection equations for pollutants can be derived. (p, t) denotes the spatiotemporal distribution of pollutants, where corresponds to the spatial variable and refers to the temporal variable.
Diffusion Equation The driving flow field for Diffusion is dictated by the distribution of pollutant concentrations, specifically the gradient field of the pollutants. Thus, the Diffusion equation de- scribes the phenomenon where pollutants always diffuse from areas of higher concentration to areas of lower concentration.
0X Öt
k x
(14)
Advection Equation The driving flow field for Advection is the natural wind field. Consequently, the Advection equation describes the phenomenon where pollutants propagate in the direction the wind, with the magnitude of propagation dependent on the wind speed.
ax
0) = 0.
(15)
0t
The Method of Lines (MOL) can discretize the PDEs into grid of location-specine ODEs for solving Eq and [5] Schiesser 2012). According to Chapman Chapman (2015), the discrete equation can be drived as follow:
Discrete Diffusion Equation
dX2 dt
k. Σ
Ganlij](x Xj),
(16)
(EN())
15

---

Published as a conference paper at ICLR 2025
where Ganl represents the pollutants transport weight from V; to V3 under the diffusion scenario, denotes diffusion coefficient, and (i) denotes the set of neighbours of V,
Discrete Advection Equation
d.X; dt
Σ Gadllik]X, + Σ GadrlijJy, Vj-+i
(17)
where Guvvli represents the transport of pollutants from V; to Vú under the advection scenario. The first term indicates that the pollutant concentration at V, decreases due the wind blowing pollutants from Vú to Vj- while the second term reflects the contribution of pollutants transported by the wind from other stations to V4-
A.3 PROOF OF CLOSED SYSTEM'S CONSERVATION PROPERTY
In this section, we use the Discrete Diffusion Equation 16) as example to illustrate the mass conservation law in a closed system. By applying the Method of Lines (MOL) for spatial discretiza- tion, we derive the spatially discrete form of Eq 14 Next, we demonstrate the validity of Eq. 5
dX; dt
k LLinX k-
Σ
Garlij(( - X2), vi
JEN(
Given X2))).
Proof:
N dx, k Ganlij((x X2))
dt
JEN()
k
Ganlyy)
Σ
Gan X
JEN(()
JEN(K)
k
X
Gaillis] - Σ
Ganlltj]
==1
JEN(()
JEN(1)
k
X,d;
k
Ganlis]
(EN()
=0
The final equality holds because the number of times X, is added depends on how many vertices is connected to, which is represented by its degree, d,
follows the same principle as the discrete Advection equation, meaning the discrete Diffusion- Advection Equation also adheres to the mass conservation law.
A.4 GEOSPATIAL GRAPH
is essential to clarify the conditions under which pollutants can propagate. According to the settings in Wang al. (2020), pollutants cannot propagate when the distance is too great or when geographical barriers such as mountains exist between stations. The formula is as follows:
GGeo-Spatial [ij] H(da
diz) H(me m.g)
dij ||; Pill
mij sup (h[Ap: + (1 - A)p=} Max()))),,M())))) AE(0,1)
16

---

Published as a conference paper at ICLR 2025
In this context, H(-) denotes the Heaviside step function, which outputs if the value is greater than O and if it is less than 0. The symbol h(-) represents elevation, while P: denotes the latitude and longitude of station i. Utilizing topographic data along with specified geographical coordinates, can determine the elevation of all regions nationwide. By doing so, we can establish a basic graph structure G. Upon this foundation, by constructing a Diffusion graph and an Advection graph, and subsequently building GNN Blocks for each, we can approximate the Laplacian operator required by the DAE.
A.S DATASETS DESCRIPTION
Beijing dataset comprises data from 35 air quality monitoring stations (measuring PM2.5. PM10- O3. 20,, SO₂, and CO), and metcorological reanalysis data (temperature, pressure, humidity, wind speed, and wind direction), organized from January 1, 2017, to March 31, 2018. Meteorological data for each station is obtained from the nearest grid point. Missing air quality data is initially filled using information from the nearest station, and gaps of less than 5 hours are interpolated along the time dimension. Gaps exceeding 5 hours are removed to preserve data integrity. The dataset is divided chronologically 7:1:2 ratio for training, validation, and testing.
grid format. The dataset contains hourly observations
KnowAir dataset includes PM2.s data and 17 metcorological attributes from 184 cities across China, with observations recorded at three-hour intervals from January 1, 2015, to December 31, 2018. Unlike the Beijing dataset, the KnowAir dataset is divided chronologically in a 2:1:1 ratio due to its ample four-year data span (Wang et al. 2020).
A.6 EVALUATION METRICS
Let (#1,..., Fm) represents the ground truth, and à (.1 - fm)) represents the predicted pollutant concentrations. The evaluation metrics we used in this paper are defined as follows:
Mean Absolute Error (MAE)
m
MAB(2,
m71
1=1
Root Mean Square Error (RMSE)
1
RMSE(2,
[x; 2.)2
Symmetric Mean Absolute Percentage Error (SMAPE)
SMAPE(r,
1=1
A.7 IMPLEMENTS DETAILS
All experiments are conducted using PyTorch 2.3.0 and executed on an NVIDIA GeForce RTX 3090 GPU, utilizing the Adam optimizer. The batch size is set to 32, and the initial learning rate 0.005, which decays at specific intervals with decay rate of 0. A GRU-based RNN encoder is employed for the Coefficient Estimator and the encoders both the Physics Dynamics and Data-Driven Dy- namics. For the ODE solver, adopt the dopri5 numerical integration method in combination with the adjoint method (Chen et al.) 2018). For Dynamics Fusion, X and X2 are set to and 0.8, re- spectively, to differentiate distinet dynamics, and the number of GNN layers is set to 3. The solver's relative tolerance (rtol) and absolute tolerance (atol) are set to le-3.
17

---

Published as a conference paper at ICLR 2025
A.8 NUMERICAL STABILITY
Given that DualOED utilizes an ODE solver in its dual branches and incorporates complex struc- tures like GNN Fusion, it is necessary to verify the model's numerical stability. The training loss curves on the Beijing and KnowAir datasets (Fig- 8) demonstrate that Air-DualODE, achieves con- vergence. Besides, our model incorporates normalization techniques, such as Layer Normalization, which contributes numerical stability.
Training loss per epach on Beijing dataset
Training
KnowAir dataset
Train Loss
0.56
Train Loss
0.54
0.95
0.48
0.90
0.40
0.85
10
Epoch
Epech
Figure 8: The training curves the two datasets experimentally demonstrate that Air-DualODE achieves convergence.
A.9 THE CASE STUDY OF KNOWAIR
Similar to F we visualized sample from the KnowAir dataset. Since KnowAir national- level dataset, we selected time points with two-day interval. Notably, within the purple circle, pollutant concentrations are influenced by the southwest wind, resulting in a gradual decrease in pollutant levels across mainland China. The case studies in Fig/ō and Fig.9) demonstrate that, re- gardless of the scale of the pollutant prediction scenario, Air- Air-DualODR effectively captures the behavior of the Advection equation, thanks to its dual-branch framework.
China 2018-6-23 0:00
NO
China 2018-6-2 0:00
NOR
southwest
KO
southwest
S0
KO
KD
CHINA
CHINA
East Colne
Fost Caice
Figure 9: Visualization of predicted PM25 concentrations under advection. The heatmap shows
PM2s concentration, while arrows represent wind direction.
A. MORE EXPERIMENT RESULTS
To highlight the superiority of Air-DualODE, we provide more comprehensive comparisons by eval- uating prediction performance for one-day, two-day, and three-day intervals. Table 4 is more com- plete than Table in Section 4.2, providing additional short-term prediction results.
At the period-of time level, from Table it is evident that Air-DualODE outperforms all competitive baselines across all three time intervals within three days. To further evaluate its performance over step error, provide the MAE results at 12 individual time steps. From Table 5 Air-DualODE
18

---

Published as a conference paper at ICLR 2025
Table 4: More experiment results on Beijing and Air datasets. The bold and underlined font show the best and the second best result respectively.
1-24h
24-48h
48-72h
Dataset
Model
MAE
RMSE
SMAPE
MAE
RMSE
SMAPE
MAE
RMSE
SMAPE
LatentDD
37.28
54.31
0.70
46,08
67.11
0.83
46.67
68. 82
0.84
PM25GNN
38.58
56.72
0.73
47.29
70.43
0.86
49.82
72.14
0.87
Beijing
Airformer
35.88
55.01
0.69
45.62
65.61
0.81
46.73
68.60
0.83
AirPhyNet
35.69
58.35
0.68
45.06
66.83
0.81
47.42
68.52
0.83
Air-DualODE
32.70
45.15
LatentODE
16.82
26.93
0.39
20.56
31.35
0.47
22.59
33.86
0.52
PM25GNN
15.99
25.53
0.37
20.39
31.33
0.45
21.59
32.99
0.48
KnowAir
Airformer
15.63
25.51
0.36
20.37
31,52
0.45
21.49
33.02
0.47
AirPhyNet
17.74
27.75
0.40
21.72
32.41
0.48
24.47
34.75
0.53
Air- Air-DualODE
32.32
Table 5: Three-Day MAE Comparison at Each Step on Beijing and Know Datasets. The bold and underlined font show the best and the second best result respectively.
Dataset
Model
1st
3rd
5th
7th
9th
11th
13th
15th
17th
19th
21st
23rd
LatemrODE
28.06
34.55
39.38
42.58
45.90
46.37
46.57
46.68
46.61
46.58
46.73
PM25GNN
27.45
36.11
41.59
43.09
45.34
47.33
47.85
48.37
49.62
49.66
49.45
49.33
Beijing
Airfomer
23.88
31.94
38.17
41.76
44.18
45.37
45.72
45.83
46.32
46.63
46.46
46.79
AirPhyNet
23.17
31.66
37.76
40.8%
43,95
44.27
45.61
45.67
46.13
46.65
46.82
47.96
Air-DualODE
18.57
29.68
36,00
39.80
42.00
42.85
43.21
43.76
44.39
44.80
45.23
45.71
LatenrODE
14.26
16,11
17.37
18.23
19.26
20.23
20.83
21
21.82
22.39
22.74
22.98
PM25GNN
10.45
15.60
17.35
17.98
19.03
20.30
20.82
20.85 00.85
21.08
21.66
21.88 21.82
21.65
KirwAir
Airformer
10.28
14.66
16.74
18.01
19.15
19.98
20.54
20.81
2..12
21 34
2..669
21.79
AirPhyNet
14.41
17.18
18.56
19.21
20.27
21.38
22.06
22.44
23.17
24.05
24.71
25.23
Air DuaODE
10.26
14.65
16.59
17.66
18.54
19.33
19.82
20.17
20.47
20.79
21.02
21.12
also surpasses all baselines in the step-by-step comparison. These results demonstrate its strong short-term and long-term prediction capabilities.
A.11 HYPERPARAMETER SENSITIVE ANALYSIS
To examine the robustness of Air-DualODE different hyperparameters, we selected the number of layers n in the GNN Fusion module (Section 3.4) and the coefficient y in the loss function (Eq.??) for sensitivity analysis. As shown in Fig [0 both hyperparameters exhibit low sensitivity on the two datasets, achieving consistently good performance.
A.12 ODE SOLVER SENSITIVE ANALYSIS
To examine the robustness and runtime differences Air-DualODE across different solvers, we trained the model using three different solvers (Euler, RK4, and Dopri5) on two datasets. The inference time refers to the runtime on the entire test dataset. The results are as follows:
Table 6: Sensitivity and Runtime study of different ODE solver
datasets.
Solver
MAE
RMSE
MAPE
Inference time(s)
Beijing-euler
41.23
63.09
0.74
0.9
Beijing-rk4
40.80
62.90
0.74
Beijing-dopriS
40.32
62.04
0.74
1.98
Know
18.92
30.77
0.42
11.4
Know Air-rk4
18.94
30.42
0.42
18.9
Know CnowAir-dopriS
18.64
29.37
0.42
22.5
19

---

Published as a conference paper at ICLR 2025
MAE
MAE
MAD
MAB
RMSE:
RMSE
RMSS
SMAPE
SMAPE
SMADE
0.3
Figure 10: Sensitivity of MAE, RMSE and SMAPE to and y on Beijing and KnowAir.
As shown Table 6 Air-DualODE demonstrates a certain level of robustness to different solver types. Dopri5, adaptive high-order ODE solver, achieves the best performance both datasets. Although it incurs a slight increase in runtime, it provides a corresponding improvement in accuracy.
A.13 VISUALIZATION OF ADVECTION GRAPH
As mentioned in Section 3.2.1, the construction of the Advection Graph is defined Eq To better understand the dynamic nature of Gadv. visualized the wind speed and direction of several stations in Beijing at different time, along with their corresponding Gadv. as shown in Fig
Visualization of Gany
Visualization of Gady
40.8
40.8
Labitude
40.4
Latitude
40.4
40.0
40.0
39.8
39.6
39.4
39.4
115.5
116.0
117.5
115.5
116.0
116.5
117.0
117.5
Longitude
Longitude
Figure 11: Visualization Gady in Beijing. The blue arrows indicate wind direction, and their lengths represent wind speed each station. The orange arrows illustrate the direct edges of Gulv-
In Fig 11 the changes in the Advection Graph depend on variations in wind speed and direction. Specifically, wind direction influences the connectivity between nodes, while wind speed determines the weight of each edge.
20

---

Published as a conference paper at ICLR 2025
A.14 THE DETAILS OF FD
FD represents the ODE function of the data-driven branch, incorporating Spatial-MSA structure as shown in 2 (FD: Masked Attention Brentioo-Based ODE Fusion). The following section provides details about the formula for FD.
Q,K, V = Projectioni (z"),
z = zD + (oatiallMSA(Q.K,,) dz D LN(ZZ= + MLP((L(ZZ))) =F
di
Among these, Q, K, and V obtained through linear projection of Z The spatiotemporal dependencies are then captured using the carefully designed Spatial-MSA and a residual connection in the form of data-driven derivative. Layer normalization is applied ensure numerical stability.
We consider air pollutant propagation as spatiotemporal dynamic system, Neural ODE, serving a bridge between dynamic systems and neural networks, is better suited for modeling the spatiotem- poral dynamics of pollutants compared traditional sequence models like RNN and Transformer.
A.15 VISUALIZATION OF SUDDEN CHANGES' RESULTS
To validate whether Air- Air-DualODE can predict upward or downward trends during sudden changes described in Section 4.1) we visualized sudden changes at specific stations, as shown in Fig [2 The visualization results demonstrate that Air-DualODE can predict the overall upward and down- ward trends, providing important support for downstream decision-maken ever, in some cases, such as those illustrated in Fig 12a and Fig [2b] the peak pollutant concentra- tions were not well predicted. Although Air-DualODE outperforms other methods in the quantitative analysis of sudden changes prediction, shown in Table I predicting sudden changes remains significant challenge.
by policymakers. How-
Visualization Statien-6i 2018.3
Visualization Station-149 in
1.00
Creund Truth
Ground buth
Inlurant con.orttr Allet
03
02
43
20
30
Time
(a) Sudden change's upward trend.
(b) Sudden change's 's upward trend.
Visualization of Station:117 in 2018.1
Visualization of Station 13 in 2018
Nudidon
202
233
Polutint
300
Ialufant
Time
Timm
(c) Sudden change's 's downward trend.
(d) Sudden change's downward trend.
Figure 12: Sudden change visualization on KnowAir dataset.
21

---

Published as a conference paper at ICLR 2025
A.16 ABLATION STUDY ON GNN FUSION
To experimentally demonstrate the effectiveness of GNN Fusion, we replaced GNN Fusion with peexDlaee other variants. Among these, MLP Fusion maps from R. NN2D - NR while Equivalent MLP Fusion uses an equivalent number of MLP layers as GNN Fusion. Additionally, we tested Pool- ing Fusion, which applies mean or sum pooling to each station's representations. The results are presented in Table
Table 7: Effect of GNN Fusion in Dynamics Fusion.
Beijing RMSE
KnowAir RMSE
Methods
MAE
SMAPE
MAE
SMAPE
MLP Fusion
41.49
63.14
0.76
21.50
32.92
0.56
Equivalent MLP Fusion
41.45
63.23
0.75
21.42
32.83
0.56
Mcan-Pooling Fusion
43.23
68.88
0.80
20.03
31.22
0.45
Sum-Pooling Fusion
43.12
68.30
0.80
19.97
31.10
0.44
Air-DuaLODE
40.32
62.04
0.74
18.64
29.37
0.42
According Table we observe that both MLP-based Fusion and Pooling Fusion exhibit degra- dation on the Beijing dataset because the Beijing dataset is city-level, where stations are relatively close each other. Since Pooling Fusion does not consider representations from other stations, its performance degrades more significantly than that of MLP-based Fusion. Conversely, on the KnowAir dataset, we observe that MLP-based fusion methods exhibit sigmificant degradation. This is because, during fusion, these methods incorporate representations from distant nodes, which act as noise for predicting the target station. Such noisy representations fail to provide useful infor- mation and instead interfere with prediction accuracy. Additionally, since Pooling fusion methods do not consider representations from other stations, its results outperform the two MI.P-based fu- sion methods. These two extreme fusions highlight the importance of incorporating representations from neighboring stations during fusion. Based on this intuition, we designed GNN Fusion, which leverages the geospatial graph G to fuse representations from nearby stations. The results in Table 7 demonstrate this design choice is effective.
A.17 DISCUSSIONS
Sudden change Evaluation. We use the current definition of sudden changes and the correspond-
ing metrics to ensure consistency with existing studies (Liang et al., [023] Hettige et al. 2024).
However, they may not be the most appropriate ones. In future work, we plan explore alternative
evaluation metrics, such as mean directional accuracy (MDA) (Van den Burg & Williams 2020).
and adopt new sudden change definitions based on change point detection and anomaly detection Wu et al. 2025).
algorithms (Witzke et al., 2023 Shentu et al., 2025
Numerical instability when removing layer normalization. In our experiments, we observe that Air-DualODE's numerical stability relies on Layer Normalization when using the Dopri5 ODE solver, an adaptive solver Neural ODE. Without it, gradient explosion and NaN occur after a few training epochs, preventing the model from converging stably. This dependency on Layer Normal- ization is closely related to solving the dual dynamics with the adaptive ODE solver, as we do not observe this phenomenon when using either single dynamics branch alone or simpler, fixed-step ODE solvers (c.g., Euler and RK4). The dual dynamics, with their inconsistent numerical ranges, lead to instability during adaptive forward and backward propagation. However, Layer Normaliza- tion ensures consistency of the numerical ranges in the dual dynamics' adaptive solving during each iteration, thereby ensuring numerical stability.
Comparison with AirPhyNet.
Air-DualODE models open air systems, making it more realistic compared to the closed system assumption of AirPhyNet.
22

---

Published as a conference paper at ICLR 2025
Table 8: The difference between Air-DualODE and AirPhyNet.
Difference
NirPhyNet
Air-DualODE
Air system modeling
Closed system
Open system
Physics-guided approaches
Explicit physical equation on latent space
Explicit physical equation on explicit space and GNN fusion
Matching physies latent representations
No mechanism to ensure
Decay-TCI.
Spatial module
GNN-bused differential equation
NODE with Spatial-MSA and BA-DAE
Temporal module
GNN-based differential equation
NODE and BA-DAB
Computational efficiency
Slow
Fast
Air- Di-DuulODE applies physical equations directly to explicit variables, which is more reasonable than applying them to latent variables that no longer hold actual physical meanings.
Air-DuaIODE employs Decay-TCL. for alignment to address the mismatch between explicit phys- ical equations and data-driven latent representations, whereas AirPhyNet overlooks this issue.
AirPhyNet relies only GNN-based differential equations model spatiotemporal dependen- cies, whereas Air-DualODE incorporates multiple components, including BA-DAE for modeling in the explicit space and NODE with Spatial-MSA for capturing additional dependencies in the latent space.
Air-DualODE outperforms AirPhyNet terms of computational efficiency and scalability to larger number of nodes.
A.18 LIMITATIONS
Generalization other domains. this paper, Air-DualODE is specifically designed for air pollutant prediction. To apply it to spatiotemporal prediction tasks in other domains (Guo et al. 2014 Liu et al.] 2018) (c.g.. traffic forecasting and water quality prediction), the physical branch would require adaptive modifications.
Handling sudden changes. As mentioned in Appendix A.15 predicting sudden changes remains challenge in air quality prediction. This is issue that demands significant attention because sudden changes in pollutant concentrations are critical to societal activities and production. In fact, many spatiotemporal dependencies underlying sudden changes cannot be effectively captured by deterministic methods or physical equations, as such changes are often influenced by uncertainties in weather and human interventions. In the future, we plan explore and address this challenge further by incorporating probabilistic approaches like variational inference into the data-driven branch. We leave this as an important and promising future research direction.
23


Reviews:
[{"review_id": "Ys1pxLTDIm", "reviewer": "Reviewer_6Tsm", "summary": "This paper proposes Air-DualODE, a novel approach for air quality prediction that combines physics-based and data-driven methods using dual Neural ODEs. The physics branch implements a modified diffusion-advection equation with a correction term for open systems (BA-DAE). In contrast, the data-driven branch employs masked attention-based Neural ODEs to capture unknown dynamics. The two branches are temporally aligned using a decaying contrastive learning scheme and fused in latent space using GNN, demonstrating superior performance on city-scale (Beijing) and national-scale (KnowAir) datasets.", "questions": "Q1. Given that real air pollution sources (industrial activity, vehicle emissions) and sinks (forests, lakes) exhibit complex non-linear relationships, why did you simplify the correction term $\\beta X$ as a linear term? What is the physical justification for setting $\\beta$'s range to $[−1, +∞)$?\n\nQ2. While you claim that the Physics branch explicitly models physical phenomena, how is this physical interpretability preserved when projecting into latent space?\n\nQ3. Regarding the temporal alignment process using Decay-TCL, how do the chosen values of $\\lambda_1 = 1$ and $\\lambda_2 =0.8$ guarantee physically meaningful alignment? What is the physical significance of using time-decaying weights?\n\nQ4. Why specifically choose Spatial-MSA in the Data-Driven branch? How does this align with a physics-informed approach?\n\nQ5. The authors justify GNN fusion based on 'distance-dependent influence', but isn't this characteristic already considered in the Physics branch?\n\nQ6. How can the authors justify the performance on the national-scale KnowAir dataset when case studies are limited to the Beijing dataset?\n\nQ7. How does the visualization of $\\beta$ values correspond to actual observed pollution source/sink data?\"\n\nQ8. Can authors perform sensitivity analysis for different ranges of $\\beta$ values?\n\nQ9. The authors seem to only consider the DOPRI5 ODE solver. Could they analyze performance and runtime differences when using simpler methods like Euler or RK4?\n\nQ10. The paper should reference and compare with recent work on climate modeling using diffusion and diffusion-advection equations in neural ODE frameworks [1,2]. Can authors clarify their position by analyzing similarities and differences in their approach to diffusion and advection?\n\nQ11. How does the intentional violation of conservation law in BA-DAE affect numerical stability, particularly for ODE solvers?\n\nQ12. How do you ensure that the BA-DAE in the Physics branch and Neural ODE in the Data-driven branch operate in the same state space?\n\nQ14. Is 'Physics-Informed' appropriate in the title? Would 'Physics-guided' or 'Physics-inspired' be more accurate, given that this might be confused with traditional PINN approaches?\n\nQ15. Can authors provide more details about the RNN used in the Coefficient Estimator?\n\nQ16. Figure 2 lacks several elements mentioned in the text, particularly α from equation 6. The relationship between Dynamics fusion and Section 3.4 equations needs clarification.\n\nQ17. Can authors provide visualizations or distribution analyses showing how Gdiff and Gadv change dynamically with wind speed and direction?\n\nQ18. While criticizing the computational cost of existing physics-based methods, how does your dual branch architecture with a complex fusion mechanism improve efficiency? Doesn't using two ODE solvers increase computational burden?\n\nQ19. Can the authors include the number of forward evaluations (NFE) comparisons in Table 2's ablation studies?\n\nQ20. The authors should cover related work on GNNs that redesign the diffusion equation [3,4] and its variations[5,6] using NODE. Can you discuss more about what the authors' methods have in common and what they differ from?\n\n> [1] Choi, Hwangyong, et al. \"Climate modeling with neural advection-diffusion equation.\" Knowledge and Information Systems 65.6 (2023): 2403-2427.\n> \n> [2] Hwang, Jeehyun, et al. \"Climate modeling with neural diffusion equations.\" 2021 IEEE International Conference on Data Mining (ICDM). IEEE, 2021.\n>\n> [3] Wang, Yifei, et al. \"Dissecting the diffusion process in linear graph convolutional networks.\" Advances in Neural Information Processing Systems 34 (2021): 5758-5769.\n>\n> [4] Chamberlain, Ben, et al. \"Grand: Graph neural diffusion.\" International conference on machine learning. PMLR, 2021.\n>\n> [5] Thorpe, Matthew, et al. \"GRAND++: Graph neural diffusion with a source term.\" ICLR (2022).\n>\n> [6] Choi, Jeongwhan, et al. \"Gread: Graph neural reaction-diffusion networks.\" International Conference on Machine Learning. PMLR, 2023.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The paper addresses the limitations of pure physics-based and pure data-driven approaches by proposing a hybrid framework that attempts to leverage the advantages of both methods.\n\n2. The introduction of BA-DAE with a correction term represents an attempt to model open system dynamics, which is more realistic for air quality prediction than traditional closed system assumptions.\n\n3. The model achieves state-of-the-art performance across different spatial scales while maintaining some level of interpretability through its physics branch.", "weaknesses": "1. The paper oversimplifies complex air pollution dynamics using a linear correction term (βX) without proper theoretical justification, undermining its claim of accurate open system modeling.\n\n2. The approach loses physical interpretability when projecting to latent space and violates conservation laws, raising concerns about numerical stability and contradicting the paper's emphasis on physics-informed modeling.\n\n3. The computational efficiency claims are questionable as the dual branch architecture with multiple ODE solvers likely increases computational burden rather than reducing it.\n\n4. The experimental validation is limited, with case studies confined to Beijing data and lacking crucial analyses such as parameter sensitivity testing and solver comparisons.\n\n5. The technical documentation is incomplete, with key mathematical elements missing from figures and insufficient details about architectural choices, making reproducibility challenging.", "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": "bFILgrO9MG", "reviewer": "Reviewer_NWR1", "summary": "This paper introduces Air-DualODE, a novel physics-informed model for predicting air quality that combines the strengths of physics-based and data-driven approaches. Traditional physics-based models are computationally intensive and rely on closed-system assumptions, while data-driven models often lack critical physical insights. Air-DualODE addresses these limitations with a dual-branch design using Neural Ordinary Differential Equations (Neural ODEs): one branch incorporates open-system physical equations to model spatiotemporal dependencies, while the second captures additional dependencies in a data-driven manner. The two branches are temporally synchronized and fused for enhanced predictive accuracy. Experimental results show that Air-DualODE outperforms existing methods in predicting pollutant levels across different spatial areas, making it a robust tool for air quality forecasting.", "questions": "1) How does Air-DualODE perform in regions with sparse or inconsistent air quality data? Are there mechanisms in place to handle data gaps, or do you recommend a minimum data density for effective predictions?\n\n2) Has Air-DualODE been tested on pollutants other than those mentioned in the paper? Could this framework be adapted to predict other types of environmental data, such as water quality?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1) Very well written paper combined with appropriate plots.\n\n2) Recognizing that air pollution transport occurs in an open system, the authors redefine the diffusion-advection equation in explicit spaces. This new formulation, termed BA-DAE, aligns the physical equations more accurately with real-world pollutant transport in open air environments, enhancing the model's applicability and reliability.\n\n3) The proposed model, Air-DualODE, uniquely integrates both Physics Dynamics and Data-Driven Dynamics. This dual-branch structure leverages the advantages of physical models (to capture foundational physical behaviors) and data-driven insights (to adapt to complex patterns not covered by physics alone). This approach represents the first dual-dynamics deep learning model specifically tailored for air quality prediction in open systems.\n\n4) Experimental results demonstrate that Air-DualODE outperforms existing models, achieving state-of-the-art accuracy in forecasting pollutant concentrations across diverse spatial scales, from city-wide to national levels.", "weaknesses": "The dual-branch structure in Air-DualODE, while innovative, adds complexity to the model, potentially making it less interpretable than simpler models. This may pose challenges for stakeholders, such as policymakers, who require clear explanations of how predictions are made. But this is not a very important point neither does it offer any reason to not accept this paper.", "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": "uXTRkVcV9N", "reviewer": "Reviewer_v3np", "summary": "To enhance air quality prediction in open systems, this paper proposes a dual Neural ODE architecture named Air-DualODE. It combines a boundary-aware diffusion-advection Equation (BA-DAE) for physical dynamics with a Neural ODE employing masked spatial attention for data-driven dynamics. The framework also incorporates a fusion mechanism that temporally aligns and merges outputs from these dynamics in a shared latent space. Experimental results on both city- and national-level datasets demonstrate state-of-the-art performance.", "questions": "1.Why using the GNN Fusion after temporal alignment? What about other simple structure like MLP?\n2.Is there other data should be included for geospatial graph construction?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "1.This paper addresses a significant problem, i.e. air quality prediction in open systems. In particular, the model thoroughly considers the non-conservation of pollutant concentration within the region of interest, introducing the BA-DAE to effectively model sources and sinks within the area.\n2.The author propose an interesting Air-DualODE framework that models known physical equations and unknown spatiotemporal dependencies separately, then aligns and fuses them in the latent space. This approach highlights the guiding role of physical knowledge within the model while allowing it to capture unknown spatiotemporal dependencies that may not be described by physical equations.\n3.In experiments, the model achieves superior performance across multiple metrics on different spatial scales datasets. Besides, the provided code enhance this paepr’s reproducibility.\n4.The writing and structure of this paper are clear and easy to understand.", "weaknesses": "1.The discrete diffusion and advection equations are not thoroughly described in this paper. Highly recommend fully explaining in the appendix.\n2.Please elaborate the role of spatial-MSA in data-driven branch presented in Section 3.3.\n3.Check some typos in this paper. For example, in Section 4.2 Table 1, \"Beijing1718\" should be “Beijing”.", "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": "ui3DPmcmG0", "reviewer": "Reviewer_joi5", "summary": "The authors propose a model called Air-DualODE that integrates dual branches of Neural Ordinary Differential Equations (ODEs) to enhance air quality predictions. To address the mismatch between explicit physical equations and implicit learned representations, the paper introduces a hybrid model consisting of two branches.\n\nThe Physics Branch utilizes the Boundary-Aware Diffusion-Advection Equation (BA-DAE) to capture the spatiotemporal dependencies of pollutants. Meanwhile, the Data-Driven Branch captures dependencies not addressed by the physical equations. These dual representations are skillfully fused using a dynamics fusion module with decaying temporal alignment, which enhances prediction accuracy. \n\nThe effectiveness of this method is demonstrated on two widely-used air quality prediction datasets, showing that the model outperforms other existing methods while offering strong interpretability compared to purely data-driven approaches.", "questions": "- Errors in Figure and Table Text: There are textual and informational errors in the figures and tables. In Figure 2, \"Pollutant Contentration\" should be corrected to \"Pollutant Concentration,\" and there is an error in the expression for Wind speed. Additionally, the last row in Table 2 should be labeled as \"Air-DualODE.\" without “w/o”.\n\n- Incomplete and Redundant Equations in the Appendix: The equations following lines 690-692 in the appendix are incomplete and appear to be redundant with the content in lines 219-220 of the main text. Could these be revised for clarity and completeness?\n\n- Detailed Derivation of Equation 3 and $F^D$: It would be beneficial to provide a more detailed derivation of Equation 3 in the appendix to enhance understanding. Additionally, the $F^D$ formula is not provided. Could you include a more thorough description to facilitate reader comprehension?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Enhanced Physical Modeling: This paper innovatively transforms closed-system physical equations into open-system physical equations based on existing Physics-Informed Neural Networks (PINNs). This modification aligns more closely with real-world conditions and improves interpretability in boundary regions.\n\n- Effective Integration of Knowledge: The DualODE model skillfully integrates physical knowledge with data-driven methods to bridge the gap between physical equations and real-world data. This dual approach ensures that the model captures dependencies that are not addressed by physical equations alone.\n\n- Advanced Fusion Technique: By employing dynamic fusion to combine the representations from the two ODE models, and further refining the output through a Graph Neural Network (GNN), the model effectively captures real-world patterns. This approach outperforms the simplistic method of directly concatenating the two representations, leading to more accurate and realistic predictions.", "weaknesses": "- Insufficient Detail on Data-Driven Neural ODE: The paper does not provide a detailed description of the data-driven Neural Ordinary Differential Equations (ODE). Additionally, the necessity of using a Neural ODE model as the data-driven approach is not clearly justified, which may leave readers questioning its selection over other potential methods.\n\n- Limited Results Presentation: The results presented in the paper are limited to predictions for 3 days and scenarios of sudden changes. Predictions for shorter time frames, such as 1-day and 2-day forecasts, are not included. Furthermore, there are discrepancies between the results reported for the reproduction of other works and those in the original publications, which could undermine the credibility of the comparative analysis.\n\n- Language Errors in Figures and Tables: There are minor language errors in the figures and tables that could cause confusion. These errors necessitate careful cross-referencing with the accompanying text to ensure clarity and accurate interpretation of the data.", "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": "lQ2v56zcFV", "reviewer": "Reviewer_iuiU", "summary": "This paper presents Air-DualODE, a hybrid model that integrates physics-based and data-driven techniques. The proposed model addresses the limitations of traditional physics-based and data-driven models, particularly the assumptions about closed systems and the mismatch between explicit physical equations and learned representations. The model involves two main components, where one component applies open-system physical equations to model spatiotemporal dependencies and learn physical dynamics, while the second component identifies the unmodeled dependencies using a fully data-driven approach. The model achieves superior performance  on two real-world datasets at city scale and national scale, compared to the other baselines in this domain.", "questions": "1. The introduction and role of the $\\beta$ term in Equation 6 is not fully clear. Could you provide more details including mathematical justifcation on how this term is derived and its exact significance within the  framework? \n\n2. Could you provide more theoretical justification and intuition behind the Decay-TCL mechanism? Specifically, how do the hyperparameters influence the alignment and fusion process between the physics-based and data-driven dynamics?\n\n3. Could you elaborate  on how the GNN fusion mechanism effectively balances the two components?\n\n4. How scalable is Air-DualODE to larger datasets with thousands of nodes? Could you provide a detailed complexity analysis  of your model as the number of locations (nodes) increases ?\n\n5. Could you provide more clarification on how the predictions can be interpreted and provide specific examples on how these outputs help policy makers or environmental scientists in their decision making process?\n\n6. The proposed model and AirPhyNet (Hettige et al., 2024) employ a similar hybrid architecture of physics based and data driven models , combining GNNs and differential equation solvers. Additionally, both approaches emphasize interpretability through case studies that link predictions to real-world physical phenomena, such as pollutant dispersion influenced by environmental factors like wind speed and direction. Could you elaborate on how your model differs methodologically and in its interpretive capabilities and what specific aspects of your work provide novelty beyond what is addressed in AirPhyNet?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper addresses a significant and timely problem.\n2. The write-up style and the narrative of the paper is clear, structured, and professional. \n3. The reported results are very strong compared to the existing baselines.", "weaknesses": "1. The paper does not adequately address the scalability of the proposed model. Given that the approach involves multiple components like ODE solvers and GNNs, the computational cost could become prohibitive when applied to larger datasets with thousands of locations. (ex: Dataset used in AirFormer (Liang et al., 2023)).  \n\n2. The overall approach and the methodology of this work closely resembles prior work like AirPhyNet (Hettige et al., 2024), which also integrates physics-based diffusion-advection processes with neural networks for air quality prediction, raising concerns about novelty and significant contributions.\n\n3. Eventhough the work emphasizes about the interpretability of the model, there is limited explanation about how the model’s output could be interpreted in terms of real-world physical phenomena.\n\n4. All the baselines used in the experiments are either fully data driven or hybrid models. Advanced physics based models are not included as baselines and the performance is not compared with the proposed model. Can you evaluate your model’s performance against specific physics-based models, such as Community Multiscale Air Quality (CMAQ) , Weather Reaserch Forecasting (WRF), AERMOD to provide a more comprehensive understanding of how your hybrid model compares in terms of accuracy and computational efficiency?", "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)