Eulerian Neural Network Informed by Chemical Transport for Air Quality Forecasting
Authors: Xukai Zhang
Shuliang Wang
Guangyin Jin
Ziqiang Yuan
Hanning Yuan
Sijie Ruan
Venue: NeurIPS 2025 poster ()
Keywords: air quality prediction
chemical transport modeling
Eulerian neural network
Primary area: machine_learning_for_sciences
TL;DR: Embedding physics to improve national air quality forecasts

Abstract: Air pollution remains one of the most critical environmental challenges globally, posing severe threats to public health, ecological sustainability, and climate governance. While existing physics-based and data-driven models have made progress in air quality forecasting, they often struggle to jointly capture the complex spatiotemporal dynamics and ensure spatial continuity of pollutant distributions. In this study, we introduce CTENet, a novel chemical transport deep learning model that embeds the Advection-Diffusion-Reaction equation into a Physics-Informed Neural Network (PINN) framework using an Eulerian representation to model the spatiotemporal evolution of pollutants. Extensive experiments on two real-world datasets demonstrate that CTENet consistently outperforms state-of-the-art (SOTA) baselines, achieving a remarkable RMSE improvement of 45.8% on the USA dataset and 21.0% on the China dataset.

Eulerian Neural Network Informed by Chemical Transport for Air Quality Forecasting
Xukai Zhang', Shuliang Wang', Guangyin Jin', Ziqiang Yuan', Hanning Yuan' Sijie Ruan
Beijing Institute of Technology. 'Sapienza University of Rome
(xkzhang, aiiiinni,iiiinn........ yhnggiinnnn...nn...... yhn6, sjruan) www.eiiiuuut.....eeeoeeoe cn jinguangyin9600maln com
Abstract
Air pollution remains one of the most critical environmental challenges globally, posing severe threats to public health, ecological sustainability, and climate gover- nance. While existing physics- bhysccs-based and data-driven models have made progress in air quality forecasting. they often struggle to jointly capture the complex spa- tiotemporal dynamics and ensure spatial continuity of pollutant distributions. In this study, we introduce CTENet, novel chemical transport deep learning model Difi---iiiiio-oaeatttttee
equation into Neural Network (PINN) framework using an Fulerian representation to model the spatiotemporal evolution of pollutants. Extensive experiments on two real- world datasets demonstrate that CTENet consistently outperforms state-of-the-ar (SOTA) baselines, achieving remarkable RMSE improvement of 45.8% the USA dataset and 21.0% on the China dataset.
that embeds the
Introduction
Air quality refers the concentration of pollutantssuch as PM2... PM ozone (O3), sulfur diox- ide (SO2), nitrogen oxides (NO,), and carbon monoxide (CO), and their impact on human health, ecosystems, and the environment. Exposure to these pollution is closely associated with respiratory and cardiovascular diseases, as well as increased risks of premature mortality[[] 22, B]] Accurate air quality prediction is crucial for mitigating health risks, guiding public health, shaping policies, and enhancing environmental monitoring in smart, sustainable cities.
In recent years, extensive research has explored various modeling approaches for air quality pre- diction. Traditional physics-based models solve partial differential equations (PDEs) to simulate pollutant transport driven by atmospheric dynamics [[] [J]. While these models offer strong in- terpretability and theoretical grounding, they are often computationally expensive and sensitive to input accuracy. In contrast, simple data-driven models learn patterns directly from historical pol- lutant and metcorological observations [[8, Ø]] but they may suffer from limited generalization in unseen or changing environments. Recent deep learning methods have further advanced this di- rection by modeling air quality as a multivariate time series at discrete stations, or by constructing graph representations of station networks to capture spatiotemporal dependencies [R, B.J. To improve predictive performance. these models typically incorporate auxiliary meteorological variablessuch as temperature and humiditycollected from ground-based monitoring sites. addition, some mul- timodal approaches attempt to incorporate complementary information from heterogeneous sources to improve task understanding, robustness, and generalization [[XX, [С]]. Beyond these data-driven models, physics-guides deep learning aims to combine the strengths of both approaches by incor- porating physical knowledge into data-driven architectures(12 [T] This hybrid paradigm seeks to enhance model accuracy and generalizability while ensuring physicall, plausible predictions.
SSijic Ruan and Hanning Yuan are the corresponding authors of this paper.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Despite considerable progress, several core challenges in air quality forecasting persist:
Structural mismatch between continuous Eulerian dynamics and discrete pollutant representations: Whether not they incor- porate physical knowledge, most deep learning methods for air quality prediction adopt station- based modeling paradigms, where data are rep- resented as multivariate time series or discrete graphs|IL I.S, IT, IIZ, [& as illustrated in Fig- Ma.. While these approaches effectively cap- ture temporal dependencies and localized sta- tion interactions, they neglect the spatial conti- nuity modeling of pollutant distributions. This limits their capacity to model the smooth and dynamic spatiotemporal evolution of air pollu- tion over larger regions, shown in Figure Additionally, the physical processes of diffusion and advection are inherently continuous and occur over continuous Eulerian space, rather than being naturally represented on discrete station graphs. Modeling these processes on graphs leads to structural mismatch.
Station-based
|
Fulerian Modeling
Modeling
Pilation
Weather
Meteorology
(a) Discrete pollutant representations
(b) continuous Eulerian representations
Figure 1: Two approaches to air pollution repre- sentation.
Incomplete representation of physicochemical drivers: Another critical limitation lies in the insufficient modeling of atmospheric chemical transformation processes and their meteorological drivers. While physical transport phenomena like advection and diffusion have received some at- tention in hybrid models[[33 [I%], the incorporation of complex atmospheric chemistrysuch photo- chemical reactions and gas-to-particle conversion [20] that generate secondary pollutants (e.g., ozone and Preaannm limited which are strongly modulated by meteorological factors such as solar radiation boundary layer height, vertical temperature gradients. These variables are crucial for accurately simulating chemical reactions and phase transitions in secondary pollutant formation.
Unlike prior physics-guided models that rely on nuution-based graph-based data, this paper pro- poses a novel approach performing Eulerian representation of pollutant evolution, enabling effective modeling of large-scale spatiotemporal diffusion, advection, and chemical transformations. also integrates continuous meteorological data, including satellite-derive inversion, to enrich the model- ing of underlying physicochemical dynamics.
Specifically, we propose the Chemical Transport Eulerian Network (CTENet), a deep learning archi- tecture guided by chemical transport modeling (CTM) principles. CTENet embeds the Advection- biffusion-Reactionn (ADR) equation within a custom-designed neural network that operates on an Eulerian spatial domain, enabling direct simulation of pollutant transport and transformation over continuous space and time. The framework consists of several key modules: (I) the Eulerian Pol- lution Encoder, which encodes pollutant concentration fields within the continuous spatial domain; (2) the Meteorological Encoder, responsible for integrating high-dimensional meteorological data, including wind vector fields; (3) the Fulerian ADR Decoder, which decodes pollutant evolution through a tion, we employ modular and replaceable Spatiotemporal Sequence Predictor that functions as Wind Predictor, Meteorology Predictor, and Pollutant Predictor within the overall framework.
differential equation network reflecting the ADR processes. In addi-
In summary, the main contributions of this paper are as follows:
We introduce novel neural network, CTENet, which integrates a spatiotemporal sequence prediction network with a custom differential equation module, effectively capturing spa- tiotemporal variations in air quality. To the best of our knowledge, this is the first PINN approach for air quality prediction that adopts a spatial continuity perspective under the Eulerian framework.
We integrate CTM with deep learning under an Eulerian framework to improve the model- ing of spatiotemporal continuity in air quality data, explicitly simulating diffusion, advec- tion, and reaction over continuous spatial domains for enhanced interpretability.
We evaluate CTENet on real-world air quality datasets from China and the USA, achieving about 45,8% and 1..0% lower RMSE respectively compared to leading basclines.

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2
Related Work
Air Quality Prediction. Air quality prediction is fundamental for smart city development and urban planning, attracting extensive research efforts. Existing methods generally fall into two cat- egories: physics-based models and data-driven models. The former category characterizes air pol- lution dynamicssuch as diffusion, advection, and chemical transfommationthruugh differential equa- tions solved by numerical methods like finite difference or finite element methods [8L, [S]. While these models offer interpretability grounded in physical laws, they often require substantial compu- tational resources and depend on high-quality input data. Data-driven models leverage historical pollutant data and contextual information, including meteorological variables, to Iearn spatiotem- poral dependencies. Early machine learning methods such random forests and support vector machines showed initial success [21] but were limited in capturing complex spatiotemporal patterns. With advances in deep learning, architectures combining convolutional and recurrent networks (e.g.. CNN-LSTM) [ and graph-based networks [ [6] have significantly improved prediction accu- racy by modeling spatial and temporal dependencies more effectively. Attention-base based models [[С]] and recent methods like Airformer [[&]] further enhance large-scale spatial correlation modeling, but they do not explicitly model the continuous spatial distribution of pollutants. In addition, some studies, such as STENN [23] and AirRadar [!Z!], emphasize modeling the spatial continuity of atmo- spheric processes, exhibiting strong capability in reconstructing and reasoning about pollutant fields, although they focus on inferring pollutant distributions rather than forecasting.
Physics Guided Deep Learning. Recent advancements have explored the integration of physical principles into deep learning architectures [37] GS). A prevalent approach involves incorporating physical knowledge into the loss function, wherein deviations from known physical laws are penal- ized to promote physical consistency and enhance model generalizability [69, 200]. However, the effectiveness of this method critically depends on the accuracy of the underlying physical knowl- edge. Inaccuracies can introduce inductive biases, thereby hindering the models representational capacity. Alternatively, hybrid frameworks have been proposed that embed scientific knowledge into specific components of the model architecture, enabling a more flexible and potentially more robust integration of physics into the learning process [MT, [2]). In recent air quality forecasting studies, physics-guided frameworks have been increasingly adopted to better capture the underlying dynamics of pollutant transport and transformation. For example, some studies have incorporated advection-diffusion mechanisms and fluid dynamics constraints into learning models [I2]]. Other works introduced state-space concepts inspired by physical systems represent temporal evolution [Σ]. Meanwhile, several approaches [ [ have modeled air quality dynamics as graph-bascd systems constrained by advectiondiffusion processes, enhancing interpretability and generalization. However, the physical processes are limited to the graph edges, without modeling spatial continuity.
3 Problem Formulation
Given historical pollutant concentration data Pı:r E RTxXCxXN from N observation stations located spatial coordinates S [(h,, ( and the corresponding continuous meteorological data MiT E R TCx x XW where CM denotes the number of channels, including those for east-west and south wind components well as other metcorological variables.
We aim to predict the pollutant concentrations at all station locations over the future time period from T + 1 to T + +,, denoted by PT41.+77 Formally, the task is to learn the function F:
Prt1t+ = F(P.r,,,,)) P E CTXC
(1)
4 Methodology
4.1 Model Overview
In this paper, we propose the Chemical Transport Eulerian Network (CTENet), as illustrated in Figure 21 The framework comprises three key components: the Fulerian Pollution Encoder, the Me- teorological Encoder, and the Fulerian ADR Decoder. Additionally, a replaceable Spatiotemporal Sequence Predictor functions the Wind, Meteorology, and Pollutant Predictor in the framework.
3

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Meteorology Wind
Pollutant
Pollutant Predictor
Prodictor
PT.1.Ter
Query
Advoction
output
Metcorological Encoder
Eulerian ADR Decoder
Cony
Diffusion
Meteonoooyy Predictor
Wind Predictor
pa
Spatiotemporal
Sequence
Eulerian ADR Decoder
Reaction
Predictor
R(X+
Metcorology Prodictor
Predictor
Fulerian ADR Decoder
(c)
input
Pollutant Map
Ixl Conv
Extract
Eulerian Pollution Encoder
RBF
T
Meteorological Data
Pollutant Data
(a)
X
steps
steps
Most recent time step
Co
Select recent time step
M 1/T
P 1:T
Update time step
(d)
Figure 2: Structure of CTENet. Pollutant data is processed by the (a) Eulerian Pollution Encoder, while meteorological data are fed into the (b) Meteorological Encoder. The outputs are jointly passed to the multi-layer (c) Eulerian ADR Decoder, followed by a query step to generate the final output. (d) Spatiotemporal Sequence Predictor represents the general building block used in the three forecasting modules.
4.2 Eulerian Pollution Encoder
At each time step, pollutant concentrations from monitoring stations are assigned to their spatial locations, averaging values when multiple stations fall within the same pixel. However, these mon- itoring stations are sparsely distributed and cannot fully cover the entire spatial domain, as shown Figure For locations without monitoring stations, we perform Radial Basis Function (RBF) interpolation to generate continuous Eulerian representation of pollutant distribution X, as shown Figure This continuous representation facilitates the modeling of spatial pollutant dynamics and serves as a proxy for learning air quality prediction over the entire domain.
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Figure 3: A sample of ground truth for PM2.5 concentrations across the continental United States
Figure 4: A sample of RBF interpolation result for PM2.5 concentrations across the continental United States
Compared to other interpolation methods, RBF is particularly well-suited for modeling air pollu- tant data due to its flexibility, smoothness, and ability to handle irregularly distributed monitoring stations. It provides a kernel-based and distance-basse the complex, non-linear spatial variability often observed in environmental data, making it ideal for constructing continuous inputs for deep learning models.
interpolation framework that adapts well to
The fundamental idea of RBF interpolation is infer a continuous field from discrete observations by expressing the spatial relationships among data points through radial basis functions. Specifically.

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the interpolated value at a spatial location x is computed as:
rbf(x) = E x Φ (1x x
LE (I....;7)
Here, x, represents the position of the i-th known data point, and ||x x denotes the Euclidean distance between x and x The function d(-) is the chosen radial basis function, and X, is the interpolation weight for each data point. To determine the weights A [1 AA we solve
the following linear system based on the known observations y, at the locations x!
(
yr E (1,...,7)
where 1
(1xx;0 - xx001 is the kernel matrix constructed from pairwise distances between known data points, and (t) [M] ................................ yr ]T is the vector of known pollutant values. In this paper, we adopt the Multiquadric function as the radial basis function, defined as:
<(r) = 2
where r
||x x; and c > is a smoothing parameter that controls the flatness of the basis function. The Multiquadric kernel is particularly effective in capturing smooth spatial patterns in complex environmental fields.
Meteorological Encoder
Meteorological data M is processed by extracting wind features for ADR modeling and applying channel-wise transformation to allow interactions among heterogeneous meteorological variables, yielding compact yet informative feature representation. Below, we detail these two processing methods.
First, we extract the wind-related channels from M. denoted as W E RTx2XxXW where the two channels represent the eastwest and northsouth components of horizontal wind. These wind features are essential for capturing directional wind patterns that influence pollutant dispersion.
In parallel, apply 1 x convolution over the full meteorological tensor M, acting the channel dimension, to reduce the number of channels while enhancing the representational capacity:
F Coneix ( F TT
(5)
where Cf denotes the number of output channels the transformed meteorological representation. Since future weather conditions have significant impact on air quality, meteorological forecasting integrated into the model. The Meteorological Encoder includes two specialized components: the Wind Predictor, which forecasts wind vectors W, and the Meteorology Predictor, which predicts gen- eral meteorological features F. Each predictor can adopt any Spatiotemporal Sequence Predictor, such as ConyLSTM | or TAU [45]]. The inputs to these predictors are four-dimensional tensors of shape (T, C, H, W' shown in Figure E (d), where C is the number of channels specific each predictor depending on the input variable types. Each predictor generates a one-step-ahead forecast, producing an output with a time dimension of The predicted outputs from both components subsequently incorporated into the Eulerian ADR Decoder to enhance predictive performance and ensure better alignment between future meteorological conditions and pollutant dispersion patterns.
4.4 Eulerian ADR Decoder
Eulerian ADR Decoder integrates crucial domain knowledge from atmospheric sciences, particularly Chemical Transport Models (A))))AG) simulate pollutant dynamics grounded in physical and chemical principles. Central to CTM is the AddvetionDiffusioRReeeem governs the spatiotemporal evolution of pollutants in the atmosphere.
(ADR) equation[H7],, which
The main idea of ADR equation to describe how pollutant concentrations change over time and space due to four fundamental mechanisms: advection, diffusion, chemical reaction, and external
5

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source term such as sources and sinks. The continuous form of ADR equation can be expressed as:
0x
VX
R(X)
S
(6)
Ot
Adoection
Diffusion
Reaction
Source
where ky is the diffusion coefficient. The mathematical analysis of ADR equation can be found in Appendix Hld the boundary conditions in .
To enable numerical implementation, discretize the continuous ADR equation using the im-
plicit forward-time central-space (FTCS) finite difference method 188 to approximate the spatial and temporal derivatives.
Spatial discretization: We apply the central difference method 19), which allows us to approxi- mate spatial gradients and diffusion terms in a numerically stable manner:
X + 1,
X 1,51 X + 1] X(i,j -1) 1]
VX= ~
2Ar
2Ay
X 1, X 1. + X(. 1] + X(i.j - 1] 4X
V V
(8)
4ArAy
Here, [ j] refers to the pollutant concentration at the grid point (i,j), and Ar, Dy are the spatial resolutions in the longitude and latitude directions, respectively.
Temporal discretization: We adopt the explicit Euler method [9 for temporal discretization. The explicit Euler update for the pollutant concentration X is given by:
Xr+1= Xr At (-Wr- VXT + kg V + R(XT) + Sr)
where At is the time step size, Xr is the concentration at time step T, and X7+1 is the updated concentration at the next time step.
For different terms, we adopt explicit and implicit modeling strategies based real-world consider- ations. The following introduces each term in equation g:
Advection term is represented as Ar+1 = Wr VXT, modeling the transport of pollutants due to the wind. The advection term accounts for the movement of pollutants along the direction of the wind, simulating the large-scale, directional transport across the spatial domain. Explicitly modeling advection allows the system to capture the influence of wind fields on pollutant movement, making it especially crucial for accurate spatiotemporal predictions in areas where wind plays a significant role in pollutant dispersion. This is partieularly important in scenarios involving strong wind fields or long-range pollutant transport, where pollutants can travel vast distances within short periods.
Diffusion term is represented as DT+1 kg V²XT, where the diffusion coefficient ke is learnable here to capture dynamic diffusion behavior during training. This term models the random spread of pollutants due to molecular motion or turbulence, accounting for the gradual diffusion of pollutants across areas with varying concentration levels. The diffusion term is essential for capturing local mixing effects and pollutant dispersion in regions with weak wind or when pollutants encounter ob- stacles that hinder direct transport. By incorporating a learnable diffusion coefficient, the model can adapt to different environmental conditions and capture the dynamic nature of pollutant dispersion.
Reaction term models nonlinear chemical transformations among pollutants, such as oxidation reactions, photochemical reactions, and secondary particulate matter formation. These reactions are highly sensitive to atmospheric conditions, including temperature, solar radiation, and humidity, among others. Instead of attempting to model thousands of complex chemical reaction equations, which are difficult to construct accurately, we opt for more flexible approach. Meteorological features Fi.r are mapped through a sigmoid activation, yielding modulation coefficients constrained to the range (0, 1). which can be interpreted as a form environment sernmeen--guiddd soft attention applied along the channel dimension. Specifically, the chemical reaction term is computed by clement-wiss, multiplying the pollutant concentrations with the corresponding modulation coefficients, as shown below:
(10) Source term is represented as S7 is not explicitly parameterized in our model, but im- plicitly captured by the deep neural network. This allows the model to flexibly learn complex spa- tiotemporal emission patternssuch as anthropogenic and biogenic sources depositionbased on
R(X++1) = o XT

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observational data and meteorological inputs, without relying on fixed or analytical formulations of typically inaccessible or sparsely available factors.
Additionally, intermediate representation Xr+1 is generated by applying the Pollutant Predictor to the historical sequence X1.r. The Pollutant Predictor shares a similar architecture with the Wind Predictor and the Meteorology Predictor.
XT+1 = Pollutant Predictor (XuT Fi:T), Xr+1 RICCXXXHX
(11)
The vertical bar denotes concatenation along the channel dimension.
In contrast to the additive formulation in equation A we employ a modified approach where the com- ponents are concatenated along the channel dimension, allowing the model flexibly learn complex, nonlinear interactions between features while preserving their individual information before fusion. In addition, we concatenate the predicted future state X1+1, which serves as a proxy for the true future state, with the meteorological features Pr, providing the model with richer context that goes beyond the traditional ADR modeling capabilities. Finally, a x convolution is applied to generate the final prediction XT+1:
X = Convixi (xr ||
At (Artal) ( R(X++1)) || Xr+1 || Fr)
(12)
4.5 Query and Optimization
Upon obtaining the predicted Fulerian representation X E RRIXCxxHXW, we query the values at the station locations to obtain the final output as:
Pr+1 Query(XT+1..3)
Pr+i E RIXCxxN
(13)
CTENet performs recursive multi-step forecasting of pollutant concentrations over a future horizon. At each iteration, it takes the past 7" time steps of input to predict the pollutant concentration X, and uses sliding window mechanism advance the input sequence by one step rep--eelacing the oldest input with the latest prediction. By applying this process iteratively, CTENet generates the multi-step prediction sequence PT+1 E RCxXNN
For optimization, we minimize the discrepancy between the predicted and ground-truth pollutant concentrations at the station locations over the entire forecast horizon. Specifically, the model, pa- rameterized by O trained by minimizing RMSE loss:
T+r
N
(())
Σ
Σ
p(
(14)
TN
where p(") and p(n)) denote the ground truth and predicted pollutant concentration vectors at station and time t.
5 Experiments
We compare our proposed CTENet with the representative and SOTA models to evaluate their effec- tiveness real-world pollutant datasets. Following the common practice in previous related studies, this study focuses on PM2.5 concentration as the target variable, with meteorological data serving as auxiliary variables. Unless otherwise specified, TAU [A5] is used the default Spatiotem- poral Sequence Predictor type in all experiments. Further experiments details are provided in the Appendix 0 and
The pollutant monitoring station datasets are collected from China[[8]] and the United States/[[0], covering the entire year of 2018. Simultaneously. obtain meteorological data from the National Centers for Environmental Prediction (NCEP) [5II] for the same time and locations.
Dataset:
Baselines: We choose the baseline methods from three categories: (1) Statistical models, including
Historical Average (HA) and Vector Auto-Regression (52)) (2) Graph-based models, includ- ing STGCN [53]] DCRNN [Н4], GTS [b5], AirFormer [|] AirPhyNet [[B), and PM2 GNN [[6]]

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(3) Spatiotemporal sequence prediction models, including TAU. M To ensure fair comparisobn with site-based models, we assign the pixel value at each grid cell all stations located within that cell. More details can be found in Appendix C
Each dataset is divided into training, validation, and test sets in 7 1.5 1.5 ratio. We use data from the past 72 hours to predict pollutant concentrations for the next 24, 48, and 72 hours. Our implementation code is released for public use
Implement Details:
5.1
Main Results
Table @ presents a performance comparison between CTENet and several baseline methods across two datasets. We can draw the following observations: (1) CTENet outperforms the other base- line models in terms of overall performance, regardless of whether ConyLSTM or TAU is used as the prediction module. Specifically, compared to the rest-performing baseline, our best CTENet variant achieves an average MAE improvement of 33.1% 1% and an RMSE improvement of 45.8% in the USA, while in China, it achieves an average MAE improvement of 15.8% and an RMSE im- provement of 21.0%. This substantial improvement underscores the effectiveness and superiority of the CTENet framework. Notably, when using TAU as the predictor, CTENet achieves optimal results in most cases; (2) traph-basee deep learning based methods clearly outperform traditional statistical methods in prediction accuracy, demonstrating the powerful capability of deep learning in learning complex spatiotemporal relationships and modeling nonlinear patterns; (3) Spatio-temports deep learning models originally developed for traffic forecasting, such as GTS and STGCN, exhibit suboptimal performance in our experiments. This may be attributed to their lack of specialized ar- chitectural adaptations for processing lige--imensional meteorological inputs. sequence prediction models, such as TAU, show great potential in fine- aeeegraiied air quality prediction tasks, performing on par with SOTA graph- models.
(4) Spatiotemporal
Table 1: PM2.5 prediction performance: the best-performin result is highlighted in bold, while the
second-best best underlined for easy comparison.
USA
China Data
Methods
24h
480
72h
24h
48h
72h
MAE
RMSE
RMSE
MAE
RMSE
RMSE
RMSE
MAE
RMSE
HA
5.30
11.57
3.66
12.54
3.59
13.23
21.64
38,03
22.76
39.12
23.38
40,03
VAR
6.32
14.41
5.78
12.74
5.76
12.94
24.74
39.85
25.43
41.85
26.66
44.14
STGEN
4.29
9.03
4.51
9.03
4.63
9.08
31.43
43.72
31.91
44.06
32.69
44.75
DCRNN
5.40
14.50
5.42
12.81
5.38
13.48
28.14
49.81
27.45
47.36
27.39
47.63
GTS
5.57
14.65
5.60
14.32
5.61
14,18
23.46
41.70
23.50
42.53
23.85
44.41
AirFormer
4.05
10.44
4.40
10.74
4.60
10.89
19.09
36.08
20.89
38.42
21.85
39.61
AirPhyNNet
4.47
11.36
4.79
11.40
4.94
11.48
18.75
36.35
19.97
37.16
20.74
37.64
PMNN-GNN
4.38
9.77
4.63
9.66
4.76
9.63
17.71
33.25
19.12
34.16
19.73
34.53
TAU
4.71
4.94
13.56
5.22
13,90
15.85
27.35
26.85
CTENer w/ COnVLSTM
4.12
8.46
4.31
8.66
4,43
8.84
13.79
23.14
14.44
23.79
15.28
24.47
CTENet TAU
2.66
2.99
3,10
5.00
16.99
13.28
22.60
26.74
% Best Improvement
34,43
46.02
32.06
46.18
32.68
44.86
31.24
36.60
13.97
17.36
2.04
8.86
Overall, these experimental results validate the exceptional performance of CTENet in air quality prediction tasks. By combining a CTM-based pollutant model with an Eulerian spatiotemporal mod- eling framework. CTENet effectively leverages spatial continuity and temporal dynamics, enabling more accurate and physically consistent air quality predictions.
5.2 Ablation Study
To evaluate the individual contributions of each component in the model for 24-hour predictions, we conduct a systematic ablation study by progressively removing components. Figure 5 presents the ablation results for the ADR terms, while ablation experiments related to the interpolation method are provided in Appendix R
Effect of Advection and Difussion term: Full CTENet outperforms models with the removal of the diffusion or advection modules on both the China and USA datasets, highlighting the benefits of incorporating physical knowledge. The removal of the diffusion term leads relatively minor degradation in performance, particularly in the China dataset, possibly because the multilayer CNNs in the predictor are able to partially compensate for its absence by implicitly modeling local spatial dependencies. The theoretical proof is provided in Appendix (B. In contrast, removing the Advection
8

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CTENet Diffusion Reaction Advection
China USA
30
12
25
19
9
RM5
18
RMSE
15
10
16
64
China
Hidden Dimension
Figure 5: Ablation study ADR terms.
Figure Hidden dimension analysis on RMSE.
module causes a more significant performance decline, emphasizing the difficulty standard deep networks face in learning pollutant transport due to wind, and the necessity of integrating domain- specific physical mechanisms into the model architecture.
Effect of Reaction term: Removing the reaction term results in noticeable performance decline, even though historical meteorological features remain part of the input. This indicates that the benefit of reaction term lies not in simply incorporating additional features, but in its role dynamic weighting mechanism. By adaptively modulating the influence meteorological inputs, the reaction term enhances the model's s ability represent complex pollutant dynamics, highlighting the value of structured chemical integration in CTENet.
5.3 Hidden Dimension Analysis
In this experiment, we evaluate the RMSE performance of CTENet under different hidden dimension settings for two datasets, as illustrated in Figure X The results show that increasing the hidden dimension generally leads lower RMSE both datasets, indicating improved model performance. However, the improvement is not indefinite: for the China dataset, the RMSE plateaus beyond hidden dimension of 32, while the USA dataset achieves its lowest RMSE at 64. This suggests that increasing the hidden dimension yields diminishing returns, model complexity eventually exceeds the intrinsic structure of the data, potentially leading overfitting. Moreover, the optimal hidden dimension varies across datasets, likely due to differences in data complexity or noise characteristics.
5.4 Case Study
To demonstrate the interpretabil- ity and physical consistency of our model, we present a case study based predicted PM2.5 concentrations over cen- tral China. Figure Ø displays the spatial distribution of PM2.5 at four consecutive time steps. In the highlighted red box, north- west winds are observed first three time steps. As a result, the pollutant mass shifts toward the southeast in the subsequent frame, indicating that the trans-
10
X
the
2018 06:00 2018-05.22 25 2:00
Figure 7: Visualization of PM2.5 concentrations as heatmaps. The first row shows reference distributions X overlaid with wind streamlines, and the second row shows model predictions X. port of PM2.s is consistent with the wind direction. This spatial displacement of pollution illustrates how CTENet captures the dynamic advection of pollutants, consistent with atmospheric physical principles, and reflects the model's capacity to learn meaningful spatiotemporal interactions.
6 Conclusion and Future Work
In this work, we propose the CTENet network for air quality prediction, which integrates physics- guided models with deep learning techniques to effectively model the spatiotemporal evolution of pollutants. Comprehensive experiments several benchmark datasets have validated the effec-

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tiveness of the proposed method, achieving remarkable RMSE improvement of 45.8% on the USA dataset and 21.0% on the China dataset. Our current formulation lacks uncertainty quantifica- tion, limiting its applicability in risk-sensitive scenarios such as probabilistic forecasting or decision- making under uncertainty. Future work will explore integrating uncertainty-awarr mechanisms to improve reliability and interpretability.
Acknowledgments and Disclosure of Funding
This research is supported by National Natural Science Foundation of China (No. 62306033, 42371480).
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Justification: The core method development in this research does not involve LLMS as any important, original, or non-standard components.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, non-standard components.
Please refer to our LLM for what should or should not be described.
20

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A
Notation
Table 2: Summary of notations used in the paper
Symbol
Description
N
Number of observation stations
T
Input time step and prediction time step, respectively Total number of grid points in the latitude and longitude dimensions Historical pollutant concentration data at time
H,W
P
X
Spatially interpolated historical pollutant concentration data at time Continuous historical meteorological data at time
M
Cp
Number pollutant types and the number of pollutant channels Number of meteorological channels Number of meteorological feature channels. Continuous wind vector at time
CM
CF
W.
Fr
Continuous meteorological features at time /
S.
Source Term at time Function modeling the chemical transformation of pollutants
R(-)
kg
Learnable diffusion coefficient
@
Element-wise product
a()
Sigmoid function: (xx) Ite-z used to squash values between 0 and 1.
1
Channel-wise concatenation
B Proof of Convolution Kernel Fitting Diffusion
The diffusion term he V X is formulated as the product of the diffusion coefficient ka and the Laplacian operator of the pollutant concentration X, whose discrete form is defined in Equation (8). In this section, we demonstrate how a 3 3 convolution operation in convolutional neural network can approximate the discretized Laplacian operator. The convolution operation can be viewed as weighted summation spatial data using sliding window, which can be expressed as:
Ybijj- = Σ Σ
K|m n| X(i + m..j + n]
(15)
n=-1
where K the convolution kernel, X 5] the input data, and Y(i. j] the result of the convolution operation. The model ultimately learns an appropriate convolution kernel K, which, when close to following form, causes convolution operation to approximate the Laplacian operator:
1
K =
1
-
(16)
0
1
0
This completes the proof. related empirical demonstration can be found in [56].
C Baselines Details
We compare our CTENet with SOTA models for long-term graph sequence prediction and spa- tiotemporal sequence prediction models, some of which are specifically optimized for air quality prediction.
НA uses the average of the historical observations as the predictions. This simple baseline can provide insights into the effectiveness of more complex models.
21

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VAR [22|
is classical statistical model that was originally proposed for macroeconomic analysis. utilizees past values of the multivariate time series to make predictions. captures temporal dependencies without considering the spatial aspect.
STGCN [ح3]]]
integrates both spatial and temporal dependencies, leveraging graph-based convolutions to predict air quality over time.
DCRNN [ improves on STGCN by incorporating recurrent layers and diffusion-based
graph convolutions to capture dynamic relationships in both spatial and temporal domains.
GTS |>>] proposes method for simultancously learning the graph structure and forecast- ing multivariate time series using GNNs, where the graph is unknown, by optimizing probabilistic graph model.
Airformer [TS] is novel Transformer-bact based model designed to predict nationwide air quality in China with fine spatial granularity, incorporating a two-stage learning process to efficiently capture spatio-temporal representations and intrinsic uncertainty.
AirPhyNet [[3] is a novel approach that integrates physical principles of air particle move- ment (diffusion and advection) into neural network architecture on graph-based data struc- tures, improving air quality prediction by capturing spatio-temporal relationships and pro- viding interpretable, physically meaningful predictions.
PM2.5-GNN [16)] is a novel graph-based model designed to capture both fine- mme-grained and long-term dependencies PM2.5 forecasting, incorporating critical domain knowledge and validated on real-world data, with an online deployment for free forecasting service.
TAU BSS decomposes temporal attention into intra-frame statical attention and inter-frame dynamical attention to efficiently capture both spatial dependencies within single frame and temporal dependencies across frames, enabling parallelization of the temporal module in spatiotemporal predictive learning. .ffer referring to [57,, 158], we have also decided to adopt TAU as one of the predictors in our model.
The key hyperparameters in other baselines are also tuned for the best performance.
D Experimental Setup
Datasets Details: We conduct extensive experiments on two real-world pollutant monitoring station datasets collected from China and the United States, covering the entire year of 2018, to evaluate the performance of our proposed CTENee.. The Chinese dataset[[8]] includes 480 primary monitoring stations. The United States dataset[[50] consists of 365 monitoring stations across the contiguous United States. Both datasets provide hourly air quality. Simultaneously. we obtain meteorological data from the National Centers for Environmental Prediction (NCEP) [[]]] for the same time and locations, covering 354 channels with information on temperature, humidity, wind speed, potential height, vertical velocity, sunlight, cloud cover, and other variables at different isobaric levels, with spatial resolution of 0.5ř in both latitude and longitude.
Implementation Details: To align the meteorological data temporally, treat each 6-hour period as one time step, averaging the pollutant concentrations within that period. During model training, employ techniques such regularization, gradient clipping, and progressive learning to improve training efficiency.
E Loss and Evaluation Metrics
Let x = (I1...., Fm)) represent the ground truth, and x = (#1...., fm) represent the predicted pollutant concentrations.
The loss function used in this paper are defined as follows:
Mean Squared Error (MSE) Loss
MSELoss(x,XX)
Σ (x. 2,
(17)
1=1
The evaluation metrics used in this paper are defined as follows:
22

---

Mean Absolute Error (MAE)
MAE(x, x)
24 24
(18)
=1
Root Mean Square Error (RMSE)
RMSE(X, x)
(x; - 8;)2
(19)
m
1=1
Ablation Study of Interpolation
As shown in & To investigate the effectiveness of the interpolation method used in our model, we conduct an ablation study comparing three variants: (I) our default RBF interpolation, (2) nearest- neighbor interpolation, and (3) a variant without any interpolation. As shown in Figure N. the RBF interpolation achieves the best performance among the three settings, demonstrating its ability to pro- vide smoother and more physically consistent spatial inputs, which benefits downstream prediction. Nearest-neighhor interpolation, while simple, introduces artifacts due to abrupt transitions between points. Removing interpolation altogether results in degraded performance, primarily because the model fails effectively capture spatial continuity and interactions when operating on sparsely and irregularly distributed observational data.
RBF Interpolation w/ Nearest Interpolation Interpolation
12
11
MAE
RMSE
Figure 8: Ablation Study of Interpolation.
G
Computational Complexity Analysis
The main computational cost of the proposed algorithm arises from the Eulerian ADR Decoder, including the computation of the three ADR terms and the three Predictor modules. The ADR terms consist of advection, diffusion, and reaction, each with computational complexity of O(B - -W). Each of the three Predictors (e.g., TAU3) contributes computational complexity W), where B is the batch size, 7 is the length of the input sequence, and D is the hidden dimension. Summing the complexities of all components, the total computational complexity is:
O(B И W + OBB T D2 H -W) O(B H.-HH W)
(20)
For reference, the computational complexity of graph- based methods is approximately O(B T D2 N)). where N is the number of spatial nodes in the graph,which is typically smaller than H W. as shown Table On a single RTX 4090 GPU, the complete training process can completed within 3 to 6 hours, depending on the data volume and forecast horizon. More importantly, during inference, generating 72-hour forecast takes less than 0.15 seconds, which is fast enough to meet the requirements of most real-time applications.
23

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Table Comparison of Theoretical Computational Complexity between CTENet and graph-based
Methods,
Methods
CTENet
Graph-based Methods
Theoretical Complexity
O D* H. W)
O T DD N)
China Dataset
O(B.T. D² .00 130)
O T D² D48)
USA Dataset
O(B -.....D.... -2. 140)
O T D2 -3655))
H
Mathematical Analysis of the ADR Equation
The ADR equation & satisfies the existence, uniqueness, and stability conditions, making it well- posed problem for modeling physical processes. A formal proof of these properties is provided below.
The ADR equation is second-order linear parabolic PDE in both space and time. The well-known theory existence for parabolic equations asserts that, for smooth (analytic) coefficients and appropriate initial and boundary conditions, there always exists at least one solution in a finite domain.
Existence and Uniqueness:
To prove the uniqueness of the solution for the ADR equation, we follow the standard method of analysis for parabolic partial differential equations. By using the energy method, we assume two solutions X, (f) and (t) satisfy the ADR equation. Defining the difference w(t) (t) X2)),, we derive the following equation for w:
dw V.. ko V2w R((x))
(21)
dt
Applying the energy method by multiplying both sides by and integrating over the spatial domain, we obtain:
di [ w² de -ko
d
Vwol dx
(22)
This shows that Jq 2 dx decreasing over time, implying that w(t) tends to zero, which leads to the conclusion Xi(t)= X2)). Hence, the ADR equation has unique solution.
Stability: For stability, we consider small perturbations the initial condition. Let X, and X₃ be two initial conditions such that (Xo X1 < The difference between the solutions (t( and Xt(t) is given by:
w(t) = (t) (t())
(23)
Using the energy method again, we find that the difference in the solutions remains bounded and decreases over time:
J w² w dx - / (X0--11)
x2) dr
(24)
Thus, small changes in the initial condition lead to small changes in the solution, demonstrating the stability of the ADR equation. This confirms that the ADR equation is a well-posed problem under the given initial and boundary conditions.
I Boundary conditions
In the absence of external pollutant data beyond the study area, we apply Neumann boundary condi- tions [28] to maintain the stability and convergence of the numerical method. To minimize boundary effects, the computational domain is set least 300 km away from the nearest monitoring station.

---

Neumann conditions enforce a zero prooggradeent (no-fluxx condition at the boundaries of the domain:
0x
ax on the boundary
(25)
By
J Statistical Significance Analysis
In this appendix, derive approximate probability that method A performs better than method
B, based solely on their respective summary statistics: mean absolute error (MAE) and root mean square error (RMSE). Let these values be denoted MAEA, RMSEA for method A and MAEg, RMSEg for method B.
Table M can be interpreted as paired tests between our method and each baseline model.
(B) denote the predictions of methods A and B Let y/ denote the ground truth value and (4) 94 sample i. We define the absolute prediction error as:
(f = ((A) Vi d (B)
V/
(26)
We aim to estimate the probability that d( < d, ie., method A outperforms B on randomly drawn sample. We assume the errors dfA) and df independent and approximately follow Gaussian distributions:
det² N(H,, ah ~N(UB,
(27)
with parameters estimated from available statistics:
MA RJ MAEA,
MB 2 MAEв,
(28)
03 ~ RMSE MAE of ~ RMSE; MAE
(29)
Define the difference in absolute errors:
Ad, = -1) ~ MA, +
(30)
Standardizing this with a normal variable:
Z Ad, (un - MA) 1),
(31)
Vo +
The probability that method A outperforms method is then given by:
P(d)A) < c (B)
1IB nI
= +
VOa + +
MAE MAEA
$
(32)
(RMSE? MAE + (RMSE?, MAE?))
where ((-) denotes the standard normal cumulative distribution function. As result, the likelihood that method performs better than method B samples is:
n/2 - np
P(A better than B) x
+
(33)
Vnp(1
25

---

For all evaluations, each experiment includes at least 174 sequences. with each sequence containing observations from no fewer than 365 stations, as detailed in Appendix D Each station provides temporal sequence of observations, which is treated as an independent sample for statistical evalu- ation. Thus, the total number of samples = 174 x 365. Based on our analysis, our algorithm outperforms the best SOTA baseline with statistical significance at confidence level of at least 99.5%.
26


Reviews:
[{"review_id": "6sabdbNFoC", "reviewer": "Reviewer_M8eB", "summary": "The paper introduces CTENet (Chemical Transport Eulerian Network), a physics-informed deep learning model that integrates the Advection-Diffusion-Reaction (ADR) equation into a neural network framework. CTENet uses an Eulerian representation to model pollutant spatiotemporal dynamics, addressing limitations of discrete station-based or graph-based methods by capturing spatial continuity and complex chemical transformations.", "questions": "Question 1 : How can CTENet be extended to provide probabilistic forecasts (e.g., prediction intervals) for health-risk alerts?\n\nQuesiton 2: How does CTENet handle coastal/mountainous regions where Neumann boundary conditions may fail?\n\nQuestion 3: What steps are needed to deploy CTENet in real-time air quality monitoring systems?", "limitations": "There are two improved suggestions: \n1. Integrate Monte Carlo dropout or Bayesian neural networks to output uncertainty estimates alongside predictions.\n2. Validate with real-world decision-making scenarios (e.g., hospitalization risk thresholds).", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. CTENet embeds real atmospheric processes (like wind-driven advection) into its neural network, making predictions more physically realistic.\n\n2. It outperforms existing models by up to 45.8% on real-world datasets (USA/China), proving its effectiveness.\n\nWeaknesses:\n\n1.  The model gives single-point predictions without confidence intervals, limiting its use in high-risk scenarios.\n\n2. It assumes basic chemical reactions, potentially missing complex pollutant interactions in extreme pollution events.", "was_revised": true, "final_justification": "After careful consideration of the author’s response and the opinions of the other reviewers, I maintain my initial judgment.", "rebuttal": "Dear reviewer,\n\nWe sincerely thank you for your thoughtful and constructive feedback. We appreciate the recognition of CTENet's strengths in embedding real atmospheric dynamics, achieving strong empirical performance, and enhancing physical realism. Below, we address your specific questions and concerns.\n\n**To W1, Q1, and L1:**\n\nThank you for your thoughtful comment. CTENet was not originally designed with the goal of producing uncertainty-aware predictions; instead, our focus was on achieving accurate and timely point forecasts. We acknowledge that the ability to quantify predictive uncertainty is crucial in health-critical applications, where decision robustness is essential. \n\nImportantly, the architecture of CTENet is flexible enough to accommodate uncertainty estimation methods. As you suggested in Limitation 1, techniques such as Monte Carlo Dropout, deep ensembles, or Bayesian neural networks can be integrated with minimal structural modification. We believe that coupling these approaches with CTENet’s physics-informed design can enable reliable prediction intervals or probabilistic outputs, thereby enhancing its applicability to high-risk scenarios.\n\nWe once again appreciate this valuable suggestion, and we plan to actively explore this direction in future work to further improve CTENet’s reliability in real-world decision-making contexts.\n\n**To W2:**\n\nThank you for your insightful and technically informed comment. Indeed, as you correctly noted, the interactions between nitrogen oxides and particulate matter during extreme pollution episodes are governed by complex chemical processes. We are pleased to see your attention to this important aspect.\n\nAs you pointed out, our current model does not explicitly encode the detailed reaction equations typically used in atmospheric chemistry. This is a deliberate design choice: rather than reconstructing the full reaction pathways under specific environmental conditions (e.g., photochemical reactions influenced by solar radiation, temperature, and humidity), our modeling objective is to capture the eventual atmospheric chemical equilibrium that results from a given initial pollution state. In this formulation, the final pollutant concentrations can be treated as a nonlinear function of the initial state.\n\nThis is precisely where the strength of our method lies. By incorporating ideas from chemical transport modeling (CTM) and integrating them with a data-driven framework, CTENet is capable of learning an effective end-to-end mapping from initial to equilibrium pollutant states, without requiring the explicit simulation of each reaction mechanism. This enables the model to bypass the need for precise specification of all environmental factors, while still capturing key dependencies. We will clarify and emphasize this modeling philosophy more explicitly in the revised version.\n\n**To Q2:**\nThank you for the insightful question. In coastal and mountainous regions, the evolution of pollutant concentrations often deviates from standard assumptions due to spatial heterogeneity, making classical Neumann boundary conditions potentially inadequate. CTENet addresses this challenge not by relying solely on boundary condition assumptions, but by explicitly incorporating heterogeneous terrain information into the modeling process.\n\nSpecifically, the meteorological data used by CTENet includes several key variables that capture complex topographic and coastal features:\n- *Orography* (Channel 249): directly encodes surface elevation, allowing the model to learn the impact of terrain height on pollutant dispersion.\n- *MSLP (Eta model reduction)* (Channel 246): represents sea-level-reduced pressure, effectively correcting for elevation-induced pressure differences.\n- *Surface pressure* (Channel 248): captures actual pressure at the surface, which varies significantly with altitude and terrain type.\n\nTo capture coastal influences:\n- *Land-sea mask* (Channel 338): a binary mask (1 for land, 0 for sea) identifying coastal boundaries.\n- *Land-sea coverage (nearest neighbor)* (Channel 354): provides a more refined classification using nearest-neighbor interpolation to distinguish land and sea grid points.\n\nBy integrating these features into its deep learning architecture, CTENet is able to model pollutant dynamics more faithfully across spatially heterogeneous domains, including complex coastal and mountainous areas, without relying on potentially fragile boundary condition approximations. *A more detailed discussion of how CTENet leverages these variables to handle topographic and coastal complexity will be provided in the full paper.*\n\n**To Q3:**\nTo deploy CTENet in real-time air quality monitoring systems, the following steps are needed:\n1. *Data Preparation*: Collect and preprocess multi-source air quality and meteorological data, as described in the paper.\n2. *Model Training*: Train CTENet on historical data to learn spatiotemporal correlations. On an RTX 4090 GPU, the full training process can be completed within **3 to 6 hours**, depending on the data volume and forecast horizon.\n3. *System Integration*: Integrate the trained model into the real-time data processing pipeline.\n4. *Real-time Inference*: Feed incoming meteorological forecasts and sensor readings into CTENet to generate predictions. For example, generating a 72-hour PM$_{2.5}$ forecast over the China takes less than **0.15 seconds** on an RTX 4090 GPU. Therefore, we believe that even on a CPU-based system, the model can still produce forecasts within a few minutes, which is acceptable for most real-time applications.\n5. *Model Update*: Periodically fine-tune the model using recent observations to adapt to seasonal or structural changes in emission patterns and meteorological drivers.\n\n**To L2:**\nWe sincerely thank you for recognizing the effectiveness of our forecasting system. All of our experiments are conducted using real-world air quality and meteorological data collected from ground-based sensors across the continental United States and China, ensuring practical relevance.\n\nWe appreciate the suggestion to validate CTENet within real-world decision-making contexts, such as assessing forecasts against hospitalization risk thresholds. While this represents an important and meaningful direction, we consider such analyses to be downstream tasks that extend beyond the scope of the present work. Nevertheless, the system can be adapted to support such integration. Prior studies have quantitatively linked air pollution to adverse health outcomes. For instance, Bell et al. demonstrated significant associations between specific PM$_{2.5}$ components (e.g., sulfate, elemental carbon) and increased hospitalization rates for cardiovascular and respiratory diseases [1]. Building on these literature, future work could extend CTENet by coupling its forecasts with health impact models, enabling validation under operational scenarios such as early warning issuance or risk-based alert systems, such as forecasting the likelihood of hospitalization threshold exceedances or issuing early warnings for at-risk communities.\n\n> [1]Bell, M. L., Ebisu, K., Peng, R. D., Samet, J. M., & Dominici, F. (2009). \"Hospital admissions and chemical composition of fine particle air pollution.\" JAMA, 300(11), 1127–1135."}, {"review_id": "BNDCg6sKFt", "reviewer": "Reviewer_rRLU", "summary": "This paper combines Eulerian representation with Physics-Informed Neural Networks (PINNs) to propose CTENet, an air quality forecasting model. It converts sparse station data into spatial fields via Radial Basis Function (RBF) interpolation and embeds the Advection-Diffusion-Reaction (ADR) equation to simulate pollutant dynamics. While CTENet shows improved accuracy over baselines, its core methodology primarily integrates existing components—RBF interpolation, modular predictors, and ADR physics constraints—without introducing fundamentally novel mechanisms. The approach reframes established concepts for air quality tasks, but its architectural innovations appear incremental rather than transformative.", "questions": "please see in Weakness", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. Experimental Reproducibility\nExtensive validation on real-world datasets demonstrates 45.8% and 21.0% RMSE reduction against 9 SOTA baselines. Ablation studies robustly justify design choices, and open-sourced code ensures reproducibility. Statistical significance testing (99.5% confidence) further strengthens reliability of this model.\n\n2. Physics-Driven Spatial Continuity\nRadial Basis Function (RBF) interpolation effectively converts sparse station data into continuous fields, resolving structural mismatches in graph-based methods and outperforming nearest-neighbor alternatives. The Neumann boundary conditions enhance stability and physical plausibility in predictions.\n\nWeakness:\n\n1. Limited Originality: CTENet combines established techniques—RBF interpolation, modular predictors, and physics-informed neural networks (PINNs)—within an Advection-Diffusion-Reaction (ADR) framework. However, this hybrid design are similar to prior works. Thus, it has incremental contributions.\n\n2. Novelty: The claimed advances in are not supported by methodological innovation. The observed performance gains largely arise from architectural adjustments—such as the dynamic weighting of reaction terms (Eq. 12)—rather than from a new algorithmic framework. Moreover, spatial continuity has already been effectively addressed in prior work, including AirFormer (§2.1), which diminishes the claimed novelty.\n\n3. Limited scalability: As confirmed by computational complexity analysis in Appendix G, CTENet incurs significantly higher computational demands than graph-based methods. The theoretical overhead grows drastically due to its grid-based architecture, with empirical tests (Table 3) showing 21.7× higher computations for China and 26.8× for the USA dataset compared to graph models. Consequently, its complexity impedes real-time forecasting feasibility, and no mitigation strategies are proposed for this bottleneck.", "was_revised": true, "final_justification": "After reading all reviewers‘s comments, I will give borderline accept to this paper and I begin to understand that air quality is a task difficult to fomalize. The main contribution lies in integrating the vector field method with PINN framework, and incoporating chemical constriants is novel. As serval initial works in this task, I think this paper provides reference in how to formalize this task.", "rebuttal": "Dear reviewer,\n\nWe sincerely thank the you for your thoughtful evaluation and constructive feedback. We appreciate the acknowledgment of our physics-driven spatial continuity modeling. We now address your raised concerns in detail:\n\n**To W1:**\n\nThank you for your comments regarding the originality of our work. We apologize for any confusion caused by our previous explanation. While it is true that CTENet incorporates established techniques such as RBF interpolation, modular predictors, and physics-informed neural networks (PINNs), we would like to clarify that these components are not the main novelty of our work. As highlighted in the Introduction, the primary contributions of CTENet are as follows:\n\n1. *Eulerian spatial modeling:* \n\nPrior works on air quality prediction typically treat the problem as a set of discrete monitoring stations, overlooking the spatial continuity of pollutant fields. To the best of our knowledge, CTENet is the first PINN-based framework to model air quality prediction directly in continuous space. Furthermore, although some of the prior works [1,2] incorporate physical constraints, but they are still based on discrete station-level modeling. As a result, advection and diffusion are modeled via edge-weighted graph operators, which do not reflect the continuous nature of atmospheric transport. In contrast, CTENet models pollution dynamics over a spatially continuous domain, enabling more realistic and physically consistent forecasting.\n\nIn this process, the use of RBF interpolation serves purely as a **preprocessing step** to convert sparse station data into a gridded format compatible with subsequent convolutional operations. We also provide comparisons with alternative interpolation methods—as well as a non-interpolated setting—in Appendix F. The results support the effectiveness and robustness of our choice.\n\n2. *Incorporation of chemical reactions:*\n\nUnlike prior PINN-based approaches that focus solely on advection and diffusion, CTENet explicitly incorporates a chemical transport module—an element absent from existing physics-informed air quality models. This design allows CTENet to account for a range of additional environmental factors that significantly influence pollutant evolution, including solar radiation, atmospheric stability, terrain elevation, and even soil moisture. These drivers enable the modeling of essential processes such as *photochemical reactions* and *gas-to-particle conversion*, which are known to cause nonlinear transformations among pollutants. As a result, CTENet can capture dynamic equilibrium shifts between gaseous and particulate species, enabling more realistic and chemically consistent multi-pollutant forecasting.\n\nTo the best of our knowledge, CTENet is the first to implement a physics-informed neural network based on the Eulerian framework for air quality prediction. We will revise the paper to more clearly highlight these contributions—both in terms of methodology and their implications for real-world data-driven environmental modeling.\n\n> [1] Kethmi Hirushini Hettige, Jiahao Ji, Shili Xiang, Cheng Long, Gao Cong, and Jingyuan Wang.Airphynet: Harnessing physics-guided neural networks for air quality prediction. arXiv preprint arXiv:2402.03784, 2024.381\n> \n> [2] Jindong Tian, Yuxuan Liang, Ronghui Xu, Peng Chen, Chenjuan Guo, Aoying Zhou, Lujia Pan,Zhongwen Rao, and Bin Yang. Air quality prediction with physics-guided dual neural odes in open systems. In The Thirteenth International Conference on Learning Representations, 2025.\n\n**To W2:**\n\nWe apologize for any confusion caused by our explanation. We would like to clarify that the observed performance improvements are not solely due to architectural adjustments. As mentioned earlier, the main novelty of our work lies in several key aspects that distinguish it from prior approaches:\n\n1. From the perspective of spatial continuity, we employ Eulerian spatial modeling to more accurately capture the evolution of pollutants in space. This approach allows for a more robust representation of pollutant dispersion and transformation across regions using NN, which goes beyond what has been addressed in earlier works. \n\n    - Although AirFormer captures *spatial correlation* by encoding relative positions of neighboring stations, it does **not** model *spatial continuity* in any physically meaningful sense. In fact, AirFormer's design reinforces the importance of spatial structure in air quality prediction, yet it remains fundamentally a discrete, station-based model. It lacks any mechanism to simulate continuous spatial transport processes such as advection and diffusion, which are central to our Eulerian PDE-based framework. This distinction is also reflected in empirical performance. As shown in our experiments (§5.1), CTENet consistently outperforms AirFormer across multiple datasets and metrics, highlighting the effectiveness of physically grounded, continuous-domain modeling over discrete attention-based approaches.\n\n2. We account for the intricate chemical transformations of pollutants, distinguishing our model from previous works that focused primarily on physical processes. This inclusion of chemical processes enhances the predictive capability of CTENet, making it more suitable for real-world applications.\n\nWe will revise the manuscript to better highlight these distinctions and to clarify how CTENet advances the state of the art both in terms of formulation and real-world applicability.\n\n**To W3:**\n\nThank you for your detailed and professional comment regarding scalability. As you correctly pointed out, and as analyzed in Appendix G, CTENet employs a grid-based architecture that incurs higher computational complexity compared to graph-based models. We fully acknowledge this trade-off between physical fidelity and computational efficiency.\n\n- We believe that in large-scale operational settings—such as enterprise-level or national-scale deployments—the modest increase in computational cost is acceptable given the substantial performance gains CTENet offers over existing methods. For example, on the USA dataset, CTENet achieves an average RMSE reduction of **45.8%** compared to strong graph-based baselines.\n\n- Despite the increased theoretical complexity, CTENet remains highly practical for real-world deployment. On a single RTX 4090 GPU, the complete training process can be completed within **3 to 6 hours**, depending on the data volume and forecast horizon. More importantly, during inference, generating a 72-hour forecast over the China dataset takes less than **0.15 seconds**, which is fast enough to meet the requirements of most real-time applications.\n\nThat said, we agree that scalability is a meaningful consideration, and we are actively exploring architectural optimizations such as mixed-precision computation and model pruning to further reduce CTENet’s computational footprint in future work."}, {"review_id": "rWPup56Vbu", "reviewer": "Reviewer_sPyn", "summary": "The paper introduces a deep learning model that learns continuous time dynamics from irregularly sampled data for chemical transport. The method embeds  Advection-Diffusion-Reaction (ADR) equation into a Physics-Informed Neural Network (PINN) framework to simulate the spatiotemporal evolution of pollutants over continuous space and time.", "questions": "- How does the method handle noise?\n- Can the method be applied with other data sources such as social vulnerability?\n- How does the approach vary for varying data grid resolutions?", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- This method has potential to improve real-world application of air quality forecasting. \n- The proposed method shows high prediction accuracy which outperforms  baselines with an average RMSE improvement of 45.8% in the USA dataset and 21.0% in the China dataset\n- The method integrates Chemical Transport Modeling Principles which allows for direct simulation of pollutant transport and transformation grounded in physical and chemical principles, enhancing interpretability.\n- The method includes a modular and replaceable components for various forecasting modules.\n- The method handles data sampled with irregular grid.\n\nWeaknesses:\n- The paper states limitation that the current formulation lacks uncertainty quantification.\n- Computational complexity seems higher than graph-based methods.", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer,\n\nWe sincerely thank you for your time and constructive comments. We are glad that our work's practical potential for improving air quality forecasting has been recognized. \n\nBelow, we provide detailed responses to the concerns raised and clarify the novelty and significance of our contributions within the ML context.\n\n**To W1:**\n\nThank you for your thoughtful comment. CTENet was not originally designed with the goal of producing uncertainty-aware predictions; instead, our focus was on achieving accurate and timely point forecasts. We acknowledge that the ability to quantify predictive uncertainty is crucial in health-critical applications, where decision robustness is essential. \n\nImportantly, the architecture of CTENet is flexible enough to accommodate uncertainty estimation methods. Techniques such as Monte Carlo Dropout, deep ensembles, or Bayesian neural networks can be integrated with minimal structural modification. We believe that coupling these approaches with CTENet’s physics-informed design can enable reliable prediction intervals or probabilistic outputs, thereby enhancing its applicability to high-risk scenarios.\n\n**To W2:**\n\nThank you for your detailed and professional comment regarding scalability. As you correctly pointed out, and as analyzed in Appendix G, CTENet employs a grid-based architecture that incurs higher computational complexity compared to graph-based models. We fully acknowledge this trade-off between physical fidelity and computational efficiency.\n\n- We believe that in large-scale operational settings—such as enterprise-level or national-scale deployments—the modest increase in computational cost is acceptable given the substantial performance gains CTENet offers over existing methods. For example, on the USA dataset, CTENet achieves an average RMSE reduction of **45.8%** compared to strong graph-based baselines.\n\n- Despite the increased theoretical complexity, CTENet remains highly practical for real-world deployment. On a single RTX 4090 GPU, the complete training process can be completed within **3 to 6 hours**, depending on the data volume and forecast horizon. More importantly, during inference, generating a 72-hour forecast over the China dataset takes less than **0.15 seconds**, which is fast enough to meet the requirements of most real-time applications.\n\nThat said, we agree that scalability is a meaningful consideration, and we are actively exploring architectural optimizations such as mixed-precision computation and model pruning to further reduce CTENet’s computational footprint in future work.\n\n**To Q1:**\n\nThank you for the thoughtful question.\n\nWe acknowledge that observational noise—especially in low-cost sensor data—is a critical challenge in air quality modeling. CTENet does not explicitly model uncertainty or input-level noise. However, it benefits from several indirect mechanisms that enhance robustness.\n\n1.First, the incorporation of a chemically inspired transport-reaction model serves as a strong structural constraint. By encouraging the model to follow physically plausible pollutant dynamics, it helps suppress overfitting to spurious or noisy fluctuations in the observations—particularly those that violate mass conservation or chemical balance.\n\n2.Second, standard training practices such as early stopping, dropout, and gradient clipping are used to reduce the model's sensitivity to label noise.\n\nThat said, CTENet currently does not include dedicated noise modeling components such as uncertainty-aware loss functions or observation-level confidence weighting. We see this as a meaningful direction for future work—for example, by incorporating heteroscedastic noise models, robust PDE solvers, or sensor-aware weighting schemes.\n\nWe appreciate you raising this important point.\n\n**To Q2:**\n\nThank you for the insightful question. While social vulnerability data is not directly relevant to the physical processes modeled by CTENet—such as pollutant transport and chemical transformation—the framework is modular and can be extended to incorporate additional data sources in downstream applications.\n\nFor example, once pollutant concentrations are forecasted, social vulnerability indices (e.g., age distribution, income level, access to healthcare) can be integrated in post-processing or risk assessment layers to support health impact forecasting, environmental justice analyses, or targeted early warning systems. In such contexts, CTENet's outputs could serve as physically grounded inputs to broader decision-support frameworks where social vulnerability plays a critical role.\n\nTherefore, although CTENet does not require social data for concentration prediction, it can be readily combined with such datasets in multi-task or multi-stage systems aimed at population-level risk mitigation.\n\n**To Q3:**\nThank you for your detailed and professional comment. CTENet is inherently resolution-adaptive and supports deployment across multiple grid scales, from regional to national levels. Since the model operates on both discrete station observations and continuous meteorological fields, its effective resolution is typically aligned with that of the meteorological input to ensure optimal performance.\n\nWhen the input data is provided on a coarser or finer grid, the overall model architecture remains unchanged. However, certain hyperparameters—particularly those related to the numerical implementation of the PDE constraints—are adjusted to match the spatial resolution. For instance, spatial step sizes used in the calculation of gradients and Laplacians are resolution-dependent, and must be set accordingly to preserve the stability and accuracy of the PDE-based loss terms."}, {"review_id": "JfQicS3EqU", "reviewer": "Reviewer_i3iM", "summary": "The paper considers the problem of forecasting future pollutant concetrations from past observations, where data is not in a grid. The paper proposes a transport equation that predicts the dynamics through advection, diffusion, reaction and source terms; and suggests off-the-shelf neural networks to model them. The method achieves dramatic performance improvements of 30%-50%.", "questions": "- What does “eulerian” mean?\n- eq 6 is missing the X div(w) component: why?\n- Do the competing methods also use the meteorological data? What data does each method use?\n- In table 1 the VAR and GTS results on USA 24h are 6.32/14.41 and 5.57/14.65. This does not seem to make sense: how can you have lower MAE but higher MSE?\n- It’s a bit unclear what is being learnt: can you clarify?\n- It seems that wind is the driving force behind the good results. Do you predict the wind, and if so, how accurate is that?", "limitations": "No issues", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The clarity of the paper is generally weak. The method description, math and writing is not sufficiently clear on describing the model. The method is described mostly in terms of what neural networks are used in what parts of the pipeline, which makes the paper read like a technical report or an applied use-case of ML. I don’t think I could reproduce this method for it’s vagueness. I also have hard time aligning fig2 to the method description. \n\nThe method has little ML novelty, and I’m not sure what is novel in general. The forecasting problem is appropriately modelled through the transport equations and through off-the-shelf neural networks. This seems to be very useful for this particular problem, but I’m not sure what ML significance or impact this has. I don’t think there is enough ML contributions for a ML conference.\n\nThe method is a black box, and there is not enough insight given to what the model learns, how and why. We only see one short time-series of the predictions, and no decomposition of the signal to the different equation terms.\n\nI’m a bit confused why the method performs so well. I suspect that the performance largely depends on using meteorological data to drive the forecasts. This is a good idea, but I’m not sure if the competing methods have access to this, which makes comparisons somewhat unfair. I’m also not sure if simply using extra data source is particularly interesting as the main contribution: obviously using more data is helpful, but is this scientifically interesting?  I might have misunderstood something, and I’d be happy to hear author’s comments about this. The paper also claims novelty of support sparse data through RBF interpolation. I’m not sure if the spatial continuity really is an open problem; and the solution of RBF interpolation seems overly simplistic. \n\nThe results lack statistical significance testing or standard deviations. \n\nDespite the good results, I’m suggesting rejection. The ML significance and contributions are too thin, the presentation and experiments are insufficient.", "was_revised": true, "final_justification": "I'm increasing my score to 3 based on the comprehensive and clarifying response by the authors, which increases my confidence in the work. This is a solid contribution that shows excellent performance improvements, but I'm still leaning on rejection due to limited novelty, issues in presentational clarity, and black-boxiness. As an applied work this paper would greatly improve from illustrating how the forecasts behave, and how the forecasts decompose into the different components in practise.", "rebuttal": "Dear reviewer,\n\nWe thank you for the positive feedback on our manuscript, particularly for highlighting the clarity of our problem setting and methodology, as well as the physical grounding of our model components. \n\nWe address the your concerns as follows:\n\n**To W1: About Clarity and Reproducibility**\n\n1. *Clarity and Style*\n\nWe apologize for any lack of clarity in the method description. The modular approach we adopted in describing CTENet is intentional, as it aligns with prior works in NeurIPS(such as [1,2]). These works also decompose complex systems into well-defined modules, which aids in presenting the model in a more structured manner. We will strive to improve the clarity of the model description, ensuring that the individual components and their relationships are more explicitly defined.\n\n2. *Reproducibility*\n\nWe provide implementation details in Section 5 and Appendix D, including data, model settings, and training procedures. The source code is publicly available (link in Abstract), and we’re happy to share the processed dataset upon request to support reproducibility. \n\n> [1] Zhang, Y., Liu, Q., & Chen, H. (2024). Suitable is the Best: Task-Oriented Knowledge Fusion in Vulnerability Detection. In Proceedings of the 38th Conference on Neural Information Processing Systems (NeurIPS 2024).\n> \n> [2] Wang, L., Zhao, M., & Xu, J. (2024). Neural Krylov Iteration for Accelerating Linear System Solving. In Proceedings of the 38th Conference on Neural Information Processing Systems (NeurIPS 2024).\n\n**To Q1: About Eulerian Definition**\n\nThank you for your question.\n\nAs stated in Section 1 Introduction of the paper, Eulerian model[3] refers to the fact that our model operates on a **fixed, gridded spatial domain**. In contrast to prior air quality forecasting approaches—many of which, whether physics-guided or not, rely on discrete station-level modeling—CTENet models the continuous spatial evolution of pollutant concentrations over a structured and uniform Eulerian grid. This gridded representation is crucial for capturing fine-grained spatial correlations and for enforcing physically consistent processes—such as advection and diffusion—that are inherently defined in continuous space.\n\n>[3] Johnson JB. An introduction to atmospheric pollutant dispersion modelling. Environmental Sciences Proceedings. 2022 Jul 14;19(1):18.\n\n**To Q2: About Missing Term**\n\nWe apologize for the confusion. \n\nEquation 6 presents the continuous form of the Advection–Diffusion–Reaction (ADR) equation. As in references [44, 45, 11, 17] cited in our paper, the equation does **not** include the additional term $X \\cdot \\mathrm{div}(\\vec{W})$.\n\nThis is because atmospheric wind speeds on Earth are typically below 30 m/s, which is significantly lower than the speed of sound in air (~343 m/s). Under such conditions, the flow can be reasonably approximated as *incompressible*, so that $\\mathrm{div}(\\vec{W})=0$ . This assumption is widely adopted in air quality modeling and greatly simplifies the advection operator.\n\n\n**To W2 & W4.1: About Fairness in Data Usage**\n\nThank you for the question about fairness in Data Usage!\n\nAs described in Section 5 (Experiments), all baseline models are provided with the same set of meteorological data as CTENet, including the full 354-channel input. For station-based baselines that cannot directly process gridded inputs, we extract the corresponding meteorological features from the grid cell that contains each station's location (see Section 5: Baselines). This design ensures a fair and controlled comparison across all methods. Therefore, the performance gain of CTENet is attributable to its architectural design rather than differences in input data.\n\n**To W4.2 & Q3: About Novelty and RBF**\n\nWe sincerely apologize if our presentation led to any misunderstanding of the central Novelty of our work. Here is my detailed explanation:\n\n1. Previous air quality forecasting network often operate on sparse station data and neglect spatial continuity, despite pollution being inherently continuous. CTENet is, to the best of our knowledge, the first to introduce continuous pollutant modeling into air quality prediction via deep learning, using chemical transport priors within a PINN-style architecture.\n\n2. Regarding RBF interpolation, our aim is not optimal interpolation accuracy but a fast, parameter-free method that ensures stable and efficient training. While more accurate methods exist, they are computationally costly and beyond the scope of this work. RBF provides a continuous spatial estimate from discrete stations, enabling convolutional processing in our architecture. As shown in Appendix F, we compare RBF with alternatives and include an ablation without interpolation, validating the robustness of our choice.\n\n**To Q4: About Evaluation Metrics**\n\nThank you for the helpful comment. We believe there may be a misunderstanding between MAE and RMSE (likely referred to as MSE in your note). These metrics capture different aspects of error—MAE measures average error magnitude, while RMSE penalizes larger deviations more heavily.\nAs shown in the toy example below:\n\n|Model|x₁|x₂|x₃|x₄|MAE|RMSE|\n|-|-|-|-|-|-|-|\n|Ground Truth|4|4|5|5|||\n|baseline1|5|5|4|4|1|1|\n|baseline2|4|4|5|8|0.75|1.5|\n\nBaseline 2 has lower MAE but higher RMSE, due to the outlier at x₄. Thus, discrepancies between MAE and RMSE are expected and informative. We hope this clears up any confusion.\n\n**To W3&Q5: On Interpretability and Physical Decomposition**\n\nWe sincerely apologize for the confusion this may have caused.\n\n1.  *On Interpretability*\n\nWhile deep learning models are often criticized as black boxes, CTENet is specifically designed to move beyond black-box behavior by embedding domain knowledge from chemical transport modeling (CTM) into the learning architecture. In particular, the model learns:\n\n - How meteorological variables interact with air pollutant dynamics — including temperature, wind, and solar radiation effects — enabling the network to capture nonlinear, spatial-temporal dependencies that are difficult to model explicitly using traditional CTMs.\n - How to dynamically integrate multiple predictive sources — including physical components such as diffusion, advection, and chemical reactions, as well as purely data-driven residual models. This fusion is learned through a neural attention-based weighting mechanism that adapts over space and time.\n\n2.  *On Signal Decomposition*\n\nAs shown in Figure 2, our model dynamically fuses four predictive components—diffusion, advection, chemical reaction, and a data-driven correction—each reflecting a distinct physical or empirical process. These components are treated as separate input channels, but their integration is handled implicitly through learned non-linear mappings. As a result, the final prediction is not a simple additive combination and cannot be directly decomposed as in traditional PDE solvers. This design allows the model to flexibly capture complex, context-dependent interactions that are challenging to encode explicitly.\n\n\n**To W5: About statistical Significance**\n\nWe thank you for your helpful suggestion regarding the statistical robustness of our results. In response:\n\n1. *Statistical Significance Analysis:*\n\nAs documented in Appendix J, we conducted statistical significance analysis to assess the reliability of the performance differences.\n\n2. *Standard Deviations:*\n\nDue to time constraints, we conducted five independent runs (with different random seeds) for the best-performing CTENet variant and the two strongest baselines. The mean and standard deviation of the results are reported below:\n\n|Model|China 24h MAE|China 24h RMSE|China 48h MAE|China 48h RMSE|China 72h MAE|China 72h RMSE|USA 24h MAE|USA 24h RMSE|USA 48h MAE|USA 48h RMSE|USA 72h MAE|USA 72h RMSE|\n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|CTENet|11.07±0.16|17.38±0.26|12.17±0.67|19.12±2.05|14.10±1.60|22.56±3.10|2.45±0.12|4.28±0.36|2.71±0.22|4.45±0.33|2.83±0.33|4.62±0.69|\n|TAU|16.08±0.20|27.98±1.01|16.60±0.75|27.93±0.84|15.92±1.43|25.38±3.83|4.54±0.15|10.00±1.49|4.40±0.40|9.80±2.11|6.00±1.15|13.10±2.04|\n|PM$_{2.5}$GNN|17.79±0.24|33.03±0.45|19.20±0.28|33.93±0.63|19.81±0.32|34.28±0.74|4.54±0.14|10.19±0.39|4.83±0.16|10.10±0.41|4.96±0.15|10.08±0.44|\n\nMoreover, two-sided t-tests confirm that CTENet's improvements over both baselines are statistically significant across nearly all metrics (p < 0.005), reinforcing the robustness of the observed performance gains.\n\n**To Q6: About Wind Prediction**\n\nThank you for the insightful question !\n\nWe confirm that our model does include a lightweight wind forecasting component to predict future wind fields. As described in the paper, the Wind Predictor module is implemented using a Spatiotemporal Sequence Predictor, which takes historical wind fields as input and predicts wind fields at the next time step. These predicted wind components are then passed into the **Eulerian ADR Decoder** to simulate wind-driven advection.\n\nWe believe our wind predictor is sufficiently accurate for the intended use case, for the following reasons:\n\n1. *Ablation studies(Section 5.2)* demonstrate the importance of advection: removing the advection module (which depends on wind prediction) leads to significant performance degradation. Specifically, the RMSE increases by **10.6%** and **21.8%** on the two datasets, respectively, when advection is ablated.\n\n2. A *case study (Section 5.4)* visually demonstrates the role of wind-driven advection in pollutant transport, further validating the model’s ability to learn physically meaningful wind patterns that align with observed pollutant dispersion behavior.\n\n**In Summary**\n\nWe are grateful for your thoughtful comments and suggestions. We have conducted additional experiments in response to your feedback and are in the process of further revising the manuscript accordingly. We fully respect your judgment and would greatly appreciate any further guidance you may have."}]

Author rebuttal:

Meta-review: This paper proposes a novel, physics guided method for air pollution prediction. The method embeds the Advection-Diffusion-Reaction equation into a PINN using an Eulerian representation to model the spatiotemporal evolution of pollutants, this helps capturing the continuous space information, which the authors claimed to be particularly novel (no existing studies have done so). The experimental results show great improvement, even more than 40% under some settings.  

Most reviewers are positive, agreeing that the paper makes an interesting contribution. There is one reviewer who gave a negative rating, but the rating was upgraded from 2 to 3 after rebuttal. Based on the overall review ratings and some detailed discussions during the rebuttal stage, the AC agrees with the majority and would like to suggest an acceptance.
Decision: Accept (poster)