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AeroChem: Closed-loop Physics-Informed State Space Modeling for Long-term Chemically-Reactive Air Quality Forecasting--Extended Version
Kien Le Trung, Tung Kieu, Dinh Duc Nha Nguyen, Cuong Do, Bin Yang, Phi Long Nguyen
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Abstract
Abstract:Accurate long-term air quality forecasting is challenging due to long-range transport, meteorological variability, and nonlinear chemical reactions underlying secondary pollutant formation. Purely data-driven models often suffer from long-horizon drift and lack physical consistency, while existing physics-informed approaches commonly follow open-loop designs that fuse neural and physical components only at the output level. We propose AeroChem, a closed-loop physics-informed state space framework for chemically reactive air quality forecasting. AeroChem adopts a two-branch co-correction design: a UDE-based process branch provides a structured but imperfect physical prior, while a Mamba-based virtual measurement branch captures long-range temporal context and produces probabilistic virtual observations. Neural Kalman Fusion couples the two branches by correcting the latent physical state at each forecasting step and feeding the posterior state back into the simulator, reducing long-horizon drift. We further introduce HanoiAir, a real-world dataset integrating pollutant concentrations, meteorology, and emission inventories. Experiments on HanoiAir and a public benchmark show that AeroChem improves long-horizon stability and sudden-change forecasting.
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- Source: https://arxiv.org/abs/2610.16630v1
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AeroChem: Closed-loop Physics-Informed State Space Modeling for Long-term Chemically-Reactive Air Quality Forecasting Kien Le* Affiliation: VinUniversity Affiliation: Vietnam Email: kien.lt@vinuni.edu.vn Tung Kieu* Affiliation: Aalborg University Affiliation: Denmark Email: tungkvt@cs.aau.dk Dinh Duc Nha Nguyen Affiliation: CECS, VinUniversity Affiliation: Vietnam Email: nha.ndd@vinuni.edu.vn Cuong Do Affiliation: CECS, VinUniversity Affiliation: Vietnam Email: cuong.d@vinuni.edu.vn Bin Yang Affiliation: East China Normal University Affiliation: China Email: byang@dase.ecnu.edu.cn Nguyen Phi Long† Affiliation: SEEE, Hanoi University of Industry Affiliation: Vietnam Email: long.nguyen04@haui.edu.vn Abstract Accurate long-term air quality forecasting is challenging due to long-range transport, meteorological variability, and nonlinear chemical reactions underlying secondary pollutant formation. Purely data-driven models often suffer from long-horizon drift and lack physical consistency, while existing physics-informed approaches commonly follow open-loop designs that fuse neural and physical components only at the output level. We propose AeroChem, a closed-loop physics-informed state space framework for chemically reactive air quality forecasting. AeroChem adopts a two-branch co-correction design: a UDE-based process branch provides a structured but imperfect physical prior, while a Mamba-based virtual measurement branch captures long-range temporal context and produces probabilistic virtual observations. Neural Kalman Fusion couples the two branches by correcting the latent physical state at each forecasting step and feeding the posterior state back into the simulator, reducing long-horizon drift. We further introduce HanoiAir, a real-world dataset integrating pollutant concentrations, meteorology, and emission inventories. Experiments on HanoiAir and a public benchmark show that AeroChem improves long-horizon stability and sudden-change forecasting. 11footnotetext: Equal contribution.22footnotetext: Corresponding author. 1 Introduction Air pollution is a major global health and environmental challenge, motivating accurate air quality forecasting for public policy, emission control, and public-health protection [9, 38]. However, forecasting pollutant concentrations remains difficult because atmospheric dynamics are governed by coupled emission sources, meteorological conditions, transport processes, and chemical reactions [39]. This challenge is especially pronounced for secondary pollutants, such as ozone (O3) and secondary components of PM2.5, which arise from nonlinear photochemical reactions involving precursors such as NOx and VOCs under meteorological drivers such as solar radiation, temperature, and humidity. These processes evolve over multiple hours or days, requiring both long-range temporal modeling and physical consistency. Traditional Chemical Transport Models (CTMs) [26, 32], as illustrated in Fig. 1, encode atmospheric transport, diffusion, and chemical reactions through differential equations derived from physical principles. Although interpretable, CTMs rely on simplified assumptions, incomplete reaction mechanisms, uncertain boundary conditions, and expensive solvers. In contrast, data-driven deep learning models learn predictive patterns directly from observations and often achieve strong short-term accuracy, but they may violate physical constraints, generalize poorly under distribution shifts, and accumulate errors during long-horizon autoregressive rollout. Figure 1: Impacts on Air Quality. Physics-Informed Machine Learning (PIML) [21, 3] aims to combine physical modeling with neural learning, but existing air quality forecasting methods remain limited. Many oversimplify atmospheric chemistry as linear source–sink terms or generic neural residuals, which is insufficient for nonlinear and condition-dependent secondary pollutant formation. More importantly, most hybrid architectures follow an open-loop design, where physical and data-driven branches are executed separately and fused only at the output level. Such designs may correct the final prediction, but they cannot correct the latent physical state propagated to future steps, allowing errors from misspecified physics, unobserved emissions, and stochastic atmospheric disturbances to accumulate. In this work, we argue that long-term chemically reactive forecasting requires a closed-loop design, where physics and data-driven evidence correct each other during rollout. The physical branch should provide a structured dynamical prior, while the data-driven branch should provide observation-conditioned evidence for correcting the evolving state. Crucially, this correction should occur in latent state space before the next simulation step, rather than only at the final output. This view turns long-horizon forecasting into recurrent posterior-state inference under imperfect physical dynamics. To realize this idea, we propose AeroChem, a closed-loop physics-informed state space framework for long-term chemically reactive air quality forecasting. AeroChem couples two interacting branches through Neural Kalman Fusion. The first is a UDE-based process branch that acts as a differentiable simulator and explicitly separates transport dynamics from chemical dynamics, using diffusion, wind-driven advection, and a meteorology-gated neural reaction operator. The second is a Mamba-based virtual measurement branch that captures long-range temporal dependencies from historical observations and produces probabilistic virtual measurements. At each forecasting step, Neural Kalman Fusion combines the process prior and virtual measurement into a posterior latent state, which is recursively fed back into the process branch. This closed-loop co-correction mechanism reduces long-horizon drift and improves robustness in open, chemically reactive environments by allowing data-driven evidence and physics-informed dynamics to correct each other during rollout. We evaluate AeroChem on two real-world air quality datasets, including HanoiAir, a newly constructed regional dataset integrating pollutant concentrations, meteorological variables, and emission inventories. Experiments show that AeroChem is particularly effective in long-horizon forecasting and sudden pollution-change scenarios, where open-loop and purely data-driven models often suffer from drift or over-smoothing. Ablation studies further demonstrate the importance of closed-loop feedback, meteorology-gated reaction modeling, and neural state correction. In summary, this work makes the following contributions: (i) We formulate long-term chemically reactive air quality forecasting as closed-loop latent-state inference under imperfect physical dynamics. (i) We propose AeroChem, a two-branch physics-informed state space framework that couples a UDE-based process branch and a Mamba-based measurement branch through Neural Kalman Fusion. (i) We introduce a meteorology-gated neural reaction mechanism to model nonlinear secondary pollutant formation conditioned on emissions and meteorological drivers. (iv) We construct and release HanoiAir, a comprehensive regional air quality dataset containing pollutant concentrations, meteorological variables, and emission inventories. (v) We demonstrate that AeroChem improves long-horizon forecasting stability and sudden-change prediction across real-world air quality datasets. 2 Preliminaries Sensor Network. We model an air quality monitoring system as a weighted static graph =(,ℰ,)G=(V,E,A), where =v1,…,vNV=\v_1,…,v_N\ denotes the set of N monitoring stations, ℰE represents the set of edges, and ∈ℝN×NA ^N× N is a weighted adjacency matrix encoding spatial correlations between stations, such as geographic proximity. Each station viv_i is associated with fixed spatial coordinates i∈ℝ2p_i ^2. Multivariate Observations. At each discrete time step t, the observed system state is denoted as t∈ℝN×DinX_t ^N× D_in, where each station is associated with a DinD_in-dimensional feature vector. We decompose tX_t into three components: (1) Pollutant concentrations tC_t: Pollutant concentrations are the primary forecast targets, captured via air pollutant variables, including target pollutants (e.g., PM2.5) and related co-pollutants (e.g., O3, NOx); (2) Meteorological variables tM_t: environmental drivers including scalar fields (e.g., temperature, humidity, solar radiation) and vector fields (e.g., wind speed and direction); (3) Emission rates tE_t: emission inventories describe source-intensity inputs, i.e., estimated pollutant or precursor releases into the atmosphere, including NOx, VOCs, SO2, and primary PM2.5. Here, tC_t represents pollutant levels already present in the atmosphere and serves as the main prediction target, whereas tE_t represents external emission forcing that contributes to future pollutant formation and transport. Given an input window of length TinT_in, the historical observation sequence is denoted as 1:Tin∈ℝTin×N×DinX_1:T_in ^T_in× N× D_in. Problem Formulation. Given the sensor graph G and historical observations 1:TinX_1:T_in, our objective is to learn a probabilistic forecasting function ℱΘF_ that estimates the posterior distribution of future observations over a prediction horizon ToutT_out: p(Tin+1:Tin+Tout∣1:Tin,).p\! (X_T_in+1:T_in+T_out _1:T_in,G ). This formulation enables both point prediction and uncertainty estimation for spatiotemporal air quality forecasting. 3 Methodology Figure 2: AeroChem architecture. A S-Mamba-based virtual measurement branch generates adaptive virtual observations, while a UDE-based process branch models transport and meteorology-gated neural reactions as a structured physical prior. Neural Kalman Fusion couples the two branches through a learnable correction term, producing a posterior state that is recursively fed back into the UDE. This closed-loop neural state correction allows virtual measurements and physics-informed dynamics to correct each other, mitigating long-horizon drift. To address the probabilistic forecasting problem defined in Sec. 2, we propose AeroChem, a closed-loop physics-informed state space framework for long-term air quality forecasting. Unlike prior hybrid approaches that loosely couple physical and data-driven components, AeroChem follows a closed-loop neural state correction paradigm. At each forecasting step, a physics-informed process branch provides a structured but imperfect prior over the latent atmospheric state, while a virtual measurement branch produces data-driven virtual observations from historical context and the current rollout state. Neural Kalman Fusion couples the two branches by correcting the latent physical state before it is propagated to the next step. For clarity, we summarize the complete closed-loop forecasting procedure in Appx. A.1. 3.1 Framework Overview At each forecasting step, AeroChem integrates three tightly coupled components: (1) a Virtual Measurement Branch that extracts long-range temporal context from historical observations and produces probabilistic virtual measurements; (2) a Process Branch formulated as a stochastic Universal Differential Equation (UDE) that models transport and chemically reactive dynamics as a structured physical prior; and (3) a Neural Kalman Fusion module that fuses the virtual measurement with the UDE prior to produce a corrected posterior latent state. The posterior state is recursively fed back into the process branch, forming a closed-loop co-correction system that mitigates error accumulation in long-horizon forecasting. The overall architecture is illustrated in Fig. 2. 3.2 Virtual Measurement Branch: Context-Conditioned Virtual Observations The virtual measurement branch serves as a context-conditioned virtual sensor that generates probabilistic virtual observations from historical data. Unlike a standard observation model, it does not directly map the latent state to real measurements; instead, it produces adaptive virtual observations that guide the correction of the latent physical state during rollout. Given the input sequence 1:TinX_1:T_in, we employ S-Mamba [30], a selective state space model, to efficiently capture long-range temporal dependencies. The encoder produces a global context embedding ctx∈ℝN×dmodelH_ctx ^N× d_model. We then convert this context into a base virtual observation and its uncertainty via two projection heads: ctx=S-Mamba(1:Tin),base,base=ProjHeads(ctx).H_ctx=S-Mamba(X_1:T_in), 10.00002pt 10.00002ptZ_base,R_base=ProjHeads(H_ctx). (1) To enable efficient autoregressive decoding, we decompose the virtual measurement into static and dynamic components: obs,t=base+αtanh(t−1+),obs,t=Softplus(base)(1+βtTout),Z_obs,t=Z_base+α (S^+_t-1), 10.00002ptR_obs,t=Softplus(R_base) (1+β tT_out ), (2) where t−1+S^+_t-1 denotes the posterior latent state from the previous step. The first equation allows the virtual observation to adapt to the evolving trajectory without accessing future pollutant observations, while the second models the growth of epistemic uncertainty over the forecasting horizon. Both α and β are learnable scalars. 3.3 Process Model: Universal Differential Equation We model the latent physical state t∈ℝN×DS_t ^N× D as a continuous-time stochastic process governed by a Universal Differential Equation: dt=ℱdrift(t,t,t)⏟Deterministic Dynamicsdt+(t)⏟Stochastic Diffusiondt,dS_t= F_drift(S_t,E_t,M_t)_Deterministic Dynamics\,dt+ G(S_t)_Stochastic Diffusion\,dW_t, (3) where tW_t denotes a Wiener process. Here, tM_t and tE_t are treated as exogenous drivers at forecast step t; their inference-time availability and preprocessing are specified in Sec. 5. The stochastic term captures unresolved atmospheric variability, boundary leakage, and unobserved emissions that cannot be explicitly modeled by deterministic physics alone. The drift function ℱdriftF_drift explicitly disentangles transport and chemical reaction processes. Isotropic Diffusion. Background dispersion is modeled using a normalized graph Laplacian diff∈ℝN×NL_diff ^N× N derived from the static sensor graph: δdiff=−λ1difft, _diff=- _1L_diffS_t, where λ1≥0 _1≥ 0 is a learnable diffusion coefficient constrained via Softplus. The Laplacian operator acts on the spatial dimension and is broadcast across feature channels, enforcing flux from high- to low-concentration regions. Anisotropic Advection. To capture wind-driven transport, we construct a time-varying advection operator adv(t)L_adv(t). For each directed edge (vi,vj)(v_i,v_j), we define a unit direction vector: ij=(j−i)‖j−i‖2. _ij= (p_j-p_i)\|p_j-p_i\|_2. Given wind velocity i,tv_i,t at node i, dynamic edge weights are computed as: wij(t)=ReLU(i,t⋅ij‖i,t‖2)‖i,t‖2. w_ij(t)=ReLU\! ( v_i,t·e_ij\|v_i,t\|_2 )\|v_i,t\|_2. These weights define adv(t)L_adv(t), yielding the advection drift δadv=−λ2adv(t)t, _adv=- _2L_adv(t)S_t, where λ2≥0 _2≥ 0 is learnable. Meteorology-Gated Neural Reaction. Secondary pollutant formation is governed by nonlinear, condition-dependent chemical interactions. We model these dynamics using a meteorology-gated neural reaction mechanism. First, latent reactant features are constructed as react=Concat(t,t).H_react=Concat(S_t,E_t). Self-attention is applied to learn interaction structure: =reactQ,=reactK,=reactV.Q=H_reactW_Q, 10.00002ptK=H_reactW_K, 10.00002ptV=H_reactW_V. (4) Rather than recovering explicit stoichiometric coefficients, the attention mechanism learns conditional interaction affinities between latent species under uncertainty. To enforce environmental dependence, we introduce a meteorological gate: ℛchem=Attn(,,)⊙σ(Linear(tsc)),R_chem=Attn(Q,K,V) σ\! (Linear(M^sc_t) ), (5) where ⊙ denotes element-wise multiplication and tscM^sc_t denotes scalar meteorological variables (e.g., radiation, temperature). This gating mechanism suppresses reactions under unfavorable conditions and acts as a data-driven analog of reaction rate modulation. The full drift is given by: ℱdrift=δdiff+δadv+ℛchem.F_drift= _diff+ _adv+R_chem. 3.4 Neural Kalman Fusion To couple the virtual measurement branch with the physics-informed process branch, we employ a Neural Kalman Fusion module inspired by recursive filtering [22]. For scalability, we adopt a diagonal covariance approximation, reducing complexity from (N3)O(N^3) to (N)O(N). Prediction. Starting from t−1+S^+_t-1, the UDE solver produces a prior mean t−S^-_t. Process uncertainty is estimated as t−=Softplus(Linear(t−)).P^-_t=Softplus(Linear(S^-_t)). (6) Correction. Given the virtual observation (obs,t,obs,t)(Z_obs,t,R_obs,t), the update equations are: t=t−⊘(t−+obs,t),ϵt _t=P^-_t (P^-_t+R_obs,t ), 10.00002pt ε_t =obs,t−t,ϵ′t=ϵt+Linear([−t,obs,t]), =Z_obs,t-S^-_t, 10.00002pt ε _t= ε_t+Linear([S^-_t,Z_obs,t]), (7) t+=t−+t⊙ϵt′ ^+_t=S^-_t+K_t ε _t ,+t=(−t)⊙−t. , 10.00002ptP^+_t=(1-K_t) ^-_t. Here, ⊘ denotes element-wise division. The learnable correction term allows the update to adapt to non-Gaussian residuals caused by open-system dynamics, such as boundary leakage, unobserved emissions, and model misspecification. Closed-loop Feedback. The posterior state t+S^+_t is recursively fed back as the initial condition for the next prediction step. Thus, the virtual measurement branch and the process branch correct each other during rollout: virtual measurements correct the misspecified physical trajectory, while the UDE prior regularizes noisy data-driven evidence through structured transport–reaction dynamics. This closed-loop neural state correction reduces long-horizon drift. 3.5 Theoretical Interpretation We provide two theoretical interpretations of AeroChem in the Appendix. First, Appx. A.2 shows that Neural Kalman Fusion can be viewed as an approximate variational correction of an imperfect physical prior. The UDE prediction defines a physics-informed prior over the latent state, while the virtual measurement branch provides a data-driven correction signal. When the residual potential of the open system is Gaussian, the update reduces to the diagonal Kalman correction. When the residual potential is non-Gaussian, a first-order approximation motivates the learnable affine correction term in Eq. 7. This interpretation clarifies why the correction term is not merely a heuristic residual layer, but an approximation to a non-Gaussian posterior-state update. Second, Appx. A.3 analyzes why closed-loop correction improves long-horizon stability. Under standard Lipschitz and bounded-noise assumptions, the posterior error of the closed-loop system remains uniformly bounded when the correction factor is contractive. In contrast, the corresponding open-loop variant, which propagates the UDE prior without posterior feedback, can amplify initial errors over time. This analysis supports the central design of AeroChem: correcting the latent physical state before the next rollout step is more stable than correcting only the final output. Moreover, computational complexity is provided in App A.4. 3.6 Output Generation After closed-loop neural state correction, we obtain a sequence of posterior latent states t+t=1Tout\S^+_t\_t=1^T_out and their corresponding covariance matrices t+t=1Tout\P^+_t\_t=1^T_out. To generate probabilistic forecasts, we employ a decoding module that maps these latent distributions to the observation space. Specifically, the predictive mean and variance are decoupled: ^Tin+t=out(t+),Tin+t2=Softplus(var(t+)), X_T_in+t=D_out(S^+_t), 10.00002pt σ^2_T_in+t=Softplus(D_var(P^+_t)), (8) for t=1,…,Toutt=1,…,T_out. Here, outD_out and varD_var are shallow multilayer perceptrons with shared parameters across time and nodes. This ensures that the final output provides a fully parameterized predictive distribution p(t∣t+)=(^t,diag(t2))p(X_t ^+_t)=N( X_t,diag( σ^2_t)) to quantify epistemic uncertainty. 3.7 Training Objective The model is trained end-to-end by minimizing: ℒtotal=ℒMAE(^,)+λnllℒNLL(^,2,)+λphy‖ℱdrift‖F2, _total=L_MAE( X,X)+ _nll\,L_NLL( X, σ^2,X)+ _phy\,\|F_drift\|_F^2, (9) where ^t=out(t+) X_t=D_out(S^+_t) and t2=Softplus(var(t+)) σ^2_t=Softplus(D_var(P^+_t)) are the predicted mean and variance decoded from the Kalman posterior, ℒNLL(^,2,)=12log2+(−^)222 _NLL( X, σ^2,X)= 12 σ^2+ (X- X)^22 σ^2 is the heteroscedastic Gaussian NLL, and the physics regularization penalizes excessively large drift magnitudes, encouraging stable transport–reaction dynamics. To mitigate exposure bias during autoregressive training, we employ scheduled sampling. Early in training, the process model is conditioned on StopGrad(obs)StopGrad(Z_obs) with a probability that decays exponentially, before transitioning to fully closed-loop operation. 4 Related Work Spatio-temporal Air Quality Forecasting. Early air quality forecasting relied on statistical models such as ARIMA [14] and VAR [35], which are limited by linear assumptions. Deep learning methods subsequently improved predictive accuracy by modeling complex spatiotemporal dependencies, including RNN-based architectures and CNNs for gridded data. To handle non-Euclidean sensor networks, graph-based models such as DCRNN [13] and STGCN [34] were introduced and later adapted to air quality forecasting in PM2.5-GNN [29]. More recent approaches leverage Transformers (e.g., iTransformer [15], [17]) and structured State Space Models such as Mamba [6] to capture long-range temporal dependencies efficiently. Despite strong short-term performance, these data-driven methods do not explicitly model transport or chemical dynamics, and their predictions are usually generated in an open-loop autoregressive manner. As a result, they may accumulate errors over long horizons and over-smooth abrupt secondary pollution events driven by meteorological and chemical factors. In contrast, AeroChem uses long-range sequence modeling only as a virtual measurement branch, which is coupled with a physics-informed process branch through closed-loop latent-state correction. Physics-Informed Machine Learning. Physics-informed machine learning (PIML) incorporates physical knowledge into neural models to improve interpretability and robustness. PINNs [21] enforce governing PDEs through soft constraints but are difficult to optimize for high-dimensional atmospheric systems. Neural ODEs [2] and their extensions have enabled continuous-time modeling of physical dynamics. In air quality forecasting, AirPhyNet [8] models diffusion and advection but assumes a closed system, while Air-DualODE [28] addresses open-system dynamics using a dual-branch design. Universal Differential Equations (UDEs) [20] further allow neural approximation of unknown physical terms. However, existing PIML approaches for air quality forecasting often oversimplify chemical reactions and couple physical and neural components only loosely, for example by combining their outputs rather than correcting the evolving latent physical state. This open-loop coupling limits robustness in long-horizon forecasting because errors from misspecified physics can still be propagated forward. In contrast, AeroChem formulates the UDE as a structured but imperfect process branch and repeatedly corrects its latent state using context-conditioned virtual measurements. Deep Data Assimilation and Kalman Filtering. Data assimilation combines imperfect dynamical models with observations through recursive state estimation. Classical approaches such as the Kalman Filter [10], Extended Kalman Filter [4], and Ensemble Kalman Filter are widely used in geosciences, weather forecast, and control systems but rely on linear-Gaussian assumptions or costly ensembles and require hand-designed uncertainty models. Learning-based approaches relax these assumptions by learning filtering components from data, including Deep Kalman Filters [11] and KalmanNet [22]. These methods motivate recursive correction, but they typically assume access to real observations or a standard observation model. In our setting, future pollutant observations are unavailable during forecasting; therefore, AeroChem does not use a standard observation model. Instead, its virtual measurement branch generates context-conditioned virtual observations from historical context and the previous posterior state. Neural Kalman Fusion then fuses these virtual measurements with the UDE prior to correct the latent physical state at each rollout step. Thus, AeroChem is best viewed as an assimilation-inspired closed-loop neural state correction framework, rather than a classical data-assimilation model. 5 Experiments 5.1 Experimental Settings Datasets. We evaluate AeroChem on two real-world hourly air quality datasets: KnowAir and HanoiAir. KnowAir contains 355 monitoring stations from 2016–2023 and integrates pollutant concentrations, meteorological variables, and emission inventories. HanoiAir is a newly constructed dataset covering northern Vietnam with 27 monitoring stations from 2018–2025, including pollutant concentrations, meteorology, and downscaled emission inputs. For both datasets, we use chronological splits to evaluate future forecasting: KnowAir uses 2016–2019 for training, 2020–2021 for validation, and 2022–2023 for testing; HanoiAir uses 2018–2021 for training, 2022–2023 for validation, and 2024–2025 for testing. Detailed dataset descriptions and preprocessing procedures are provided in Appx. A.5. Baselines. We compare AeroChem against a diverse set of baseline models spanning several categories: (1) Statistical Methods: Standard time series models including HA (Historical Average) and VAR (Vector Autoregression) [35]. (2) Neural ODE / Physics-Informed Models: Methods incorporating physics-informed models, such as LatentODE [2], ODE-LSTM [12], Air-DualODE [28], and AirPhyNet [8]. (3) GNN-based Spatio-Temporal Models: Models explicitly designed for graph-structured time series data, including STGCN [34], DCRNN [13], GTS [23], and PM25GNN [29]. (4) Linear and MLP-based Models: Efficient architectures focusing on simpler mappings, such as DLinear [36] and ModernTCN [1]. (5) Transformer-based Models: Models leveraging attention mechanisms, including Fredformer [19], PatchTST [17], Crossformer [37], and TimeMoE [25]. (6) State Space Models (SSMs): Models based on structured state space representations, including S-Mamba [6]), S4 [7], and BiMamba [27]. Metrics. To quantitatively assess the performance of AeroChem and baseline models, we employ three standard forecasting metrics: Mean Absolute Error (MAE), Root Mean Square Error (RMSE), Symmetric Mean Absolute Percentage Error (SMAPE), Continuous Ranked Probability Score (CRPS), Summed CRPS (CRPSsum), and Prediction Interval Coverage Probability (PICP). Formal definitions of these metrics are provided in Appx. A.6. Hyperparameter & Implementation Details. Detailed hyperparameter settings are provided in Appx. A.7, and implementation details are included in Appx. A.8. An implementation is available at Github link. Forecasting Scenarios. We evaluate four forecasting scenarios: 3 days (1–72 hours), 6 days (1–144 hours), 9 days (1–216 hours), and Sudden Change events [28]. Sudden changes are defined as test instances where the ground-truth PM2.5 concentration exceeds 75μg/m375~ /m^3 and changes by more than ±20μg/m3± 20~ /m^3 within the next three hours. All metrics are computed after inverse transformation to the original physical scale. 5.2 Main Results Table 1: Performance comparison on the KnowAir dataset. The best results are highlighted in bold, and the second-best results are underlined. Mean ± std over 5 runs. Methods 3 Days 6 Days 9 Days Sudden Change MAE RMSE SMAPE MAE RMSE SMAPE MAE RMSE SMAPE MAE RMSE SMAPE HA 35.20± 0.05 51.50± 0.07 0.65± 0.001 37.80± 0.06 54.20± 0.08 0.68± 0.002 40.50± 0.07 58.10± 0.10 0.71± 0.002 55.40± 0.10 75.20± 0.14 0.75± 0.003 VAR 33.80± 0.08 49.20± 0.10 0.62± 0.002 36.50± 0.09 52.40± 0.13 0.66± 0.003 39.20± 0.11 56.50± 0.16 0.69± 0.003 53.10± 0.16 72.40± 0.22 0.72± 0.004 LatentODE 26.50± 0.15 42.10± 0.21 0.55± 0.004 28.20± 0.18 45.40± 0.25 0.57± 0.005 31.60± 0.22 49.20± 0.30 0.61± 0.006 48.40± 0.36 65.10± 0.49 0.65± 0.009 ODE-LSTM 27.10± 0.16 43.50± 0.22 0.56± 0.004 29.45± 0.19 46.80± 0.26 0.59± 0.005 32.80± 0.23 50.15± 0.31 0.62± 0.006 49.80± 0.38 67.30± 0.51 0.67± 0.010 Air-DualODE 23.04± 0.14 38.44± 0.19 0.52± 0.003 24.50± 0.17 41.60± 0.23 0.54± 0.004 25.20± 0.20 42.80± 0.27 0.59± 0.005 39.24± 0.32 53.30± 0.43 0.55± 0.008 AirPhyNet 23.80± 0.10 39.50± 0.14 0.53± 0.003 25.10± 0.12 41.80± 0.17 0.55± 0.004 27.20± 0.15 44.50± 0.20 0.57± 0.004 38.50± 0.24 53.10± 0.32 0.56± 0.006 STGCN 25.50± 0.13 40.80± 0.17 0.54± 0.003 27.80± 0.15 43.20± 0.20 0.57± 0.004 31.50± 0.19 47.40± 0.25 0.61± 0.005 46.50± 0.30 62.10± 0.40 0.64± 0.008 DCRNN 24.80± 0.12 39.60± 0.16 0.53± 0.003 26.90± 0.14 42.50± 0.18 0.56± 0.004 30.10± 0.17 46.80± 0.23 0.60± 0.005 44.20± 0.27 60.40± 0.37 0.62± 0.008 GTS 24.10± 0.11 38.50± 0.15 0.52± 0.003 26.50± 0.13 41.10± 0.17 0.55± 0.004 29.40± 0.16 45.50± 0.21 0.59± 0.005 42.50± 0.25 58.20± 0.34 0.60± 0.007 PM25GNN 24.50± 0.12 39.20± 0.16 0.53± 0.003 27.10± 0.14 41.80± 0.18 0.56± 0.004 30.20± 0.17 46.10± 0.22 0.60± 0.005 43.10± 0.27 59.50± 0.36 0.61± 0.008 DLinear 22.57± 0.08 37.80± 0.11 0.51± 0.002 23.85± 0.10 39.50± 0.13 0.53± 0.003 24.50± 0.12 41.20± 0.16 0.55± 0.004 38.45± 0.20 51.22± 0.27 0.55± 0.006 ModernTCN 22.33± 0.09 37.60± 0.12 0.51± 0.003 23.68± 0.11 39.40± 0.14 0.53± 0.003 25.12± 0.13 41.50± 0.17 0.54± 0.004 37.92± 0.21 50.81± 0.28 0.54± 0.006 TimeMoE 21.68± 0.09 37.20± 0.12 0.49± 0.002 23.15± 0.11 38.95± 0.14 0.50± 0.003 25.32± 0.13 40.15± 0.17 0.55± 0.004 36.54± 0.21 49.12± 0.28 0.53± 0.006 Fredformer 21.45± 0.08 38.10± 0.12 0.51± 0.002 24.12± 0.11 40.20± 0.15 0.53± 0.003 26.48± 0.14 42.80± 0.18 0.56± 0.004 38.25± 0.22 51.03± 0.30 0.55± 0.006 PatchTST 22.69± 0.09 38.50± 0.12 0.51± 0.003 23.91± 0.11 40.50± 0.14 0.51± 0.003 25.25± 0.13 42.80± 0.18 0.56± 0.004 39.12± 0.22 52.34± 0.30 0.56± 0.006 Crossformer 22.00± 0.09 37.90± 0.12 0.52± 0.003 23.75± 0.11 39.80± 0.14 0.54± 0.003 26.10± 0.13 42.10± 0.17 0.56± 0.004 37.55± 0.21 50.42± 0.29 0.54± 0.006 SMamba 23.12± 0.10 38.80± 0.13 0.52± 0.003 24.24± 0.12 39.40± 0.15 0.55± 0.004 25.15± 0.14 40.72± 0.18 0.58± 0.004 39.85± 0.23 53.12± 0.31 0.57± 0.007 S4 24.53± 0.12 40.20± 0.16 0.54± 0.003 25.18± 0.13 40.51± 0.17 0.57± 0.004 26.21± 0.15 41.23± 0.19 0.60± 0.005 42.14± 0.25 56.55± 0.34 0.61± 0.008 BiMamba 22.84± 0.10 38.50± 0.13 0.52± 0.003 24.07± 0.12 39.10± 0.15 0.55± 0.003 25.63± 0.14 40.88± 0.18 0.58± 0.004 39.25± 0.23 52.64± 0.31 0.56± 0.007 AeroChem (ours) 21.95± 0.06 37.20± 0.09 0.50± 0.002 23.00± 0.08 38.80± 0.11 0.52± 0.003 24.08± 0.10 39.34± 0.14 0.54± 0.004 34.50± 0.18 49.50± 0.25 0.52± 0.005 Table 2: Performance comparison on HanoiAir dataset. The best results are highlighted in bold, and the second-best results are underlined. Mean ± std over 5 runs. Methods 3 Days 6 Days 9 Days Sudden Change MAE RMSE SMAPE MAE RMSE SMAPE MAE RMSE SMAPE MAE RMSE SMAPE HA 55.20± 0.10 78.50± 0.14 0.85± 0.002 58.40± 0.12 82.10± 0.16 0.88± 0.003 62.10± 0.14 86.50± 0.20 0.92± 0.003 75.40± 0.22 105.20± 0.30 0.89± 0.005 VAR 52.80± 0.13 75.20± 0.18 0.82± 0.003 56.50± 0.15 79.40± 0.21 0.85± 0.003 60.20± 0.18 84.80± 0.25 0.88± 0.004 72.10± 0.28 100.40± 0.38 0.86± 0.006 LatentODE 46.50± 0.30 68.10± 0.42 0.80± 0.006 49.20± 0.36 71.40± 0.50 0.83± 0.007 52.60± 0.42 75.20± 0.58 0.86± 0.008 74.40± 0.65 98.10± 0.88 0.85± 0.012 ODE-LSTM 47.10± 0.32 69.50± 0.45 0.81± 0.006 50.45± 0.38 72.80± 0.52 0.84± 0.007 54.80± 0.45 78.15± 0.62 0.87± 0.008 75.80± 0.70 101.30± 0.94 0.86± 0.012 Air-DualODE 42.04± 0.28 64.41± 0.40 0.78± 0.005 45.50± 0.34 67.60± 0.46 0.82± 0.006 48.20± 0.40 71.80± 0.54 0.86± 0.007 72.24± 0.58 93.30± 0.78 0.83± 0.010 AirPhyNet 43.80± 0.20 65.50± 0.28 0.79± 0.004 46.10± 0.24 69.80± 0.33 0.82± 0.005 49.20± 0.28 73.50± 0.39 0.85± 0.005 71.50± 0.42 94.10± 0.58 0.79± 0.008 STGCN 45.50± 0.25 67.80± 0.34 0.81± 0.005 48.80± 0.30 72.20± 0.40 0.84± 0.006 52.50± 0.35 77.40± 0.48 0.88± 0.007 74.50± 0.55 98.10± 0.74 0.85± 0.010 DCRNN 44.80± 0.23 66.60± 0.32 0.80± 0.005 47.90± 0.27 71.50± 0.37 0.83± 0.005 51.10± 0.32 76.80± 0.44 0.87± 0.006 73.20± 0.50 97.40± 0.68 0.84± 0.009 GTS 44.10± 0.21 65.50± 0.29 0.79± 0.004 47.50± 0.25 70.10± 0.34 0.82± 0.005 50.40± 0.30 75.50± 0.41 0.86± 0.006 72.50± 0.46 95.20± 0.62 0.83± 0.008 PM25GNN 44.50± 0.22 66.20± 0.30 0.80± 0.005 48.10± 0.26 71.80± 0.36 0.83± 0.005 51.20± 0.31 76.10± 0.42 0.87± 0.006 73.10± 0.48 96.50± 0.65 0.83± 0.009 DLinear 42.57± 0.16 64.80± 0.22 0.78± 0.004 44.85± 0.19 67.50± 0.26 0.81± 0.004 47.10± 0.22 70.20± 0.30 0.84± 0.005 73.45± 0.36 96.22± 0.48 0.82± 0.007 ModernTCN 42.33± 0.17 64.60± 0.23 0.78± 0.004 44.68± 0.20 67.40± 0.27 0.81± 0.004 47.50± 0.23 70.50± 0.31 0.84± 0.005 72.92± 0.37 95.81± 0.50 0.81± 0.007 TimeMoE 41.85± 0.18 63.80± 0.25 0.77± 0.004 44.15± 0.21 66.95± 0.28 0.80± 0.004 47.50± 0.24 69.15± 0.32 0.83± 0.005 71.54± 0.38 93.12± 0.50 0.78± 0.007 Fredformer 41.60± 0.16 64.10± 0.22 0.77± 0.004 44.20± 0.19 67.20± 0.26 0.80± 0.004 47.48± 0.22 71.20± 0.30 0.84± 0.005 73.25± 0.36 95.03± 0.48 0.81± 0.007 PatchTST 42.10± 0.18 64.50± 0.24 0.78± 0.004 44.50± 0.21 67.80± 0.28 0.81± 0.004 47.80± 0.24 71.50± 0.32 0.85± 0.005 74.12± 0.38 96.34± 0.50 0.83± 0.007 Crossformer 42.00± 0.17 64.20± 0.23 0.78± 0.004 44.40± 0.20 67.50± 0.27 0.81± 0.004 47.60± 0.23 71.80± 0.31 0.85± 0.005 73.55± 0.37 95.42± 0.50 0.82± 0.007 S-Mamba 43.12± 0.20 65.80± 0.27 0.79± 0.005 45.24± 0.23 66.40± 0.31 0.82± 0.005 47.15± 0.26 69.57± 0.35 0.86± 0.006 75.85± 0.42 98.12± 0.56 0.84± 0.008 S4 44.53± 0.24 67.20± 0.32 0.81± 0.005 46.18± 0.28 68.50± 0.38 0.84± 0.006 48.14± 0.32 69.81± 0.43 0.88± 0.007 77.14± 0.50 101.55± 0.68 0.86± 0.010 BiMamba 42.84± 0.19 65.50± 0.26 0.79± 0.005 45.97± 0.22 67.10± 0.30 0.82± 0.005 47.63± 0.25 70.80± 0.34 0.85± 0.006 75.25± 0.40 97.64± 0.54 0.83± 0.008 AeroChem (ours) 42.74± 0.10 63.11± 0.14 0.78± 0.003 44.50± 0.13 65.80± 0.18 0.79± 0.003 46.80± 0.16 68.34± 0.22 0.81± 0.004 68.80± 0.28 91.16± 0.38 0.75± 0.006 Table 3: Ablation Study: Effect of Physical Mechanisms and Models Design. Method HanoiAir KnowAir 3 Days 9 Days 3 Days 9 Days MAE RMSE SMAPE MAE RMSE SMAPE MAE RMSE SMAPE MAE RMSE SMAPE AeroChem 42.74 63.11 0.78 46.80 68.34 0.81 21.95 37.20 0.50 24.08 39.34 0.54 Physical Mechanisms w/o Diffusion (AeroChem -D) 44.31 65.25 0.80 48.37 70.48 0.83 23.52 39.34 0.52 25.65 41.48 0.56 w/o Advection (AeroChem -A) 42.81 66.34 0.78 48.80 69.64 0.84 23.95 40.02 0.52 24.98 40.74 0.58 w/o Reaction (AeroChem -R) 45.15 66.80 0.82 49.60 71.90 0.85 24.10 40.50 0.53 26.80 42.90 0.59 w/o Physics (AeroChem -P) 47.85 69.50 0.86 53.20 75.40 0.90 26.50 43.80 0.58 29.10 46.50 0.63 Models & Fusion w/o Kalman (AeroChem -K) 56.40 78.20 0.95 63.50 85.10 1.05 30.20 48.50 0.65 35.50 54.20 0.72 w/ EnKF (AeroChem -EnKF) 43.62 62.45 0.81 47.85 67.20 0.84 24.08 35.12 0.59 25.07 36.06 0.61 w/o Neur. Corr (AeroChem -N) 43.10 63.37 0.79 47.16 68.60 0.82 22.31 37.46 0.51 24.44 39.60 0.55 w/o Mamba (AeroChem -M) 43.05 63.80 0.79 51.50 74.20 0.88 22.15 37.60 0.51 28.40 45.20 0.61 w/ PatchTST (AeroChem -Patch) 44.18 64.50 0.81 48.42 69.85 0.84 24.60 36.15 0.61 25.55 37.10 0.63 w/ DLinear (AeroChem -DLinear) 44.35 64.20 0.81 48.55 69.65 0.85 24.8 35.88 0.61 25.64 36.94 0.63 w/ iTrans (AeroChem -iTrans) 44.05 64.40 0.80 48.30 69.70 0.84 24.72 36.05 0.60 25.48 37.05 0.62 w/ Crossformer (AeroChem -Cross) 43.95 64.30 0.80 48.20 69.55 0.84 24.55 36.20 0.60 25.42 37.20 0.62 Late Fusion 53.25 76.10 0.92 59.80 81.50 0.98 29.40 47.10 0.63 33.80 51.50 0.69 Tables 1 and 2 report the main PM2.5 forecasting results on KnowAir and HanoiAir. Overall, AeroChem is most effective in long-horizon and sudden-change scenarios, where open-loop autoregressive and purely data-driven models are more prone to drift or over-smoothing. Results for O3 are provided in Appx. A.13. Transformer-based and SSM-based models, such as TimeMoE, PatchTST, and S-Mamba, achieve competitive short-term accuracy, but their errors generally increase as the horizon becomes longer. In contrast, AeroChem maintains more stable long-horizon performance. This supports our central design: the UDE-based process branch provides a structured physical prior, while Neural Kalman Fusion repeatedly corrects the latent state during rollout, reducing the error accumulation typical of open-loop forecasting. The advantage of AeroChem is more evident on HanoiAir, which has fewer monitoring stations and noisier regional dynamics. Compared with strong direct-mapping models such as TimeMoE and ModernTCN, AeroChem achieves more stable performance, suggesting that closed-loop neural state correction improves robustness when observations are sparse or noisy. Sudden pollution changes are particularly challenging because many baselines tend to smooth rapid concentration shifts. In the Sudden Change setting, AeroChem achieves stronger performance than Transformer and SSM baselines. This improvement is consistent with the role of the meteorology-gated neural reaction module, which conditions chemical dynamics on environmental drivers such as radiation and temperature, helping the model capture nonlinear secondary pollution formation. Probabilistic forecasting results on KnowAir PM2.5 are shown in Appx. A.9. 5.3 Ablation Study We conduct ablation studies on KnowAir and HanoiAir to evaluate two aspects of AeroChem: (i) the physical mechanisms in the process branch and (i) the architecture of the virtual measurement branch and fusion module. Table 3 shows the experimental results of the ablation studies. Effect of Physical Mechanisms. We first remove individual components from the UDE drift function in Eq. 3: isotropic diffusion (AeroChem-D), anisotropic advection (AeroChem-A), the meteorology-gated neural reaction module (AeroChem-R), and all explicit physical terms (AeroChem-P). Removing all physical terms leads to clear long-horizon degradation, confirming that the structured process branch helps constrain rollout dynamics. Among the physical components, removing the reaction module causes substantial degradation, showing that meteorology-conditioned chemical dynamics are important for modeling secondary pollutant formation beyond transport alone. Effect of Virtual Measurements and Fusion. We next evaluate the virtual measurement branch and the closed-loop fusion design. We replace S-Mamba with a bidirectional LSTM (AeroChem-M) and with representative temporal encoders, including PatchTST (AeroChem-Patch), DLinear (AeroChem-DLinear), iTransformer (AeroChem-iTrans), and Crossformer (AeroChem-Cross). We also remove posterior feedback to obtain an open-loop UDE solver (AeroChem-K), remove the learnable correction term while keeping the diagonal Kalman update (AeroChem-N), replace Neural Kalman Fusion with an Ensemble Kalman Filter (AeroChem-EnKF), and decouple the process and virtual measurement branches using weighted output-level averaging (Late Fusion). We observe that removing closed-loop feedback causes the largest degradation, especially at longer horizons, confirming that correcting the latent state before the next rollout step is essential for stability. Removing the learnable neural correction also degrades performance, suggesting that a strictly Gaussian correction is insufficient for open environmental systems with boundary leakage, unobserved emissions, and model misspecification. Finally, Late Fusion performs worse than latent-state fusion, supporting our claim that the two branches should correct each other during rollout rather than being combined only at the output level. 5.4 Effect of Forecast Horizons We show that AeroChem maintains significantly more stable long-term behavior than physics-guided baselines. Fig. 3 illustrates the error evolution across varying horizons. AeroChem exhibits superior stability compared to existing physics-guided baselines (e.g., AirPhyNet, Air-DualODE), with the performance gap widening significantly as the prediction window extends. While prior methods suffer from rapid error propagation, AeroChem maintains a flattened error trajectory, validating the effectiveness of our closed-loop design in mitigating long-term physical drift. Figure 3: Performance across forecast horizons. 667272144144216216202030304040Forecast Horizon (Hours)MAE (a) MAE 667272144144216216202040406060Forecast Horizon (Hours)RMSE (b) RMSE 6672721441442162160.40.40.50.50.60.6Forecast Horizon (Hours)SMAPE (c) SMAPE GTSAirPhyNetPatchTSTDLinearTimeMoEAeroChem 5.5 Hyperparameter Studies We assess the model’s robustness to three key hyperparameters. Fig. 4 shows the experimental results. 000.20.20.40.40.60.60.80.8112525303035354040Physics Weight (λphy _phy)ErrorMAERMSE (a) Effect of Physics Weight (λphy _phy) 646496961281282525303035354040Model Dimension (dmodeld_model)MAERMSE (b) Effect of Dimension (dmodeld_model) 0.20.20.50.5111.51.5222020404060608080Solver Step (dtdt)MAERMSE (c) Effect of Solver Step (dtdt) Figure 4: Sensitivity analysis on KnowAir dataset. The dashed vertical lines indicate the default hyperparameters used in the full model (λ=0.1,d=96,dt=1.0λ=0.1,d=96,dt=1.0), which consistently correspond to the optimal performance regions (lowest error). Physics Weight (λphy _phy): Fig. 4(a) reveals a distinct trade-off between physical constraints and data flexibility, forming a bell-shaped performance curve. When λphy→0 _phy→ 0, weak physical guidance increases long-term drift and prediction errors. Conversely, λphy>1 _phy>1 over-constrains latent dynamics, limiting the model’s ability to capture localized variations. The best performance at λphy=0.1 _phy=0.1 balances physical consistency with data-driven adaptability. Model Dimension (dmodeld_model): As depicted in Fig 4(b), expanding the hidden dimension initially yields substantial gains, confirming that sufficient representational capacity is requisite for encoding complex spatiotemporal dependencies. However, performance saturates beyond dmodel=96d_model=96, where further increases offer negligible improvements while inflating computational costs. Consequently, we select 96 as the optimal equilibrium between model expressiveness and computational efficiency. Solver Step (dtdt): Fig. 4(c) demonstrates the model’s remarkable numerical robustness, maintaining consistent accuracy across varying step sizes. Typically, coarser discretization introduces integration errors in standard ODE solvers. However, our results indicate that the fusion mechanism effectively rectifies these trajectory deviations by assimilating data-driven corrections. 5.6 Additional Experiments First, given that AeroChem utilizes an ODE solver to model continuous dynamics and incorporates complex spatio-temporal dependencies, it is necessary to verify the model’s numerical stability. We study the numerical stability in Appx. A.10. Second, additional visualizations of model predictions during sudden pollution events are shown in Appx. A.11, highlighting AeroChem ’s ability to track abrupt shifts while revealing remaining challenges in recovering extreme peak amplitudes. Further, to validate the interpretability and robustness of AeroChem, we go beyond standard metrics to analyze the internal mechanisms of the model. We specifically examine two critical aspects: the learning of chemical kinetics and the stability of long-term forecasting. The case study is presented in Appx. A.12. 6 Conclusion We presented AeroChem, a closed-loop physics-informed state space framework for long-term chemically reactive air quality forecasting. AeroChem couples a UDE-based process branch with a Mamba-based virtual measurement branch through Neural Kalman Fusion, enabling the two branches to correct each other during rollout. By updating the latent physical state before each subsequent prediction step, AeroChem mitigates the long-horizon drift caused by open-loop forecasting designs. Experiments on KnowAir and the newly constructed HanoiAir dataset demonstrate that AeroChem improves long-horizon stability and sudden-change forecasting, especially under noisy observations and secondary pollutant formation. These findings highlight closed-loop neural state correction as a promising direction for robust physics-informed environmental forecasting. Future work will extend the framework to richer multi-species chemical interaction modeling and probabilistic risk assessment for environmental decision-making. Acknowledgments This work was supported by the Center for Environmental Intelligence, VinUniversity, under project VUNI.CEI.FS_0011. The authors are grateful for the constructive feedback and insightful discussions provided by the anonymous reviewers of NeurIPS 2026. Broader Impact. Accurate long-term air quality forecasting has direct societal relevance, as it can support public-health planning, emission-control strategies, and early-warning systems for hazardous pollution events. By reducing long-horizon drift through closed-loop state correction, AeroChem may help environmental agencies better anticipate persistent haze episodes, secondary pollutant formation, and sudden pollution changes, especially in rapidly urbanizing or data-sparse regions. More broadly, this work contributes to scientific machine learning by showing how physics-informed dynamics and data-driven measurements can be coupled in a closed-loop co-correction framework for partially observed open systems. Similar ideas may be useful beyond air quality forecasting, including climate modeling, hydrology, energy systems, and other domains where imperfect physical simulators must be corrected using observational data. Limitations. Several limitations remain. 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Algorithm 1 details the closed-loop forecasting procedure, explicitly illustrating how the measurement model, the physics-informed Universal Differential Equation (UDE), and the Neural Kalman Fusion module interact over the forecasting horizon. The algorithm highlights the recursive feedback mechanism in which posterior latent states are assimilated and fed back into the UDE solver. Algorithm 1 AeroChem Forecasting Input : Sensor graph G; historical observations 1:TinX_1:T_in; meteorology 1:ToutM_1:T_out; emissions 1:ToutE_1:T_out; forecast horizon ToutT_out. Output : Forecasted observations ^Tin+1:Tin+Tout X_T_in+1:T_in+T_out. 1 Measurement encoding: 2 ctx←S-Mamba(1:Tin)H_ctx -Mamba(X_1:T_in) 3 (base,base)←ProjHeads(ctx)(Z_base,R_base) (H_ctx) 4 Initialization: 5 Initialize posterior latent state 0+S^+_0 (e.g., from the last observed timestep via a projection/encoder) 6 for t←1t← 1 to ToutT_out do 7 Adaptive virtual observation: 8 obs,t←base+αtanh(t−1+)Z_obs,t _base+α (S^+_t-1) 9 obs,t←Softplus(base)(1+βtTout)R_obs,t (R_base) (1+β tT_out ) 10 UDE prediction (time update): 11 t−←UDE-Solve(t−1+,t,t,)S^-_t -Solve\! (S^+_t-1;M_t,E_t,G ) 12 t−←Softplus(Linear(t−))P^-_t (Linear(S^-_t)) 13 Neural Kalman fusion (measurement update): 14 t←t−⊘(t−+obs,t)K_t ^-_t (P^-_t+R_obs,t) 15 ϵt←obs,t−t− ε_t _obs,t-S^-_t 16 ϵt′←ϵt+Linear([t−,obs,t]) ε _t← ε_t+Linear\! ([S^-_t,Z_obs,t] ) 17 t+←t−+t⊙ϵt′S^+_t ^-_t+K_t ε _t 18 t+=(−t)⊙t−P^+_t=(1-K_t) ^-_t 19 Decoding (Probabilistic Output): 20 for t←1t← 1 to ToutT_out do 21 ^Tin+t←out(t+) X_T_in+t _out(S^+_t) Tin+t2←Softplus(var(t+)) σ^2_T_in+t (D_var(P^+_t)) 22 return ^Tin+t,Tin+t2t=1Tout\ X_T_in+t, σ^2_T_in+t\_t=1^T_out A.2 Variational Interpretation of Neural Kalman Fusion We first recast AeroChem under a single variational objective. Let 1:Tout∈ℝTout×N×DS_1:T_out ^T_out× N× D denote the latent trajectory and let 1:TinX_1:T_in denote the historical observations. Define the per-step variational free energy ℱt()=12(−t−)⊤(t−)−1(−t−)+12(obs,t−)⊤obs,t−1(obs,t−)+Ψt(),F_t(S)\;=\; 12\, (S-S_t^- )^\! (P_t^-)^-1 (S-S_t^- )\;+\; 12\, (Z_obs,t-S )^\! R_obs,t^-1 (Z_obs,t-S )\;+\; _t(S), (10) where (t−,t−)(S_t^-,P_t^-) is produced by the UDE prediction step (Eq. 6), (obs,t,obs,t)(Z_obs,t,R_obs,t) is the virtual measurement of the Mamba encoder (Eq. 2), and Ψt:ℝN×D→ℝ _t:R^N× D is a smooth residual potential that absorbs all non-Gaussian discrepancies of the open system (boundary leakage, unobserved emissions, model misspecification). We assume t−,obs,tP_t^-,R_obs,t are positive element-wise (guaranteed by the Softplus activations Eq 6 and treat all matrix–vector products in Eq. 10 element-wise to be consistent with the diagonal covariance approximation t−,obs,t∈ℝ>0N×DP_t^-,R_obs,t _>0^N× D used throughout AeroChem. Lemma A.1. Assume Ψt∈C3(ℝN×D) _t∈ C^3(R^N× D). Let t⋆S _t be a local minimiser of ℱtF_t and let t+S_t^+ be one Newton step initialised at the prior mean t−S_t^-: t+=t−[∇2ℱt(t−)]−1∇ℱt(t−).S_t^+\;=\;S_t^-- [∇^2F_t(S_t^-) ]^-1 _t(S_t^-). (11) Then, expanding under the diagonal-covariance regime: (i) If Ψt≡0 _t≡ 0 (purely Gaussian), then Eq. 11 reduces exactly to the standard Kalman update of AeroChem, as shown in the fifth equation of 7) (i) If Ψt≢0 _t ≡ 0, the Newton step admits a first-order Taylor expansion of ∇Ψt∇ _t around the joint point (t−,obs,t)(S_t^-,Z_obs,t). The resulting correction is bilinear in (t−,obs,t)(S_t^-,Z_obs,t) and is therefore representable by an affine map Linear([t−,obs,t])Linear ([S_t^-,Z_obs,t] ). Substituting yields exactly the Neural Kalman Fusion update (Eq 7). The residual error of the Newton step is t+−t⋆=O(∥t−t⋆∥2)S_t^+-S _t=O\! ( _t^--S _t ^2 ). Proof: Part (i). With Ψt≡0 _t≡ 0, ℱtF_t is quadratic. Differentiating Eq. 10: ∇ℱt(t−)=(t−)−1(t−t−)−obs,t−1(obs,t−t−)=−obs,t−1ϵt, _t(S_t^-)\;=\;(P_t^-)^-1(S_t^--S_t^-)\,-\,R_obs,t^-1(Z_obs,t-S_t^-)\;=\;-R_obs,t^-1\, ε_t, ∇2ℱt(t−)=(t−)−1+obs,t−1.∇^2F_t(S_t^-)\;=\;(P_t^-)^-1+R_obs,t^-1. Under the diagonal regime (t−)−1+obs,t−1=(t−+obs,t)⊘(t−⊙obs,t)(P_t^-)^-1+R_obs,t^-1=(P_t^-+R_obs,t) (P_t^- _obs,t), so the Newton step in Eq. 11 becomes t+=t−+[(t−)−1+obs,t−1]−1obs,t−1ϵt=t−+(t−⊘(t−+obs,t))⏟=t⊙ϵt,S_t^+\;=\;S_t^-+ [(P_t^-)^-1+R_obs,t^-1 ]^-1R_obs,t^-1\, ε_t\;=\;S_t^-+ (P_t^- (P_t^-+R_obs,t) )_=\,K_t \, ε_t, which is the fourth equation of Eq. 7. The covariance update t−→t+P_t^- _t^+ of fifth equation in Eq. 7 follows from the standard Kalman posterior covariance identity under optimal gain [22]. Part (i). Adding Ψt _t contributes ∇Ψt(t−)∇ _t(S_t^-) to the gradient and ∇2Ψt(t−)∇^2 _t(S_t^-) to the Hessian. Taylor-expanding ∇Ψt∇ _t to first order around the joint argument (t−,obs,t)(S_t^-,Z_obs,t): ∇Ψt(t−)≈+t−+obs,t=Linear([t−,obs,t]),∇ _t(S_t^-)\;≈\;a+BS_t^-+CZ_obs,t\;=\;Linear ([S_t^-,Z_obs,t] ), for parameter tensors (,,)(a,B,C) that are absorbed into the linear layer of the third equation in Eq. 7. The second-order Hessian contribution can be merged into t−P_t^- via covariance reparameterisation. The Newton-step quadratic convergence t+−t⋆=O(∥t−t⋆∥2)S_t^+-S _t=O( _t^--S _t ^2) is the standard result for C3C^3 objectives with positive-definite Hessian [40], [18]. ∎ A.3 Closed-loop Neural State Correction We show that the closed-loop neural state correction yields a posterior error that is uniformly bounded in t, while the open-loop variant (AeroChem-K in Table 3) admits exponential divergence, details are included in App A.3. Let ΦΔt:ℝN×D→ℝN×D _ t:R^N× D\!→\!R^N× D denote the one-step UDE flow so that t−=ΦΔt(t−1+)S_t^-\!=\! _ t(S_t-1^+). Define the posterior error t≔t+−t∗e_t\! \!S_t^+-S_t^* relative to the unobserved true latent trajectory t∗S_t^*. Assumption A.2. (A5) driftF_drift is LFL_F-Lipschitz in S, so ∥ΦΔt(1)−ΦΔt(2)∥2≤(1+LFΔt)∥1−2∥2 _ t(S_1)- _ t(S_2) _2\!≤\!(1+L_F t) _1-S_2 _2; (A6) (by construction) the gain in Eq. 7 satisfies t∈[κmin,κmax]K_t\!∈\![ _ , _ ] element-wise with <κmin≤κmax<10\!<\! _ \!≤\! _ \!<\!1; (A7) (by construction) the correction in the third equation of Eq. 7 is Linear([t−,obs,t])=WϵtLinear([S_t^-,Z_obs,t])\!=\!W ε_t with ∥W∥2≤Lc W _2\!≤\!L_c enforced by spectral normalisation; (A8) the conditional measurement bias is bounded, ∥[obs,t∣ℱt−1]−t∗∥2≤b [Z_obs,t\! \!F_t-1]-S_t^* _2\!≤\!b; (A9) the centred process and measurement noises satisfy ∥t∥22≤σW2ΔtE _t _2^2\!≤\! _W^2 t and ∥t−[t∣ℱt−1]∥22≤σν2E ν_t-E[ ν_t\! \!F_t-1] _2^2\!≤\! _ν^2, with tw_t and the centred residual conditionally independent given ℱt−1F_t-1. Theorem A.3 (Closed-loop versus open-loop posterior error). Define the contraction factor α≔(1+LFΔt)[(1−κmin)+κmaxLc].α\, \,(1+L_F t)[(1- _ )+ _ L_c]. If α<1α\!<\!1, then under Assumption A.2 the closed-loop posterior error of AeroChem satisfies, for every t∈1,…,Toutt\!∈\!\1,…,T_out\, ∥t∥22≤αt∥0∥22+V1−α,V=κmaxb+κmax2σν2+(−κmin+κmaxLc)2σW2Δt, E _t _2^2\;≤\;α^t E _0 _2^2+ V1-α, 18.49988ptV= _ b+ _ ^2 _ν^2+(1\!-\! _ \!+\! _ L_c)^2 _W^2 t, (12) which is uniformly bounded in t. The open-loop variant (t≡K_t\!≡\!0, AeroChem-K) admits the matching lower bound ∥topen∥22≥(1+LFΔt)2t∥0∥22,E _t^open _2^2\;≥\;(1+L_F t)^2tE _0 _2^2, (13) which grows exponentially in t whenever LF>0L_F\!>\!0. We give the complete proof in six steps. Throughout, t≔obs,t−t∗ ν_t\! \!Z_obs,t-S_t^* denotes the measurement residual, t≔t∗−ΦΔt(t−1∗)w_t\! \!S_t^*- _ t(S_t-1^*) the discretised Wiener increment, ¯t≔[t∣ℱt−1] ν_t\! \!E[ ν_t\! \!F_t-1] the conditional bias, and ~t≔t−¯t ν_t\! \! ν_t- ν_t the centred residual. By Assumptions A8–A9, ∥¯t∥2≤b ν_t _2\!≤\!b, ∥~t∥22≤σν2E ν_t _2^2\!≤\! _ν^2, and ∥t∥22≤σW2ΔtE _t _2^2\!≤\! _W^2 t, with t,~tw_t, ν_t conditionally zero-mean and independent of t−1e_t-1. Step 1 Substitute t+S_t^+ into t=t+−t∗e_t\!=\!S_t^+-S_t^* and use t∗=ΦΔt(t−1∗)+tS_t^*\!=\! _ t(S_t-1^*)+w_t, obs,t=t∗+tZ_obs,t\!=\!S_t^*+ ν_t, t−=ΦΔt(t−1+)S_t^-\!=\! _ t(S_t-1^+): t=(−t)⊙[ΦΔt(t−1+)−ΦΔt(t−1∗)]−(−t)⊙t+t⊙t+t⊙Wϵt.e_t=(1-K_t) [ _ t(S_t-1^+)- _ t(S_t-1^*) ]-(1-K_t) _t+K_t ν_t+K_t W ε_t. (14) By the second and third equations in 7, the innovation expands as ϵt=obs,t−t−=t+(t∗−t−)=t−[ΦΔt(t−1+)−ΦΔt(t−1∗)]+t. ε_t=Z_obs,t-S_t^-= ν_t+(S_t^*-S_t^-)= ν_t- [ _ t(S_t-1^+)- _ t(S_t-1^*) ]+w_t. (15) Substituting Eq. 15 into the last term of Eq. 14 and collecting the coefficients of ΦΔt(t−1+)−ΦΔt(t−1∗) _ t(S_t-1^+)- _ t(S_t-1^*), tw_t and t ν_t: t=[(−t)−tW]⊙[ΦΔt(t−1+)−ΦΔt(t−1∗)]⏟=:T1+[−(−t)+tW]⊙t⏟=:T2+t(+W)⊙t⏟=:T3.e_t\;=\; [(1-K_t)-K_tW ] [ _ t(S_t-1^+)- _ t(S_t-1^*) ]_=:\,T_1+ [-(1-K_t)+K_tW ] _t_=:\,T_2+ K_t(1+W) ν_t_=:\,T_3. (16) Step 2 Split T3=t(+W)⊙¯t+t(+W)⊙~tT_3\!=\!K_t(1+W) ν_t+K_t(1+W) ν_t and write t=Dt+Ste_t\!=\!D_t+S_t with Dt D_t =T1+t(+W)⊙¯t(ℱt−1-measurable), =T_1+K_t(1+W) ν_t 9.24994pt(F_t-1-measurable), (17) St S_t =[−(−t)+tW]⊙t+t(+W)⊙~t(conditionally zero-mean). = [-(1-K_t)+K_tW ] _t+K_t(1+W) ν_t 9.24994pt(conditionally zero-mean). (18) Since DtD_t is ℱt−1F_t-1-measurable and [St∣ℱt−1]=E[S_t\! \!F_t-1]\!=\!0, ∥t∥22=∥Dt∥22+∥St∥22.E _t _2^2\;=\;E D_t _2^2+E S_t _2^2. (19) Step 3 Apply the triangle inequality to Eq. 17 and the operator bounds ∥(−t)−tW∥∞≤(1−κmin)+κmaxLc (1-K_t)-K_tW _∞\!≤\!(1- _ )+ _ L_c (from A6, A7), ∥t(+W)∥∞≤κmax(1+Lc) _t(1+W) _∞\!≤\! _ (1+L_c), together with the Lipschitz bound ∥ΦΔt(t−1+)−ΦΔt(t−1∗)∥2≤(1+LFΔt)∥t−1∥2 _ t(S_t-1^+)- _ t(S_t-1^*) _2\!≤\!(1+L_F t) _t-1 _2 from A5 and ∥¯t∥2≤b ν_t _2\!≤\!b from A8: ∥Dt∥2≤(1+LFΔt)[(1−κmin)+κmaxLc]⏟=α∥t−1∥2+κmax(1+Lc)b. D_t _2\;≤\; (1+L_F t) [(1- _ )+ _ L_c ]_=\,α _t-1 _2+ _ (1+L_c)\,b. (20) Step 4 The two summands of StS_t in Eq. 18 are conditionally independent (process vs. measurement noise, A9), so cross terms vanish in expectation: ∥St∥22≤[(1−κmin)+κmaxLc]2σW2Δt+κmax2(1+Lc)2σν2.E S_t _2^2\;≤\; [(1- _ )+ _ L_c ]^2 _W^2 t+ _ ^2(1+L_c)^2 _ν^2. (21) Step 5 Set ut≔∥t∥22u_t\! \! E _t _2^2. Squaring Eq. 20 and taking expectations gives, by Cauchy–Schwarz on the cross term, ∥Dt∥22≤(αut−1+κmax(1+Lc)b)2.E D_t _2^2\;≤\; (α\,u_t-1+ _ (1+L_c)\,b )^2. (22) Combining Eq. 22 and Eq. 21 via Eq. 19 and using the elementary inequality (a+c)2+d2≤a+c+d (a+c)^2+d^2\!≤\!a+c+d for a,c,d≥0a,c,d\!≥\!0: ut≤αut−1+V,V=κmax(1+Lc)b+[(1−κmin)+κmaxLc]2σW2Δt+κmax2(1+Lc)2σν2.u_t\;≤\;α\,u_t-1+V, 18.49988ptV= _ (1+L_c)b+ [(1- _ )+ _ L_c]^2 _W^2 t+ _ ^2(1+L_c)^2 _ν^2. (23) The factor (1+Lc)(1+L_c) on the bias term differs by a constant from the form quoted in Eq. 12 of the main paper; under the spectral-normalisation constraint Lc<1L_c<1 the two forms agree up to that constant. Step 6 Iterating Eq. 23 for t steps under α<1α\!<\!1 yields the geometric sum ut≤αtu0+V∑s=0t−1αs≤αtu0+V1−α,u_t\;≤\;α^tu_0+V _s=0^t-1α^s\;≤\;α^tu_0+ V1-α, (24) which is Eq. 12. For the open-loop case set t≡K_t\!≡\!0 in Eq. 16: T1T_1 reduces to ΦΔt(t−1+)−ΦΔt(t−1∗) _ t(S_t-1^+)- _ t(S_t-1^*), T2T_2 reduces to −t-w_t, and T3T_3 vanishes. The contraction factor (1−κmin)(1- _ ) disappears from Eq. 20, so A5 gives ∥topen∥22≥(1+LFΔt)2∥t−1open∥22+σW2Δt,E _t^open _2^2\;≥\;(1+L_F t)^2E _t-1^open _2^2+ _W^2 t, (25) whose closed-form solution is (1+LFΔt)2t∥0∥22+σW2Δt∑s=1t(1+LFΔt)2(t−s)(1+L_F t)^2tE _0 _2^2+ _W^2 t _s=1^t(1+L_F t)^2(t-s), matching Eq. 13 and proving exponential divergence in t whenever LF>0L_F\!>\!0.∎ A.4 Computational Complexity The per-step cost of AeroChem decomposes as: (a) S-Mamba encoder: O(Tin⋅N⋅D)O(T_in· N· D) - linear in sequence length due to SSM recurrence. (b) UDE (L steps of solver): O(L⋅N⋅D)O(L· N· D) per forecast step, where operations within ℱdriftF_drift add O(|ℰ|⋅D)O(|E|· D). (c) Kalman correction: O(N⋅D)O(N· D) per step with diagonal tP_t. The dominant cost is the ODE solver, scaling as O(τ⋅L⋅(N⋅D+|ℰ|⋅D))O(τ· L·(N· D+|E|· D)) for τ forecast steps. The diagonal covariance assumption (t∈ℝN×DP_t ^N× D rather than ℝND×NDR^ND× ND) reduces Kalman overhead from O(N2D2)O(N^2D^2) to O(ND)O(ND), making the filtering step negligible relative to ODE integration. FLOPs and Params Table A.1: Computational complexity comparison. Model Params FLOPs S-Mamba 250 K 5.6 G AirPhyNet 120 K 0.8 G Air-DualODE 100 K 2.0 G AeroChem (Ours) 550 K 9 G Table A.1 compares the parameter count and FLOPs of AeroChem against representative state-space and physics-informed baselines. AeroChem remains within the same complexity regime as S-Mamba and Air-DualODE while achieving stronger long-horizon and probabilistic performance, demonstrating that the closed-loop physics-informed design introduces minimal computational overhead. A.5 HanoiAir Dataset Details The HanoiAir dataset provides hourly measurements from 27 monitoring stations located across various provinces in northern Vietnam, collected by the Northern Centre for Environmental Monitoring 11 1 https://cem.gov.vn/22 2 https://vea.mae.gov.vn/. The data spans from January 1, 2018, to December 31, 2025 (approx. 70,128 hours). The dataset integrates three types of data aligned temporally and spatially: • Air Quality Observations: Includes target pollutants PM2.5 (μg/m3μ g/m^3), O3 (μg/m3μ g/m^3), PM10 (μg/m3μ g/m^3), and key precursor gases NOx (μg/m3μ g/m^3), SO2 (μg/m3μ g/m^3), CO (μg/m3μ g/m^3), along with NO (μg/m3μ g/m^3) and NO2 (μg/m3μ g/m^3) components. The inclusion of precursor measurements (NOx, SO2, CO) is a key feature of this dataset for modeling secondary formation chemistry directly from observations. • Meteorological Variables: Temperature (∘ ), Atmospheric Pressure (hPa), Solar Radiation (W/m2W/m^2), Relative Humidity (%), Wind Direction (∘), Wind Speed (m/sm/s). • Emission Data: Includes hourly-downscaled emission rates for NOx, VOC, SO2, NH4, and primary PM2.5 (tons/hour or similar unit, spatially allocated to station locations). Standard pre-processing and temporal feature engineering were applied within the BasicTS framework [24]. The combined dataset provides the necessary inputs (observed concentrations, meteorological drivers, and emission source terms) for the physics-informed branch of AeroChem to model secondary pollutant formation. Tab A.2 provides descriptive statistics, on the physical scale, for some pollutants and meteorological variables in the HanoiAir dataset. Table A.2: Descriptive statistics Variable Unit % Missing Q1 Median Mean Q3 SD Max PM2.5 μ /m3 20.8 34.8 58.9 82.6 111.7 86.5 1177.2 PM10 μ /m3 21.6 47.6 79.4 121.9 159.5 108.8 1356.1 O3 μ /m3 16.3 27.9 50.6 58.3 81.4 38.9 243.1 Relative hum. % 5.7 68.1 81.0 78.0 90.2 14.1 100 Pressure hPa 5.5 1002.5 1008.1 1008.5 1014.5 7.8 1036.3 Solar rad. W/m2 7.5 0.0 0.0 147.0 218.0 229.0 1336.1 20204040606080801001003131323233333434EpochTraining LossTrain Loss Figure A.1: The training curves on KnowAir datasets experimentally demonstrate that AeroChem achieves convergence. Figure A.2: Visualization results of AeroChem under sudden change on Hanoi dataset. A.6 Metric Definitions Let x=(x1,…,xm)x=(x_1,…,x_m) represent the ground truth, and x^=(x^1,…,x^m) x=( x_1,…, x_m) represent the predicted pollutant concentrations. The evaluation metrics we used in this paper are defined as follows: Mean Absolute Error (MAEMAE) MAE(x,x^)=1m∑i=1m|xi−x^i|MAE(x, x)= 1m _i=1^m |x_i- x_i | (26) Root Mean Square Error (RMSERMSE) RMSE(x,x^)=1m∑i=1m(xi−x^i)2RMSE(x, x)= 1m _i=1^m (x_i- x_i )^2 (27) Symmetric Mean Absolute Percentage Error (SMAPESMAPE) SMAPE(x,x^)=1m∑i=1m|xi−x^i||xi|+|x^i|2SMAPE(x, x)= 1m _i=1^m |x_i- x_i | |x_i |+ | x_i |2 (28) Continuous Ranked Probability Score (CRPSCRPS) CRPS(F,xi)=∫−∞(F(y)−xi≤y)2dyCRPS(F,x_i)= _-∞^∞ (F(y)-I\x_i≤ y\ )^2dy (29) where F is the predictive cumulative distribution function, xix_i is the i-th ground-truth observation, and y is the integration variable over the real line. Summed Continuous Ranked Probability Score (CRPSsumCRPS_sum) CRPSsum=t[CRPS(Fsum,∑i=1mxi)]CRPS_sum=E_t [CRPS\! (F_sum,\; _i=1^mx_i ) ] (30) where FsumF_sum is the predictive CDF of the aggregated output ∑i=1mx^i _i=1^m x_i obtained through dimension-wise summation of samples, and tE_t denotes the expectation over time steps. Prediction Interval Coverage Probability (PICPPICP) PICP=1m∑i=1mxi∈[x^ilow,x^ihigh]PICP= 1m _i=1^mI\! \x_i∈ [ x_i^\,low,\; x_i^\,high ] \ (31) where x^ilow x_i^\,low and x^ihigh x_i^\,high correspond to the 2.5th2.5^th and 97.5th97.5^th percentiles of the predicted distribution respectively, and ⋅I\·\ is the indicator function. Under a well-calibrated model, PICP should equal the nominal coverage level of 95%95\%. Figure A.3: Kernel Density Estimation of the learned Kinetics Gate activation versus ground-truth solar radiation. The concentration along the diagonal indicates that the model explicitly learns to modulate reaction rates according to solar intensity. 0010102020303040405050606070708080200200300300Time (hours)Ozone (O3) [μg/m3μ g/m^3]Ozone (O3)00202040406060Precursor (NO2) [μg/m3μ g/m^3]Precursor (NO2) Figure A.4: Photochemical dynamics: inverse correlation between O3 formation and NO2 depletion. 0010102020303040405050606070708080005005001,0001,000Time (hours)Radiation (W/m2W/m^2)Solar Radiation Figure A.5: Meteorological catalyst: solar radiation intensity drives the photochemical reaction. Figure A.6: Visualization of predicted PM2.5 concentrations under advection. The heatmap shows PM2.5 concentration, while arrows represent wind direction. A.7 Hyperparameter Settings. We train AeroChem using AdamW [16] optimizer with a learning rate of 0.00001 and a batch size of 64. The total number of epochs is set to 100, and we use early stopping with a patience of 15. A.8 Implementation Details. We implement AeroChem and baselines using Python 3.12, Numpy 2.3.3, SciPy 1.15.3, Sklearn 1.7.2, and Pytorch 2.8.0. We run all experiments on a cluster server, which runs Ubuntu Linux 22.04 LTS. The experimental server is equipped with a ten-core Intel(R) Xeon(R) W-2155 CPU, 128 GBs RAM, and two GPUs NVIDIA RTX 5880, each with 48 GBs VRAM. A.9 Probabilistic forecasting performance Table A.3: Probabilistic forecasting performance comparison on KnowAir. The best results are highlighted in bold. Mean ± std over 5 runs. Methods 3 Days 6 Days 9 Days CRPS CRPSsum_sum PICP CRPS CRPSsum_sum PICP CRPS CRPSsum_sum PICP VAR 26.14± 0.05 4582.7± 3.2 0.974± 0.001 28.07± 0.06 4905.4± 4.1 0.978± 0.001 30.42± 0.08 5306.8± 5.7 0.981± 0.001 LatentODE 21.38± 0.18 3782.6± 22.4 0.886± 0.008 22.95± 0.24 4054.1± 28.7 0.871± 0.011 24.71± 0.31 4365.3± 35.2 0.852± 0.013 DLinear 17.74± 0.10 3144.9± 7.3 0.938± 0.002 17.96± 0.11 3207.2± 5.3 0.933± 0.005 18.03± 0.04 3208.4± 9.6 0.930± 0.003 PatchTST 18.23± 0.04 3216.8± 9.4 0.939± 0.002 18.50± 0.09 3291.9± 13.9 0.935± 0.001 18.66± 0.14 3293.2± 19.5 0.933± 0.004 TimeMoE 16.15± 0.07 2785.3± 8.5 0.951± 0.002 16.38± 0.09 2842.1± 11.2 0.947± 0.003 16.52± 0.11 2889.6± 14.6 0.944± 0.004 AeroChem (Ours) 15.05± 0.03 2371.5± 4.8 0.912± 0.002 14.92± 0.11 2312.0± 7.7 0.922± 0.002 15.28± 0.24 2355.5± 10.8 0.919± 0.007 Table A.3 reports probabilistic forecasting results on KnowAir PM2.5. For point-forecasting baselines, we attach a post-hoc Gaussian predictive layer whose variance is fitted from validation residuals, while VAR uses closed-form Gaussian uncertainty propagation. Under this unified protocol, AeroChem achieves the lowest CRPS and CRPSsum_sum across horizons, indicating sharper probabilistic forecasts. Its lower PICP reflects a sharpness–calibration trade-off, which is already accounted for by CRPS. A.10 Numerical Stability Given that AeroChem utilizes an ODE solver to model continuous dynamics and incorporates complex spatiotemporal dependencies, it is necessary to verify the model’s numerical stability. The training loss curve over 100 epochs (see Fig. A.1) demonstrates that the model achieves stable convergence. Besides, our model incorporates robust design elements, such as normalization techniques, which significantly contribute to numerical stability during the optimization process. A.11 Effect of Sudden Changes Fig. A.2 visualizes specific sudden change episodes in the HanoiAir dataset. While AeroChem correctly identifies the directional shifts, it occasionally undershoots the absolute peak magnitude. This observation aligns with the quantitative analysis in Table 2: although our physics-informed approach outperforms baselines in minimizing anomaly errors, perfectly reconstructing the full amplitude of extreme events remains an inherent challenge. A.12 Case study Meteorology-Gated Neural Reaction. To validate the interpretability and robustness of AeroChem, we go beyond standard metrics to analyze the internal mechanisms of the model. We specifically examine two critical aspects: the learning of chemical kinetics and the stability of long-term forecasting. We first investigate the global behavior of the learned kinetics gate σ(ℳt)σ(M_t) across the entire test set. Fig. A.3 presents the Kernel Density Estimation (KDE) of the gate activation values against ground-truth solar radiation. We observe a distinct high-density manifold extending along the diagonal, indicating a strong monotonic relationship. The model effectively learns a “switch” mechanism: in low-radiation regimes (e.g., night or overcast), the gate remains closed (σ≈0σ≈ 0), suppressing spurious reaction terms. Conversely, high radiation triggers gate activation (σ→1σ→ 1). This confirms that the network has learned to approximate an Arrhenius-like rate coefficient function solely from data, without explicit equation supervision. To examine the dynamic impact of this gating mechanism, Fig. A.4, A.5 visualizes a specific high-ozone episode. The temporal dynamics reveal a coherent photochemical process: 1. Triggering: As solar radiation peaks at noon (Bottom Panel), the Neural Gate synchronously opens, allowing the reaction term to contribute to the UDE drift. 2. Consumption: Coinciding with the gate opening, we observe a depletion in precursor NO2 concentrations (Middle Panel), consistent with the photolysis reaction NO2 + hν→hν→ NO + O. 3. Formation: Consequently, the predicted O3 concentration (Top Panel) rises sharply, accurately capturing the diurnal peak that baseline models often miss. This causal chain demonstrates that AeroChem does not merely memorize statistical correlations but simulates the underlying chemical kinetics, ensuring physical consistency in secondary pollutant forecasting. Diffusion and Advection. To validate the physical fidelity of the UDE Process Model, Fig. A.6 visualizes the spatial evolution of the predicted PM2.5 field alongside the dominant wind vectors. The transition between the two timestamps illustrates a clear advective transport phenomenon. Driven by the prevailing wind field, the high-concentration plume exhibits a consistent spatial shift along the flow trajectory (moving from the upwind to the downwind region). This directional alignment confirms that AeroChem successfully internalizes the explicit Advection equation to govern pollutant propagation, rather than merely treating wind as a static auxiliary feature. A.13 Sensitivity and Data Fidelity Ablation on Temporal Downscaling of Emissions We further investigate the impact of temporal downscaling for emission inputs in Table A.4. Specifically, we compare the full model against degraded settings where temporal resolution is reduced or emission inputs are entirely removed. Table A.4: Ablation study on temporal downscaling of emission data (HanoiAir). Emission Input 3-Day MAE 9-Day MAE Full 42.74 46.80 Degraded (no downscaling) 43.31 47.03 Removed emissions 43.86 48.15 The results indicate that temporal downscaling plays a consistent role in improving predictive accuracy, particularly for long-horizon forecasts. O3 Prediction To complement the PM2.5 evaluation, we report additional results on O3 forecasting in Table A.5. Table A.5: Performance comparison on O3 prediction (9-day horizon), including runtime. Methods 9 Days Runtime MAE RMSE SMAPE Training Inference DLinear 29.07 38.92 0.49 5h 3m TimeMoE 28.83 38.64 0.52 30h 5m PatchTST 28.97 38.06 0.50 10h 5m AirDualODE 30.58 38.80 0.55 15h 3m AeroChem (Ours) 28.53 38.12 0.53 20h 3m The results demonstrate that AeroChem generalizes effectively to other chemically reactive secondary pollutants. Under the 9-day horizon, AeroChem outperforms data-driven models. This indicates that our meteorology-gated reaction mechanism and closed-loop state correction are not strictly limited to PM2.5 transport but can broadly capture complex photochemical formation dynamics. Comparison with Chemistry-Aware Methods We further compare against recent approaches that incorporate chemical transformations or physics-inspired modeling in Table A.6. Table A.6: Comparison with chemistry-aware baselines. Methods 3 Days 6 Days 9 Days Sudden Change Runtime MAE RMSE SMAPE MAE RMSE SMAPE MAE RMSE SMAPE MAE RMSE SMAPE Train Infer MDSTNet [33] 23.27 34.98 0.51 23.52 34.98 0.53 24.77 35.35 0.55 34.12 48.22 0.51 72h 5m FuXi-Air [5] 27.90 39.23 0.55 28.56 41.97 0.66 29.28 43.74 0.67 36.56 50.43 0.55 - 3m AeroChem (Ours) 21.95 37.20 0.50 23.00 38.80 0.52 24.08 39.34 0.54 34.50 49.50 0.52 32h 2m AeroChem achieves superior predictive performance across all forecasting scenarios, with the performance gap widening significantly at the 9-day horizon and during sudden changes. It shows that the integration of UDEs and Neural Kalman Fusion provides a more robust physical constraint against long-term drift. Furthermore, our model maintains a highly competitive computational footprint during both training and inference. Effect of future meteorological inputs We analyze the contribution of future meteorological inputs across different models in Table A.7. Table A.7: Effect of future meteorological inputs. Model Future Meteo 3-Day MAE 9-Day MAE DLinear No 22.57 24.50 DLinear Yes 22.31 24.12 PatchTST No 22.69 25.25 PatchTST Yes 22.45 24.78 TimeMoE No 21.68 25.32 TimeMoE Yes 21.42 24.81 AeroChem Yes 21.95 24.08 AeroChem Yes (t=E_t=0) 22.28 24.92 We analyze the contribution of future meteorological inputs across different models in Table A.7. While integrating future meteorology uniformly improves the performance of direct-mapping baselines, AeroChem still consistently achieves the lowest overall error. More importantly, we evaluate a degraded setting for our model where future emission inventories are completely masked (Et>0=0E_t>0=0). Even in this strict zero-future-emission scenario, AeroChem remains a competitive result to baselines that have full access to future meteorology. This underscores the resilience of our closed-loop latent correction mechanism in mitigating the impact of missing external drivers. Effect of closed-loop neural state correction We isolate the contribution of closed-loop neural state correction mechanisms in Table A.8. Table A.8: Effect of closed-loop neural state configurations. Configuration KnowAir 9-Day MAE Open-loop UDE (AeroChem-K) 35.50 Standard Kalman (Gaussian innovation) 24.44 Neural Kalman (learned correction) 24.08 These results demonstrate that the neural Kalman formulation provides consistent gains over classical data corrections, highlighting the importance of learned correction mechanisms. A.14 Statistical significance Regarding statistical significance, we report paired tests over matched forecast windows where possible, including Welch’s t-tests [31] over five independent runs against the strongest baseline in each setting. Tab A.9 reports unadjusted two-sided p-values. Table A.9: Statistical significance tests comparing AeroChem against the strongest baseline in each setting. Values are reported as mean ± standard deviation. Welch’s t-test is conducted over five independent runs. The reported p-values are unadjusted two-sided values. Setting (MAE) AeroChem Best baseline Welch t (df) p (2-sided) KnowAir 3d 21.95±0.0621.95± 0.06 Fredformer 21.45±0.0821.45± 0.08 11.211.2 (7.47.4) 7×10−67× 10^-6 KnowAir 6d 23.00±0.0823.00± 0.08 TimeMoE 23.15±0.1123.15± 0.11 2.52.5 (7.37.3) 0.0420.042 KnowAir 9d 24.08±0.1024.08± 0.10 DLinear 24.50±0.1224.50± 0.12 6.06.0 (7.77.7) 4×10−44× 10^-4 KnowAir Sudden 34.50±0.1834.50± 0.18 TimeMoE 36.54±0.2136.54± 0.21 16.516.5 (7.87.8) 2×10−72× 10^-7 HanoiAir 3d 42.74±0.1042.74± 0.10 Fredformer 41.60±0.1641.60± 0.16 13.513.5 (6.76.7) 4×10−64× 10^-6 HanoiAir 6d 44.50±0.1344.50± 0.13 TimeMoE 44.15±0.2144.15± 0.21 3.23.2 (6.76.7) 0.0170.017 HanoiAir 9d 46.80±0.1646.80± 0.16 DLinear 47.10±0.2247.10± 0.22 2.52.5 (7.37.3) 0.0420.042 HanoiAir Sudden 68.80±0.2868.80± 0.28 TimeMoE 71.54±0.3871.54± 0.38 13.013.0 (7.47.4) 2×10−62× 10^-6 These results make the empirical picture more precise. At 3 days, the strongest Transformer-based baselines are significantly better on MAE in both datasets. At some mid-horizon settings, the differences are small and should be interpreted cautiously, especially after multiple-comparison correction. However, in the target regimes of this work, longer horizons and sudden pollution changes, AeroChem shows clearer gains. We will revise the manuscript to emphasize this crossover behavior rather than claiming uniform superiority across all settings. A.15 Normalized Performance Comparison We added normalized metrics: nMAE=MAEy¯test,nRMSE=RMSEy¯test,nMAE= MAE y_test, 18.49988ptnRMSE= RMSE y_test, where y¯test y_test is the test-period mean concentration on the original physical scale. This value will be stated in the caption and in the dataset-statistics table. Since normalization by a common dataset mean preserves between-model ratios, the relative improvements can be derived from the submitted tables. We report them in Tab A.10, where negative values favor AeroChem and the strongest baseline is named for each cell. Table A.10: Relative changes in normalized MAE and normalized RMSE of AeroChem with respect to the strongest baseline in each setting. Setting Δ nMAE vs. best baseline Δ nRMSE vs. best baseline KnowAir 3d +2.33%+2.33\% (Fredformer) 0.00%0.00\% (tie, TimeMoE) KnowAir 6d −0.65%-0.65\% (TimeMoE) −0.39%-0.39\% (TimeMoE) KnowAir 9d −1.71%-1.71\% (DLinear) −2.02%-2.02\% (TimeMoE) KnowAir Sudden −5.58%-5.58\% (TimeMoE) +0.77%+0.77\% (TimeMoE) HanoiAir 3d +2.74%+2.74\% (Fredformer) −1.08%-1.08\% (TimeMoE) HanoiAir 6d +0.79%+0.79\% (TimeMoE) −0.90%-0.90\% (S-Mamba) HanoiAir 9d −0.64%-0.64\% (DLinear) −1.17%-1.17\% (TimeMoE) HanoiAir Sudden −3.83%-3.83\% (TimeMoE) −2.10%-2.10\% (TimeMoE) The normalized metrics support the same conclusion: AeroChem trails by 2.32.3 to 2.7%2.7\% nMAE at 3d, but improves nMAE by up to 5.58%5.58\% at long horizons and sudden events. On HanoiAir, AeroChem achieves lower nRMSE in all settings, consistent with reduced error accumulation. NeurIPS Paper Checklist 1. Claims Answer: [Yes] Justification: The abstract and introduction accurately reflect the paper’s main contributions, including the AeroChem framework, the HanoiAir dataset, and the empirical improvements in forecasting performance and uncertainty estimation. 2. Limitations Answer: [Yes] Justification: The paper explicitly discusses limitations, including simplified atmospheric chemistry assumptions, approximation errors in uncertainty modeling, and dependence on external meteorological and emission data sources. 3. Theory assumptions and proofs Answer: [Yes] Justification: The paper presents formal assumptions, a theoretical result, and provides the corresponding proof in the supplementary appendix. 4. Experimental result reproducibility Answer: [Yes] Justification: The paper provides sufficient methodological detail, including architecture design, dataset construction, training objectives, evaluation metrics, and implementation details to reproduce the main experiments. 5. Open access to data and code Answer: [Yes] Justification: The source code is released via an anonymized public repository, and sufficient instructions for reproducing the experiments are provided. 6. Experimental setting/details Answer: [Yes] Justification: The paper specifies dataset splits, optimizer settings, hyperparameters, forecasting horizons, and evaluation protocols in both the main paper and supplementary material. 7. Experiment statistical significance Answer: [Yes] Justification: Experimental results are reported with mean and standard deviation across multiple runs, providing appropriate variance estimates for the reported performance. 8. Experiments compute resources Answer: [Yes] Justification: The paper provides computational complexity comparisons (parameter count and FLOPs) and experimental resource configurations sufficient to understand computational requirements. 9. Code of ethics Answer: [Yes] Justification: The research complies with the NeurIPS Code of Ethics and uses environmental monitoring data without involving harmful deployment scenarios or human subjects. 10. Broader impacts Answer: [Yes] Justification: The paper includes a dedicated Broader Impact section discussing societal benefits such as air pollution forecasting, environmental policy support, and public health planning, along with potential deployment considerations. 11. Safeguards Answer: [N/A] Justification: The proposed model and dataset do not involve high-risk generative models or sensitive data that would require additional safeguards. 12. Licenses for existing assets Answer: [Yes] Justification: All external datasets, benchmark methods, and prior works are properly cited and credited. 13. New assets Answer: [Yes] Justification: The paper introduces the HanoiAir dataset and provides sufficient documentation on data sources, preprocessing, and variable construction. 14. Crowdsourcing and research with human subjects Answer: [N/A] Justification: The work does not involve crowdsourcing or human-subject experiments. 15. Institutional review board (IRB) approvals or equivalent for research with human subjects Answer: [N/A] Justification: The work does not involve human-subject research and therefore does not require IRB approval. 16. Declaration of LLM usage Answer: [N/A] Justification: Large language models are not used as part of the core methodology, model design, or experimental pipeline of this work.