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

cs.LGOct 12, 2026

AeroChem: Closed-loop Physics-Informed State Space Modeling for Long-term Chemically-Reactive Air Quality Forecasting--Extended Version

Authors: Kien Le Trung, Tung Kieu, Dinh Duc Nha Nguyen, Cuong Do, Bin Yang, Phi Long Nguyen

Organizations: VinUniversity Vietnam · Aalborg University Denmark · CECS, VinUniversity Vietnam · East China Normal University China · SEEE, Hanoi University of Industry Vietnam

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.

Figures & tables

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.

Methods	3 Days			6 Days			9 Days			Sudden Change
Methods	MAE	RMSE	SMAPE	MAE	RMSE	SMAPE	MAE	RMSE	SMAPE	MAE	RMSE	SMAPE
HA	35.20 $\pm$ 0.05	51.50 $\pm$ 0.07	0.65 $\pm$ 0.001	37.80 $\pm$ 0.06	54.20 $\pm$ 0.08	0.68 $\pm$ 0.002	40.50 $\pm$ 0.07	58.10 $\pm$ 0.10	0.71 $\pm$ 0.002	55.40 $\pm$ 0.10	75.20 $\pm$ 0.14	0.75 $\pm$ 0.003
VAR	33.80 $\pm$ 0.08	49.20 $\pm$ 0.10	0.62 $\pm$ 0.002	36.50 $\pm$ 0.09	52.40 $\pm$ 0.13	0.66 $\pm$ 0.003	39.20 $\pm$ 0.11	56.50 $\pm$ 0.16	0.69 $\pm$ 0.003	53.10 $\pm$ 0.16	72.40 $\pm$ 0.22	0.72 $\pm$ 0.004
LatentODE	26.50 $\pm$ 0.15	42.10 $\pm$ 0.21	0.55 $\pm$ 0.004	28.20 $\pm$ 0.18	45.40 $\pm$ 0.25	0.57 $\pm$ 0.005	31.60 $\pm$ 0.22	49.20 $\pm$ 0.30	0.61 $\pm$ 0.006	48.40 $\pm$ 0.36	65.10 $\pm$ 0.49	0.65 $\pm$ 0.009
ODE-LSTM	27.10 $\pm$ 0.16	43.50 $\pm$ 0.22	0.56 $\pm$ 0.004	29.45 $\pm$ 0.19	46.80 $\pm$ 0.26	0.59 $\pm$ 0.005	32.80 $\pm$ 0.23	50.15 $\pm$ 0.31	0.62 $\pm$ 0.006	49.80 $\pm$ 0.38	67.30 $\pm$ 0.51	0.67 $\pm$ 0.010
Air-DualODE	23.04 $\pm$ 0.14	38.44 $\pm$ 0.19	0.52 $\pm$ 0.003	24.50 $\pm$ 0.17	41.60 $\pm$ 0.23	0.54 $\pm$ 0.004	25.20 $\pm$ 0.20	42.80 $\pm$ 0.27	0.59 $\pm$ 0.005	39.24 $\pm$ 0.32	53.30 $\pm$ 0.43	0.55 $\pm$ 0.008

Table 1: Performance comparison on the KnowAir dataset. The best results are highlighted in bold , and the second-best results are underlined . Mean

\pm

std over 5 runs.

Methods	3 Days			6 Days			9 Days			Sudden Change
Methods	MAE	RMSE	SMAPE	MAE	RMSE	SMAPE	MAE	RMSE	SMAPE	MAE	RMSE	SMAPE
HA	55.20 $\pm$ 0.10	78.50 $\pm$ 0.14	0.85 $\pm$ 0.002	58.40 $\pm$ 0.12	82.10 $\pm$ 0.16	0.88 $\pm$ 0.003	62.10 $\pm$ 0.14	86.50 $\pm$ 0.20	0.92 $\pm$ 0.003	75.40 $\pm$ 0.22	105.20 $\pm$ 0.30	0.89 $\pm$ 0.005
VAR	52.80 $\pm$ 0.13	75.20 $\pm$ 0.18	0.82 $\pm$ 0.003	56.50 $\pm$ 0.15	79.40 $\pm$ 0.21	0.85 $\pm$ 0.003	60.20 $\pm$ 0.18	84.80 $\pm$ 0.25	0.88 $\pm$ 0.004	72.10 $\pm$ 0.28	100.40 $\pm$ 0.38	0.86 $\pm$ 0.006
LatentODE	46.50 $\pm$ 0.30	68.10 $\pm$ 0.42	0.80 $\pm$ 0.006	49.20 $\pm$ 0.36	71.40 $\pm$ 0.50	0.83 $\pm$ 0.007	52.60 $\pm$ 0.42	75.20 $\pm$ 0.58	0.86 $\pm$ 0.008	74.40 $\pm$ 0.65	98.10 $\pm$ 0.88	0.85 $\pm$ 0.012
ODE-LSTM	47.10 $\pm$ 0.32	69.50 $\pm$ 0.45	0.81 $\pm$ 0.006	50.45 $\pm$ 0.38	72.80 $\pm$ 0.52	0.84 $\pm$ 0.007	54.80 $\pm$ 0.45	78.15 $\pm$ 0.62	0.87 $\pm$ 0.008	75.80 $\pm$ 0.70	101.30 $\pm$ 0.94	0.86 $\pm$ 0.012
Air-DualODE	42.04 $\pm$ 0.28	64.41 $\pm$ 0.40	0.78 $\pm$ 0.005	45.50 $\pm$ 0.34	67.60 $\pm$ 0.46	0.82 $\pm$ 0.006	48.20 $\pm$ 0.40	71.80 $\pm$ 0.54	0.86 $\pm$ 0.007	72.24 $\pm$ 0.58	93.30 $\pm$ 0.78	0.83 $\pm$ 0.010

Table 2: Performance comparison on HanoiAir dataset. The best results are highlighted in bold , and the second-best results are underlined . Mean

\pm

std over 5 runs.

Physical Mechanisms
Method	HanoiAir						KnowAir
Method	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
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

Table 3: Ablation Study: Effect of Physical Mechanisms and Models Design.

Panel of Figure 3 — Figure 3: Performance across forecast horizons.

Panel of Figure 4 — Figure 4: Sensitivity analysis on KnowAir dataset. The dashed vertical lines indicate the default hyperparameters used in the full model ( $\lambda=0.1,d=96,dt=1.0$ ), which consistently correspond to the optimal performance regions (lowest error).

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

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

Appendix

Table A.1: Computational complexity comparison.

Variable	Unit	% Missing	Q1	Median	Mean	Q3	SD	Max
PM2.5	$\mu$ g/m 3	20.8	34.8	58.9	82.6	111.7	86.5	1177.2
PM10	$\mu$ g/m 3	21.6	47.6	79.4	121.9	159.5	108.8	1356.1
O3	$\mu$ g/m 3	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/m 2	7.5	0.0	0.0	147.0	218.0	229.0	1336.1

Appendix

Table A.2: Descriptive statistics

Panel of Figure 10 — Figure A.1: The training curves on KnowAir datasets experimentally demonstrate that AeroChem achieves convergence.

Panel of Figure 11 — Figure A.2: Visualization results of AeroChem under sudden change on Hanoi dataset.

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.

Panel of Figure 13 — Figure A.4 : Photochemical dynamics: inverse correlation between O 3 formation and NO 2 depletion.

Panel of Figure 14 — Figure A.5 : Meteorological catalyst: solar radiation intensity drives the photochemical reaction.

Panel of Figure 15 — Figure A.6: Visualization of predicted PM 2.5 concentrations under advection. The heatmap shows PM 2.5 concentration, while arrows represent wind direction.

Methods	3 Days			6 Days			9 Days
Methods	CRPS	CRPS ${}_{\text{sum}}$	PICP	CRPS	CRPS ${}_{\text{sum}}$	PICP	CRPS	CRPS ${}_{\text{sum}}$	PICP
VAR	26.14 $\pm$ 0.05	4582.7 $\pm$ 3.2	0.974 $\pm$ 0.001	28.07 $\pm$ 0.06	4905.4 $\pm$ 4.1	0.978 $\pm$ 0.001	30.42 $\pm$ 0.08	5306.8 $\pm$ 5.7	0.981 $\pm$ 0.001
LatentODE	21.38 $\pm$ 0.18	3782.6 $\pm$ 22.4	0.886 $\pm$ 0.008	22.95 $\pm$ 0.24	4054.1 $\pm$ 28.7	0.871 $\pm$ 0.011	24.71 $\pm$ 0.31	4365.3 $\pm$ 35.2	0.852 $\pm$ 0.013
DLinear	17.74 $\pm$ 0.10	3144.9 $\pm$ 7.3	0.938 $\pm$ 0.002	17.96 $\pm$ 0.11	3207.2 $\pm$ 5.3	0.933 $\pm$ 0.005	18.03 $\pm$ 0.04	3208.4 $\pm$ 9.6	0.930 $\pm$ 0.003
PatchTST	18.23 $\pm$ 0.04	3216.8 $\pm$ 9.4	0.939 $\pm$ 0.002	18.50 $\pm$ 0.09	3291.9 $\pm$ 13.9	0.935 $\pm$ 0.001	18.66 $\pm$ 0.14	3293.2 $\pm$ 19.5	0.933 $\pm$ 0.004
TimeMoE	16.15 $\pm$ 0.07	2785.3 $\pm$ 8.5	0.951 $\pm$ 0.002	16.38 $\pm$ 0.09	2842.1 $\pm$ 11.2	0.947 $\pm$ 0.003	16.52 $\pm$ 0.11	2889.6 $\pm$ 14.6	0.944 $\pm$ 0.004

Appendix

Table A.3: Probabilistic forecasting performance comparison on KnowAir . The best results are highlighted in bold . Mean

\pm

std over 5 runs.

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

Appendix

Table A.4 : Ablation study on temporal downscaling of emission data (HanoiAir).

Methods	9 Days			Runtime
Methods	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

Appendix

Table A.5 : Performance comparison on O 3 prediction (9-day horizon), including runtime.

Methods	3 Days			6 Days			9 Days			Sudden Change			Runtime
Methods	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

Appendix

Table A.6 : Comparison with chemistry-aware baselines.

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

Appendix

Table A.7 : Effect of future meteorological inputs.

Configuration	KnowAir 9-Day MAE
Open-loop UDE ( AeroChem -K)	35.50
Standard Kalman (Gaussian innovation)	24.44
Neural Kalman (learned correction)	24.08

Appendix

Table A.8 : Effect of closed-loop neural state configurations.

Setting (MAE)	AeroChem	Best baseline	Welch $t$ (df)	$p$ (2-sided)
KnowAir 3d	$21.95\pm 0.06$	Fredformer $21.45\pm 0.08$	$11.2$ ( $7.4$ )	$7\times 10^{-6}$
KnowAir 6d	$23.00\pm 0.08$	TimeMoE $23.15\pm 0.11$	$2.5$ ( $7.3$ )	$0.042$
KnowAir 9d	$24.08\pm 0.10$	DLinear $24.50\pm 0.12$	$6.0$ ( $7.7$ )	$4\times 10^{-4}$
KnowAir Sudden	$34.50\pm 0.18$	TimeMoE $36.54\pm 0.21$	$16.5$ ( $7.8$ )	$2\times 10^{-7}$
HanoiAir 3d	$42.74\pm 0.10$	Fredformer $41.60\pm 0.16$	$13.5$ ( $6.7$ )	$4\times 10^{-6}$
HanoiAir 6d	$44.50\pm 0.13$	TimeMoE $44.15\pm 0.21$	$3.2$ ( $6.7$ )	$0.017$

Appendix

Table A.9: Statistical significance tests comparing AeroChem against the strongest baseline in each setting. Values are reported as mean

\pm

standard deviation. Welch’s

t

-test is conducted over five independent runs. The reported

p

-values are unadjusted two-sided values.

Setting	$\Delta$ nMAE vs. best baseline	$\Delta$ nRMSE vs. best baseline
KnowAir 3d	$+2.33\%$ (Fredformer)	$0.00\%$ (tie, TimeMoE)
KnowAir 6d	$-0.65\%$ (TimeMoE)	$-0.39\%$ (TimeMoE)
KnowAir 9d	$-1.71\%$ (DLinear)	$-2.02\%$ (TimeMoE)
KnowAir Sudden	$-5.58\%$ (TimeMoE)	$+0.77\%$ (TimeMoE)
HanoiAir 3d	$+2.74\%$ (Fredformer)	$-1.08\%$ (TimeMoE)
HanoiAir 6d	$+0.79\%$ (TimeMoE)	$-0.90\%$ (S-Mamba)

Appendix

Table A.10: Relative changes in normalized MAE and normalized RMSE of AeroChem with respect to the strongest baseline in each setting.

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NeuroDDAF: Neural Dynamic Diffusion-Advection Fields with Evidential Fusion for Air Quality Forecasting

Prasanjit Dey, Soumyabrata Dev, Angela Meyer, Bianca Schoen-Phelan

Air Quality Prediction Spatiotemporal Forecasting Physics-Informed ML Spatiotemporal GNNs

Open paper AskPDF arXiv

Accurate air quality forecasting is crucial for protecting public health and guiding environmental policy, yet it remains challenging due to nonlinear spatiotemporal dynamics, wind-driven transport, and distribution shifts across regions. Physics-based models are interpretable but computationally expensive and often rely on restrictive assumptions, whereas purely data-driven models can be accurate but may lack robustness and calibrated uncertainty. To address these limitations, we propose Neural Dynamic Diffusion-Advection Fields (NeuroDDAF), a physics-informed forecasting framework that unifies neural representation learning with open-system transport modeling. NeuroDDAF integrates (i) a GRU-Graph Attention encoder to capture temporal dynamics and wind-aware spatial interactions, (ii) a Fourier-domain diffusion-advection module with learnable residuals, (iii) a wind-modulated latent Neural ODE to model continuous-time evolution under time-varying connectivity, and (iv) an evidential fusion mechanism that adaptively combines physics-guided and neural forecasts while quantifying uncertainty. Experiments on four urban datasets (Beijing, Shenzhen, Tianjin, and Ancona) across 1-3 day horizons show that NeuroDDAF consistently outperforms strong baselines, including AirPhyNet, achieving up to 9.7% reduction in RMSE and 9.4% reduction in MAE on long-term forecasts. On the Beijing dataset, NeuroDDAF attains an RMSE of 41.63

μ

g/m

^3

for 1-day prediction and 48.88

μ

g/m

^3

for 3-day prediction, representing the best performance among all compared methods. In addition, NeuroDDAF improves cross-city generalization and yields well-calibrated uncertainty estimates, as confirmed by ensemble variance analysis and case studies under varying wind conditions.

Abstract

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