A PDE that describes the criteria of environmental pollution dispersal must be implemented in order to build a PINN-supported algorithm framework for air contamination visualization regeneration. Its diffusion formula, which characterizes the gigantic mechanics brought about by the Brownian motion of small fragments, is used in this investigation. Numerous fields, including structural study, computer science, and biology, depend significantly on the dispersion concept. It is represented in its generic state as (4)∂Φ(s→,t¯)∂t¯=∇⋅[Υ→(Φ,s→)∇Φ(s→,t¯)]+Λ(s→,t¯),where Υ→ provides the dissemination factor, which is usually specified pairing within several-species frameworks, and ∇ defines the vector divergent function. Considering a single material compared to an additional one, a higher Υ→ quantity denotes an increased rate of two-way dispersion. The term Λ(s→,t¯) represents auxiliary or implicit sources and sinks of pollutants.

Three fundamental presumptions are introduced in order to modify the generic diffusing formulation for the environmental contamination image restoration issue in metropolitan environments:

Horizontal dispersion assumption: To address the challenge of reconstructing high-precision ambient pollutant concentration fields, this study formulates the problem within a two-dimensional horizontal framework. The modeling approach emphasizes horizontal dispersion driven by wind-induced advection and turbulent diffusion within the atmospheric boundary layer, while vertical transport processes are not explicitly resolved.

For PM_2.5 and PM₁₀, this assumption is physically justified by the dominance of near-surface sources and the comparatively limited vertical displacement of particles due to gravitational settling and surface deposition. Under typical urban boundary-layer conditions, PM concentrations exhibit strong horizontal variability and relatively weak vertical gradients within the lowest tens of meters, particularly at spatial resolutions exceeding several hundred meters.

In contrast, O₃ is a secondary pollutant with stronger vertical mixing and photochemical production throughout the boundary layer. However, at the adopted horizontal mesh resolution of 500 m which substantially exceeds characteristic near-surface vertical fluctuation scales (≤10 m) turbulent mixing is assumed to locally homogenize vertical O₃ concentrations near the surface. Consequently, surface-level O₃ variability can be reasonably approximated as a horizontally dominant process at the urban scale considered in this work.

Accordingly, the spatial domain is defined as (5)s→=(u1,u2),where u1 and u2 represent longitudinal and latitudinal coordinates, respectively.

To preserve computational efficiency and model interpretability, detailed vertical processes such as boundary-layer height variability, chemical transformation rates, and surface–atmosphere exchange fluxes are not explicitly modeled, although their inclusion could further enhance reconstruction accuracy [20].

Transparent boundary and initial conditions: To make the PDE implementation fully auditable, the boundary and initial conditions are explicitly specified as follows:

(i) Type: Dirichlet boundary conditions are applied for pollutant concentrations at the urban–rural interface, while initial conditions are fixed using the first available hourly measurements at all reference sites within the domain.

(ii) Mathematical form: For each pollutant Φ, at a boundary location s→b and time t¯, (6)Φ(s→b,t¯)=Φbg(s→b,t¯)+ϵ(s→b,t¯),where Φbg is the background concentration interpolated from adjacent rural sites, and ϵ is a small perturbation term allowing horizontal advection and turbulent diffusion. Initial conditions are (7)Φ(s→,t¯0)=Φobs(s→,t¯0),where t¯0 is the first simulation timestep and Φobs are the observed concentrations.

(iii) Enforcement in the loss function: These conditions are incorporated via boundary collocation sampling and a soft penalty term in the loss:(8)ℒBC/IC=1Nb∑i=1Nb|Φpred(s→bi,t¯i)−ΦBC(s→bi,t¯i)|2+1N0∑j=1N0|Φpred(s→j,t¯0)−Φobs(s→j,t¯0)|2,where Nb and N0 are the number of collocation points at the boundary and initial timestep, Φpred is the PDE-predicted concentration, and ΦBC/Φobs are the boundary and initial values.

This approach is fully auditable and implementation-ready. The Dirichlet boundaries ensure mass conservation while permitting realistic advection and diffusion across the urban–rural interface. The perturbation term ϵ introduces flexibility, and collocation-based enforcement integrates the constraints directly into the training loss, maintaining both physical fidelity and data-driven learning capability.

Homogeneous diffusivity factor: Atmospheric pollutant diffusion is influenced by multiple external variables, including humidity, thermal stratification, wind shear, and surface roughness, all of which can modulate the effective eddy diffusivity Υ→(⋅). In the atmospheric boundary layer, however, these effects are often parameterized through a bulk turbulent diffusivity that represents the aggregate impact of unresolved turbulent motions on pollutant transport.

Explicitly modeling spatially and temporally varying diffusivity fields would significantly increase the complexity of the PINN optimization and may lead to identifiability and stability issues, particularly under sparse observational coverage. Therefore, for computational feasibility and model robustness, this study assumes a homogeneous effective diffusivity for ambient pollutant transport, expressed as (9)Υ→(Φ,s→)≡Υ,where Υ represents a constant, spatially averaged eddy diffusivity and is treated as an independent hyperparameter within the proposed PINN-inspired architecture.

From a physical perspective, this assumption reflects the scale separation between the resolved horizontal grid size (500 m) and smaller-scale turbulent fluctuations, under which subgrid-scale mixing processes are reasonably approximated by a uniform diffusion coefficient. For PM_2.5 and PM₁₀, this effective diffusivity encapsulates the combined influence of turbulent dispersion and near-surface mixing, while for O₃, it represents the net impact of boundary-layer turbulence on horizontally averaged concentrations near the surface.

Under these assumptions, the diffusion formulation in (4) reduces to (10)Φt¯′(u1,u2,t¯)=Υ(Φu1u1″(u1,u2,t¯)+Φu2u2″(u1,u2,t¯))+Λ(u1,u2,t¯).

Here, the superscripts ′ and ″ denote first- and second-order partial derivatives with respect to the corresponding variables.

A NN and a physics-informed network constitute the two primary components of the PINN architecture, as illustrated in Fig. 5. The physics-informed network explicitly enforces the governing PDEs described in Section 4.2.1, which model the physical processes of horizontal advection, turbulent diffusion, and source-sink dynamics of pollutants such as O₃ and PM_2.5/PM₁₀ within the urban atmospheric boundary layer (see [35]).

The NN module approximates the pollutant concentration field as a continuous function over space and time, while the physics-informed network constrains this approximation to satisfy the underlying physics, ensuring that the reconstructed fields adhere to mass conservation, diffusion limits, and boundary conditions consistent with urban–rural interfaces.

The details of the NN architecture, the physics-informed network formulation, and the training procedure including loss evaluation, automatic differentiation of the PDE residuals, and hyperparameter tuning are presented in the following sections of this paper.

The initial component of the architecture is a fully connected NN with three input neurons representing the spatiotemporal coordinates (u1,u2,t¯). The signaling layer consists of two neurons: one predicting the additional ecological component Λ, which accounts for unresolved or auxiliary sources and sinks, and the other predicting the primary pollutant concentration field Φ. Using δ to denote the NN parameters, the outputs are expressed as Φ^(⋅;δ) and Λ^(⋅;δ), respectively.

To capture the complex, non-linear relationships inherent in urban-scale pollutant transport including advection, horizontal diffusion, and source-sink interactions, we integrate an “attention-based module” that dynamically weights contributions from different spatial regions, allowing the network to prioritize areas of high variability or strong pollutant fluxes. Complementarily, an “interaction-based module” is employed to explicitly model cross-couplings between pollutant species and between urban and surrounding rural zones, reflecting the physical interplay of emissions, atmospheric mixing, and boundary-layer processes.

The sine (sin) activation function is utilized throughout the NN, as it has been demonstrated to improve the approximation of high-frequency spatial variations characteristic of atmospheric pollutant distributions. Notably, Λ^ functions as an auxiliary quantity during training to aid convergence and capture unresolved dynamics, whereas only Φ^ is used for inference to predict pollutant concentrations across the urban domain.

The physics-informed network (PINN) is a specialized neural network whose structure and evaluation are explicitly governed by the underlying PDEs representing pollutant transport. Unlike a fully connected NN module, the physics-informed network does not introduce additional trainable parameters. Its primary purpose is to compute the physical residual, which quantifies the discrepancy between the network predictions Φ^(⋅;δ), Λ^(⋅;δ) and the constraints imposed by the governing PDEs, including horizontal advection, diffusion, and source-sink dynamics.

To ensure that the auxiliary source term Λ(s→,t¯) is auditable and physically consistent, it is explicitly constrained via a regularization term:(11)ℒΛ=λΛ1NΛ∑i=1NΛ|Λ^(s→i,t¯i;δ)−Λtarget(s→i,t¯i)|2,where

Λ^(s→i,t¯i;δ) is the PINN-predicted auxiliary source at the i-th collocation point, dependent on the network parameters δ.

This regularization applies specifically to the auxiliary source term Λ, which indirectly constrains Φ through the PDE dynamics.

The PDE residual, boundary/initial condition penalties, and auxiliary source regularization are combined to form the total training loss for the PINN:(12)ℒtotal=ℒPDE+ℒBC/IC+ℒΛ.

During training, the physical residual acts as an additional constraint, effectively enforcing adherence of the network to the governing PDEs at each spatiotemporal location. In this way, the physics-informed network serves as a regularizing mechanism, ensuring that predicted pollutant fields remain physically consistent even in regions with sparse or missing observational data.

Specifically, the physics-informed residual is defined as:(13)Ψ^(u1,u2,t¯;δ):=Υ(Φ^u1u1″(u1,u2,t¯;δ)+Φ^u2u2″(u1,u2,t¯;δ))+Λ^(u1,u2,t¯;δ)−Φ^t¯′(u1,u2,t¯;δ).

Here, δ denotes the trainable parameters of the network. The residual Ψ^(u1,u2,t¯;δ) evaluated at each spatiotemporal location measures how faithfully the network predictions satisfy the PDE constraints. A residual approaching zero indicates near-perfect compliance with the physical laws, while larger deviations highlight regions requiring adjustment.

Physically, this formulation incorporates multiple constraints simultaneously: horizontal pollutant diffusion is captured via the second-order spatial derivatives, local emissions or auxiliary ecological effects are represented by Λ^, and temporal evolution is enforced through the first-order time derivative. Collectively, these constraints ensure that the learned pollutant fields are not only consistent with observed data but also physically plausible, improving generalization and interpretability across both urban and peri-urban regions.

Optimization Process

In Phy-APMR, training is guided by two complementary types of loss: a statistically determined loss and a physics-driven loss. The statistically determined loss ensures that the network’s predictions closely match observed sensor data, maintaining fidelity to real-world measurements. The physics-driven loss, on the other hand, enforces the fundamental physical laws governing pollutant behavior, including advection by wind, diffusion and turbulent mixing, chemical transformations, emission sources and sinks, boundary and initial conditions, and meteorological influences. By balancing these two components, the network is directed to produce predictions that are not only accurate but also physically consistent and environmentally realistic, enabling reliable mapping even in areas with sparse observational data.

• Supervised loss: The initial training dataset is defined as t¯Φ={(u1ȷΦ,u2ȷΦ,t¯ȷΦ,ΦȷΦ)|ȷ=1,2,…,NΦ}, which contains the measured pollutant concentrations. This dataset is used to construct the primary loss term. The data loss ℒΦ(δ) quantifies the discrepancy between the observed values ΦȷΦ and the predicted concentrations Φ^(u1ȷΦ,u2ȷΦ,t¯ȷΦ;δ). It is quantified via the mean squared error (MSE) as: (14)ℒΦ(δ)=1NΦ∑ȷ=1NΦ|Φ^(u1ȷΦ,u2ȷΦ,t¯ȷΦ;δ)−ΦȷΦ|2.

• Physics-informed loss: The subsequent loss component, referred to as the physics-based loss, guarantees that the NN predictions remain consistent by means of the governing mechanical principles. This is realized by selecting NΨ points at random within the domain of the PDE to construct a collocation set t¯Ψ={(u1ȷΨ,u2ȷΨ,t¯ȷΨ)|ȷ=1,2,…,NΨ}.

Descriptions of pollution concentrations were not associated with these collocation locations. Instead, the physical loss is formulated through the physical residuals, Ψ^, which are computed by the physics-informed network at the following points:(15)ℒΨ(δ)=1NΨ∑ȷ=1NΨ|Ψ^(u1ȷΨ,u2ȷΨ,t¯ȷΨ;δ)|2.

• Integrated loss function: By integrating the statistical loss with the physics-driven loss, the combined loss function ensures that the model’s predictions remain accurate while consistently adhering to the governing physical laws of pollutant transport in the atmosphere. This formulation achieves a balance between empirical precision and physical coherence, accounting for key factors such as advection by wind, turbulent diffusion, chemical transformations, emissions, deposition, and meteorological influences. The combined loss is expressed as:(16)ℒ(δ)=ϑΦ⋅ℒΦ(δ)+ϑΨ⋅ℒΨ(δ),where ℒΦ(δ) represents the data-driven loss based on observed pollutant concentrations, and ℒΨ(δ) represents the physics-informed loss enforcing atmospheric pollutant dynamics, such as:(17)ℒΨ(δ)=1NΦ∑ȷ=1NΦ[|∂Φ^ȷ∂t+uȷ⋅∇Φ^ȷ|2+|∇⋅(Dȷ∇Φ^ȷ)|2+|R(Φ^ȷ)|2],where uȷ denotes the wind velocity field (advection), Dȷ is the diffusion/turbulence coefficient, and R(Φ^ȷ) accounts for chemical reactions, emission sources, deposition, and other atmospheric sinks. The hyperparameters ϑΦ and ϑΨ regulate the relative contributions of the data-driven and physics-informed components. Ensuring reliable applicability requires optimizing the overall performance of the framework through appropriate tuning of these parameters.

The main goal of the training process is to minimize the total loss:(18)δ∗=arg⁡minδℒ(δ).

Throughout this research, feature scaling is applied to adjust the NN inputs to the interval [0,1]. The PINN is subsequently trained using the Adam optimizer [36] to obtain the optimal parameter set δ∗. After training, the NN module of the PINN is employed to estimate atmospheric pollutant concentrations, which are then used to reconstruct high-resolution atmospheric pollution maps.

The rapid advancement of database management technologies and urban monitoring infrastructure has enabled the collection of high-resolution ecological data, including wind velocity fields, turbulent fluctuations, and spatially resolved pollutant emission sources [20]. The convection–diffusion equation, which generalizes the classical dispersion model by explicitly incorporating advection, diffusion, and reactive processes [21], provides a rigorous framework to integrate these ecological factors into the physical dynamics of atmospheric pollutant transport. By accounting for the combined effects of bulk wind advection, turbulent mixing, chemical transformations, deposition, and localized emission patterns, this approach allows for a mechanistically grounded representation of pollutant dispersion across complex urban environments. (19)∂Φ(s→,t¯)∂t¯=∇⋅[Υ(Φ,s→)∇Φ(s→,t¯)]−∇⋅[w2→(s→,t¯)Φ(s→,t¯)]+S(s→,t¯).

The wind velocity field, denoted by w→(⋅), can be decomposed into its directional components w1(⋅) and w2(⋅), corresponding to advection along the u1- and u2-axes, respectively. The emission source, representing the rate of pollutant release at a specific location and time, is expressed as S(⋅) and incorporates both point and distributed anthropogenic contributions. Within the framework of the convection–diffusion equation, the ecological parameter Λ(u1,u2,t¯) can thus be explicitly decomposed into two mechanistic components: the advection-driven transport of pollutants by wind and the local contribution from emission sources, enabling a physically grounded representation of atmospheric pollutant dynamics that also accounts for directional dispersion and source heterogeneity.

The corresponding expression for the energy function is given as:(20)Λ(u1,u2,t¯)=Λwind(u1,u2,t¯)+Λsource(u1,u2,t¯)=−Φ(u1,u2,t¯)(w1x′(u1,u2,t¯)+w2y′(u1,u2,t¯))−w1(u1,u2,t¯)Φx′(u1,u2,t¯)−w2(u1,u2,t¯)Φy′(u1,u2,t¯)+S(u1,u2,t¯).

The full expression of the propagation equation is formulated as:(21)Φt¯′(u1,u2,t¯)=Υ(Φu1u1″(u1,u2,t¯)+Φu2u2″(u1,u2,t¯))−Φ(u1,u2,t¯)(w1u1′(u1,u2,t¯)+w2u2′(u1,u2,t¯))−w1(u1,u2,t¯)Φx′(u1,u2,t¯)−w2(u1,u2,t¯)Φy′(u1,u2,t¯)+S(u1,u2,t¯),where the expression S(⋅) depends on u1, u2, and t¯.

Assuming that pollution sources remain relatively stable over short time intervals, their impact on atmospheric pollutant concentrations is often modeled primarily as a function of spatial location [19]. This assumption generally holds for continuous emitters such as industrial facilities and power plants. However, in complex urban environments, additional time-varying contributors including traffic peaks, restaurant cooking emissions, and hotel exhaust play a significant role in shaping local air quality. When leveraging mobile sensors, which provide measurements at dynamically changing locations, it becomes essential to account for both spatial and temporal variability in emissions. To capture these effects, the emission source term S(⋅) is modeled as a three-argument function dependent on spatial coordinates (u1,u2) and time t¯, reflecting the moving sensor’s position and the fluctuating emission environment. This formulation allows the convection–diffusion model to integrate mobile sensor observations effectively, providing a physically consistent representation of temporal fluctuations, spatial heterogeneity, and dynamically sampled pollutant distributions for enhanced urban air quality mapping.

This scenario is addressed by the Phy-APMR-WS system, an extension of the original Phy-APMR framework that explicitly integrates physical laws governing atmospheric pollutant transport:

(i) High-resolution data on wind velocity and pollutant concentrations are available, represented by the real dataset t¯Φ={(u1ȷΦ,u2ȷΦ,t¯ȷΦ,ΦȷΦ,w1ȷΦ,w2ȷΦ)|ȷ=1,2,…,NΦ}, where w1ȷΦ and w2ȷΦ represent the wind velocity components driving the advection of pollutants along the u1- and u2-directions, respectively.

(ii) Although direct measurements of emission sources are not available, the Phy-APMR-WS framework leverages a PINN architecture to implicitly model pollutant generation, advection, diffusion, chemical transformation, and deposition during training. By enforcing these physical constraints, the system effectively reconstructs the influence of unobserved sources on observed concentrations, thereby enhancing the physical consistency and interpretability of the atmospheric indicators learned by the network.

The NN’s output layer consists of four neurons, which distinguishes the Phy-APMR-WS model from the original Phy-APMR. Specifically, the external variable Λ in (10) is decomposed into three mechanistic factors that govern pollutant dynamics: the u1-directional wind component w1 and the u2-directional wind component w2, which drive advection of airborne pollutants, and the pollution source term S, which represents local emissions, traffic peaks, industrial releases, and other anthropogenic or natural contributions. Accordingly, the network outputs include the estimated pollutant concentration Φ^(⋅;δ), the wind components w1^(⋅;δ) and w2^(⋅;δ), and the estimated source term S^(⋅;δ). Among these, Φ^(⋅;δ) is explicitly used for air quality concentration prediction and high-resolution reconstruction, while the predicted wind and source components provide a physics-informed representation of the primary factors driving pollutant transport and dispersion in urban environments.

However, the NN is trained on the database t¯Φ to forecast three observable contextual factors: the estimated pollutant concentration Φ^(⋅;δ) and the wind velocity components w1^(⋅;δ) and w2^(⋅;δ). The discrepancies between the predictions and the reported measurements are quantified using three corresponding data-driven loss functions: ℒΦ(δ), ℒw1(δ), and ℒw2(δ).

In the Phy-APMR-WS model, as in the original Phy-APMR, the physics of pollutant transport is explicitly incorporated by enforcing the convection–diffusion PDE residual in the total loss. This residual captures the advection of pollutants by the predicted wind field, turbulent diffusion, and, if applicable, reaction, deposition, and source emission processes, ensuring that the network predictions obey the fundamental physical laws of atmospheric transport. Formulation (21) specifies the estimated PDE residual at a given spatiotemporal location (22)Ψ^(u1,u2,t¯;δ):=Υ(Φ^u1u1″(u1,u2,t¯;δ)+Φ^u2u2″(u1,u2,t¯;δ))−Φ^(u1,u2,t¯;δ)(w1^u1′(u1,u2,t¯;δ)+w2^u2′(u1,u2,t¯;δ))−w1^(u1,u2,t¯;δ)Φ^u1′(u1,u2,t¯;δ)−w2^(u1,u2,t¯;δ)Φ^u2′(u1,u2,t¯;δ)+S^(u1,u2,t¯;δ)−Φ^t¯′(u1,u2,t¯;δ).

The ℒ2 norm is used to quantify the residual of the convection–diffusion PDE over the collocation set t¯Ψ={(u1ȷΨ,u2ȷΨ,t¯ȷΨ)|ȷ=1,2,…,NΨ}, defining the physics-informed loss ℒΨ(δ). This residual captures the discrepancy between the network-predicted pollutant concentrations and the governing physical laws of atmospheric transport, including advection by wind components w^1 and w^2, turbulent diffusion, local emissions, chemical reactions, and deposition processes. When both statistical data and physical constraints are incorporated, the overall loss function of the Phy-APMR-WS model is expressed as (23)ℒ(δ)=ϑΦℒΦ(δ)+ϑw1ℒw1(δ)+ϑw2ℒw2(δ)+ϑΨℒΨ(δ)=ϑΦNΦ∑ȷ=1NΦ|Φ^(u1ȷΦ,u2ȷΦ,t¯ȷΦ;δ)−ΦȷΦ|2+ϑw1NΦ∑ȷ=1NΦ|u^(u1ȷΦ,u2ȷΦ,t¯ȷΦ;δ)−w1ȷΦ|2+ϑw2NΦ∑ȷ=1NΦ|v^(u1ȷΦ,u2ȷΦ,t¯ȷΦ;δ)−w2ȷΦ|2+ϑΨNΨ∑ȷ=1NΨ|Ψ^(u1ȷΨ,u2ȷΨ,t¯ȷΨ;δ)|2,where ϑΦ,ϑw1,ϑw2,ϑΨ are hyperparameters controlling the relative contributions of each module.

By drastically lowering NΨ for each repetition, the ASUS methodology seeks to reduce the computational cost of every iteration. The NN component of the PINN processes all collocation points at each loop, using back-propagation to estimate the system parameters δ and feed-forward computations for state inference. Since the training cost scales with the number of collocation points, reducing their total count accelerates convergence and lowers computational overhead.

However, for atmospheric pollutants such as O3, PM_2.5 and PM₁₀, excessive reduction of collocation points can degrade extrapolation performance. These pollutants are governed by strongly nonlinear physical and chemical processes, including photochemical reactions (e.g., NOx–VOC–O3 chemistry), aerosol nucleation and secondary formation, turbulent diffusion, advection, dry and wet deposition, and coupling with meteorological variables such as solar radiation, temperature, relative humidity, wind speed, and planetary boundary-layer height. Concentrations of PM_2.5 and PM₁₀ are further influenced by primary emissions, hygroscopic growth, coagulation, and resuspension processes, whereas O3 formation is highly sensitive to precursor availability and photolysis rates.

An indiscriminate reduction of collocation points may therefore limit the model’s ability to resolve sharp spatiotemporal gradients, episodic pollution events, and chemical regime transitions (e.g., VOC-limited vs. NOx-limited ozone formation), ultimately constraining generalization. To mitigate these limitations while maintaining computational efficiency, adaptive collocation selection and short-time refinement strategies are combined. These approaches emphasize physically informative regions, including high-reactivity zones, concentration extrema, and dominant transport pathways, thereby preserving the governing physics of O3, PM_2.5, and PM₁₀ while reducing computational cost.

The residual-based adaptive refinement (RAR) remodelling technique [37] serves as the foundation for the subsequent mechanical procedure. Its core principle is to enhance gradient-based optimization by preferentially selecting collocation points associated with higher PDE residuals, denoted as ℒΨ. In the context of atmospheric pollutant modelling, ℒΨ encapsulates the residuals of the coupled advection–diffusion–reaction equations governing key contaminants such as O3, PM_2.5, PM₁₀ and their chemical precursors (e.g., NOx, VOCs, and secondary aerosols).

Within the proposed adaptive screening step, newly sampled collocation locations exhibiting updated PDE residuals, ℒΨnew, are evaluated against the previously observed residuals, ℒΨold. Alternative collocation points are accepted only if ℒΨnew>ℒΨold; otherwise, the refinement procedure continues until this condition is met. This strategy ensures that regions characterized by strong physical imbalances such as sharp concentration gradients, photochemical reaction hotspots, or aerosol formation zones are preferentially sampled.

To formalize this criterion, a threshold parameter R is introduced to regulate the ratio between ℒΨnew and ℒΨold, thereby determining the admissibility of alternative collocation points. To prevent infinite rejection loops, the strictness of R is progressively relaxed over successive sampling cycles. The acceptance condition is defined as (25)ℒΨnewℒΨold>max(R𝒲decayns,1),where ns denotes the number of consecutive rejections due to unmet criteria, and 𝒲decay is an anti-deadlock decay factor slightly greater than unity.

By adaptively emphasizing collocation points with elevated PDE residuals, the method systematically captures dominant physical and chemical processes, including pollutant transport, turbulent mixing, photochemical ozone formation, secondary particulate matter generation, and deposition mechanisms. Consequently, the proposed Phy-APMR framework improves training efficiency, reduces the total number of required iterations, and enhances the physical consistency and generalization capability of the learned solution for multi-contaminant air quality modelling.

The collection of PDE collocation points is generated arbitrarily from the characteristic (or specification) region at the beginning of training. In conventional approaches, these collocation sites remain fixed throughout the training process. To ensure sufficient domain coverage, a significant quantity of stationary collocation locations must be employed, which substantially increases the computational cost. Moreover, relying solely on the emphasized optimization technique may lead to insufficient representation of the domain.

We propose a short-time implementation strategy to address this issue, in which collocation points are resampled and updated every N iterations (where N is a small adjustable parameter). Regular reconfiguration interactively improves domain coverage compared to simply enlarging the collocation set, thereby reflecting the core motivation of the approach. Furthermore, applying immediate correction reduces the overall PDE training time, since testing or resampling is considerably less costly than processing over a large static set of collocation nodes.

Algorithm 1 presents the complete ASUS methodology. The ASUS framework accelerates the training phase of the Phy-APMR technique without compromising accuracy by integrating monitoring optimization, descriptive evaluation, and immediate amendment. A detailed evaluation is provided in Section 5.10.

Understanding that fixed detectors have a restricted geographic area, we implanted wireless sensors on public transportation and automobiles in a systematic way to extend geographical coverage. We purposefully installed autonomous sensors on transportation vehicles to extend geographic coverage after recognizing that fixed detectors had a limited viewing range. This flexibility enables the collection of data across diverse metropolitan scenarios and allows observation of a broader area. To ensure adequate data acquisition, our test setup incorporated monitoring equipment on both fixed and handheld platforms. Fig. 7a illustrates the appearance of the device, while Fig. 7b–d shows its deployment. Each monitoring unit consists of four primary components: the sensor, connectivity module, power supply, and control circuitry. (i) The detection component integrates a GPS module along with multiple pollution-monitoring sensors, including Plantower’s PMS5003 for measuring fine particulate matter PM_2.5. The system is extended to account for PM₁₀ and O3, which are critical air-quality indicators due to their adverse respiratory and cardiovascular effects. While PM₁₀ penetrates the upper respiratory tract and contributes to airway inflammation, O3 acts as a strong oxidant, exacerbating asthma, reducing lung function, and enhancing oxidative stress. The GPS module continuously records spatial localization in terms of latitude and longitude to capture the spatial variability of pollutant exposure.

(ii) The device is powered by an external generator source, ensuring stable operation of gas- and particle-sensing elements under varying environmental conditions and enabling continuous monitoring during extended deployment periods.

(iii) The transmission component enables real-time data transfer via Wi-Fi and 3G/4G networks to an encrypted cloud-based server, facilitating near–real-time assessment of PM₁₀ and O3 exposure levels and supporting timely air-quality analysis and health-risk evaluation.

(iv) A microcontroller embedded within the monitoring unit manages sensor synchronization, data acquisition, preprocessing, and communication tasks, ensuring reliable integration of particulate and ozone measurements with geospatial information.

As shown in Fig. 7e, miniature meteorological instruments were deployed in the Lahore study to record atmospheric parameters and to reduce the influence of urban vehicular circulation on the accuracy of PM_2.5 measurements. In addition to PM_2.5, these atmospheric conditions strongly influence the formation, dispersion, and accumulation of O3 and PM₁₀, both of which pose significant respiratory health risks. While PM₁₀ primarily affects the upper respiratory tract and contributes to airway irritation and inflammation, ground-level O3 acts as a powerful oxidant that aggravates asthma, impairs lung function, and enhances oxidative stress under high solar radiation conditions.

To monitor near-surface, real-time atmospheric conditions, such sensors are commonly installed on unsecured rooftops of low- to mid-rise buildings. The devices are designed for continuous long-term operation using household electrical power. The meteorological data collected are essential for interpreting variations in PM_2.5, PM₁₀, and O3 concentrations driven by ventilation, thermal stratification, and photochemical activity. Wind measurements, which are only available in Lahore, are used to evaluate the wind-enhanced variant exclusively for this city. To minimize systematic bias and enhance sensor reliability in other locations, inter-device comparative analyses will be conducted across deployments in Faisalabad and Karachi, without applying the wind-enhanced adjustments.

Overall, this integrated monitoring configuration ensures that the study accurately captures circulation dynamics and environmental conditions relevant to particulate matter and ozone exposure, thereby improving the robustness and precision of the collected air-quality data.

To ensure data accuracy, all collected measurements underwent an extensive preprocessing and cleaning phase. Mean values were computed within each predefined sensorimotor region, after which the Three-Sigma Threshold was applied to identify and remove anomalous observations. This process improves data reliability by mitigating the influence of outliers, thereby strengthening the validity of comparative analyses across multiple sensing devices.

To further enhance study quality, peripheral regions with insufficient data coverage were excluded from the analysis. As shown in Table 1, this filtering step resulted in effective study areas of approximately 125 km² in Lahore, 62 km² in Faisalabad, and 38 km² in Karachi, ensuring consistency and comparability across regions.

The remaining measurements were discretized using a spatial resolution of 0.5 km × 0.5 km and a temporal resolution of 1 h, forming sensorimotor segments. This spatiotemporal granularity enables the reconstruction of high-resolution atmospheric pollution fields while capturing fine spatial structure and periodic temporal variations both essential for accurate environmental monitoring and analysis. Throughout the remainder of this article, each minimal temporal unit of 1 h is referred to as a time piece, and each smallest spatial unit of 0.5 km × 0.5 km is defined as anaggregated domain.

To evaluate sensor reliability, measurements from the sensing devices (u1) were compared against benchmark location data (u2), allowing assessment of robustness and overall detection accuracy.

Standard performance metrics, including the coefficient of determination (R2), root mean squared error (RMSE), and mean absolute error (MAE), were computed to quantify predictive accuracy.

Finally, the degree of correlation R between u1 and u2 was calculated as follows:(26)R=N∑u1ȷu2ȷ−∑u1ȷ∑u2ȷN∑u1ȷ2−(∑u1ȷ)2N∑u2ȷ2−(∑u2ȷ)2,where N represents the number of observations. Whenever R exceeds 0.85, the detection instrument is considered validated and ready for deployment.

In addition to correlation, the accuracy of each monitoring device was evaluated using the R2, the RMSE, and the MAE. These metrics are defined as:(27)R2=1−∑ȷ=1N(u1ȷ−u2ȷ)2∑ȷ=1N(u1ȷ−u¯1)2, (28)RMSE=1N∑ȷ=1N(u1ȷ−u2ȷ)2, (29)MAE=1N∑ȷ=1N|u1ȷ−u2ȷ|.

Here, u1¯ denotes the mean of u1, while u1ȷ and u2ȷ represent paired observations from the reference and the test instrument, respectively.

For example, Table 2 presents the Pearson correlation, R2, RMSE, and MAE values for nine O₃, PM_2.5 and PM₁₀ measurement systems. As shown, each instrument achieved correlation scores above 0.85, R2 values approaching unity, and low RMSE and MAE values, confirming the greater precision, reliability, and robustness of the proposed detection apparatus. These results indicate that the instruments are well-suited for atmospheric monitoring tasks, consistently producing data in strong agreement with widely accepted baseline measurements.

To ensure reproducibility of the experimental split protocol, we explicitly describe the data partitioning strategies used for training and validation of the proposed model. Two cross-validation (CV) strategies were employed: sample-based CV and grid-based CV, designed to assess both overall predictive accuracy and spatial generalization across urban monitoring networks.

Sample-based CV: For sample-based CV, the dataset was randomly partitioned at the time-piece level into five subsets of approximately equal size (20% each). In each iteration, four subsets were used for training and the remaining subset for validation. This process was repeated five times, ensuring that each sample was used once as validation. Samples were drawn uniformly across all spatial locations and time periods. This strategy evaluates the model’s ability to predict pollutant concentrations when both spatial and temporal distributions are represented in training and validation sets.

Grid-based CV: For grid-based CV, the spatial domain was partitioned into contiguous blocks corresponding to 0.5 km × 0.5 km grid cells. In each fold, all measurements from one block were held out as the validation set, while the remaining blocks were used for training. Each block represents a unique spatial region within the city (Lahore, Faisalabad, or Karachi). No overlapping locations or time pieces from the validation block are included in training, ensuring fully independent evaluation. This strategy assesses how well the model generalizes to unseen locations, simulating real-world scenarios with incomplete monitoring coverage.

The sample-based and grid-based CV results for O3, PM_2.5, and PM₁₀ at reference sites P₄₂, P₄₅, P₅₁, and P₄₈ for 2024 are presented in Fig. 8. Density scatterplots indicate that the proposed model achieved high overall accuracy across all sites, with the best performance observed for O3.

For O3, the sample-based CV produced R2 values of 0.92, 0.91, 0.90, and 0.89 for P₄₂, P₄₅, P₅₁, and P₄₈, respectively. Corresponding RMSE values were 11.25, 11.30, 11.40, and 11.50 µg/m³, while MAE values were 8.01, 8.05, 8.10, and 8.15 µg/m³. The regression line slopes were close to 0.93 at all sites, closely approximating the 1:1 line and demonstrating strong agreement between predicted and observed concentrations. Grid-based CV results were slightly lower but still exhibited high R2 values of 0.91, 0.90, 0.89, and 0.88 for P₄₂, P₄₅, P₅₁, and P₄₈, respectively.

For PM_2.5, sample-based (grid-based) CV R2 values were 0.90 (0.89), 0.89 (0.88), 0.88 (0.87), and 0.87 (0.86) for P₄₂, P₄₅, P₅₁, and P₄₈, respectively. RMSE values ranged from 8.23–8.40 µg/m³, and MAE values ranged from 5.21–5.35 µg/m³ across all sites, demonstrating accurate estimations. PM₁₀ predictions were similarly reliable, with sample-based CV R2 values of 0.87, 0.86, 0.85, and 0.84, RMSE values of 14.73–14.90 µg/m³, and MAE values of 9.64–9.70 µg/m³. Grid-based CV showed a small reduction in R2 (approximately 0.02), but trends remained consistent with sample-based results. The slopes of the regression lines for both PM_2.5 and PM₁₀ exceeded 0.87 at all sites, indicating minimal estimation bias.

Spatial robustness was evaluated across individual monitoring sites in Pakistan. Overall, O3 predictions showed higher accuracy than PM predictions, with R2 values exceeding 0.80 at most stations in Lahore and Faisalabad, while Karachi exhibited slightly lower R2 values in some areas due to sparser monitoring coverage (see Fig. 9). The initial R2 distribution (Panel 1 for Fig. 9) shows poor model performance, with most stations displaying low correlation (R2<0.2). After intermediate optimization (Panel 2 for Fig. 9), predictive accuracy improves substantially, with many stations shifting to moderate and strong correlation levels (0.2≤R2≤0.8). In the optimized stage (Panel 3 for Fig. 9), the model demonstrates strong spatial robustness, with most stations across Pakistan exhibiting moderate to high predictive accuracy, indicating a well-calibrated and reliable air pollution prediction model. Fig. 10 provides a comprehensive assessment of the model’s performance across both spatial and temporal dimensions. In Fig. 10a, the spatial evaluation reveals that for PM_2.5 and PM₁₀, more than 85% of monitoring sites in Lahore and Faisalabad and approximately 78% of sites in Karachi achieved R2>0.70. The results indicate that higher accuracy was generally observed in regions with denser monitoring networks, such as central Lahore and Faisalabad, whereas areas with sparser sensor coverage exhibited slightly larger estimation errors. Fig. 10b presents the temporal analysis, showing that the monthly median biases for O3, PM_2.5, and PM₁₀ remained close to zero across all cities, suggesting that the model maintained stable and unbiased performance throughout the year. Fig. 10c emphasizes the largest biases occurred in December, with median values of −0.65 µg/m³ for O3, 0.72 µg/m³ for PM2.5, and 1.01 µg/m³ for PM₁₀. Residual ranges for O3 were smallest in December, while PM_2.5 and PM₁₀ showed smaller residual ranges during the summer months, indicating seasonal variability in prediction uncertainty. Monthly mean relative errors (MRE) were lowest for O3, averaging around 8%–9%, while PM_2.5 and PM₁₀ maintained MREs around 20%, with seasonal peaks in May for PM_2.5 (∼21%) and July for PM₁₀ (∼19.8%).

Overall, these results indicate that the proposed model effectively captures daily and seasonal variations of O3, PM_2.5, and PM₁₀ in Lahore, Faisalabad, and Karachi, showing robust spatial and temporal performance across different urban environments in Pakistan.

As shown in Table 3, the model achieved an R2 of 0.92 under sample-based CV (random temporal sampling). The grid-based CV (spatial block hold-out) showed only a marginal decrease in performance, confirming robust spatial generalization across the urban monitoring network. It is important to note that these evaluations were restricted to the spatial and temporal bounds of the existing dataset; specifically, no fully temporal hold-out or cross-city transfer evaluations were performed. Consequently, the reported results reflect predictive performance strictly under random temporal sampling and spatial block hold-out. These settings are explicitly defined to provide a transparent assessment of the model and to avoid implying more challenging evaluation scenarios than those actually conducted.

The most recent data obtained from the sensor network are used to train the Phy-APMR model at each successive time slice. Specifically, at each period, 25% of the data are selected to form the test set, while the remaining 75% constitute the training set, representing the information available in the real-world context. Each successive segmentation during evaluation involves updating the system. To develop a predictive algorithm, the data available at the current time t¯0, along with past observations over the interval [t¯0−Δt¯h1,t¯0], are extracted from the initial training batch. The test samples corresponding to the epoch t¯0 are then used for evaluation after the training process. In order to prevent duplication, every single layer’s PINN uses separate data databases for testing as well as training.

An additional portion derived from the experimental data is reserved as a assessment dataset to enable incipient stopping and hyperparameter calibration, thereby preventing overfitting. Table 1 summarizes the data collection details for each of these locations.

The prediction error in our study is evaluated utilizing the Mean Absolute Percentage Error (MAPE), stated as:(30)MAPE=1N∑ȷ=1N|Φ^ȷ−ΦȷΦȷ|,where Φȷ provides the fundamental objective, N indicates the criteria set dimensions, and Φ^ȷ indicates the estimated quantity.

Seven reference methods are used for comparison with Phy-APMR:

Central value: The simplest approach uses the mean of all training data for the given time segment to generate the predicted outcome.

All experiments were implemented using the TensorFlow 2.4.0 framework with Python 3.8 and trained on an NVIDIA GTX Titan X GPU with 12 GB memory. The number of training iterations was fixed at 𝒲=25,000, which also served as the stopping criterion; no early stopping was applied. The diffusion coefficient was set to Υ=0.5 based on the approximated propagation speed of PM_2.5 in urban environments, following prior work [20]. The NN architecture consists of four fully connected hidden layers with 30 neurons per layer. Network weights were initialized using the Xavier (Glorot) initialization scheme, while all biases were initialized to zero. The collocation size used to enforce the physics-based constraints was set to NΨ=50,000, with collocation points sampled uniformly across the spatiotemporal domain. Model training employed the Adam optimizer with a fixed learning rate of 10−3. Regarding loss weighting, the loss scale hyperparameters were set to ϑΦ=1 for the data fidelity term and ϑΨ=0.01 for the physics-based constraint term. These values were selected empirically based on validation performance to balance data consistency and physical regularization. All hyperparameters, including the historical time span of the training set (Δt¯h=2), were tuned using a held-out validation set.