Advanced Graph–Physics Hybrid Framework (AGPHF) for Holistic Integration of AI-Driven Graph- and Physics- Methodologies to Promote Resilient Wastewater Management in Dynamic Real-World Conditions

Abstract

Wastewater treatment is evolving rapidly with the advent of advanced deep-learning AI, graph-based, and physics-informed approaches. This study integrates graph neural networks, physics-informed neural networks, and multi-agent reinforcement learning within a hybrid digital-twin framework, evaluated on multi-scale real-world datasets. The developed models achieved over 90% dissolved organic carbon removal and reduced aeration energy by up to 22% while maintaining process stability. The results demonstrate that graph–physics synergies not only boost operational efficiency but also reveal critical trade-offs between energy savings and hydraulic performance. Our findings establish a new benchmark for resilient, low-carbon wastewater treatment, highlighting the transformative role of data-driven system design.

1. Introduction

Wastewater treatment is undergoing a paradigmatic transformation driven by the integration of advanced artificial intelligence (AI) [1,2], graph theory, and physics-informed methods. The objective of achieving low-carbon, high-performance wastewater management has motivated extensive research in hybrid technologies that employ novel computational techniques [3,4]. The domain is increasingly characterized by methodologies that employ graph neural networks (GNNs), physics-informed neural networks (PINNs), hybrid digital twins (DTs), and optimization frameworks [5,6,7,8]. However, despite promising advances, significant impediments remain, including data sparsity, lack of comprehensive integration, and lack of robust closed-loop validation, necessitating further research [9,10]. This study seeks to investigate the following guiding question: How can graph–physics AI improve trade-offs between energy efficiency and hydraulic stability in wastewater treatment?

Previous research in wastewater treatment leveraging AI methodologies can be categorized into distinct sub-domains based on their intrinsic characteristics. Structural GNNs harness network topology to impute missing data, forecast flows, and predict emissions. Techniques such as graph convolutional networks (GCNs) have notably improved database completeness, achieving correlations surpassing traditional regression methods [11]. EcoGNN models have demonstrated efficacy in predicting greenhouse gas emissions, substantially mitigating methane and nitrous oxide release by informing optimal control strategies [12]. Nonetheless, these models require detailed graph-resolved data and exhibit sensitivity to data imbalance [13].

Knowledge graphs encode expert-driven rules and epidemiological data. Open knowledge graphs (OKGs) such as Water Health OKG effectively integrate socio-environmental data for sustainable development goal (SDG) monitoring [14]. Attack graphs facilitate cybersecurity modeling in wastewater treatment plants (WWTPs), yet these systems necessitate substantial manual curation and quantitative integration with process variables remaining sparse [15]. PI-GNNs embed physical constraints directly within graph structures, enabling robust predictions with minimal training data. Examples include pressure predictions in water distribution networks with significant computational efficiency improvements [16]. However, these methodologies predominantly focus on steady-state hydraulic conditions and lack full representation of biochemical dynamics [16].

PINNs directly address the inherent difficulties of modeling wastewater processes with partial differential equations (PDEs). These networks provide a physically consistent output without labeled data [17]. However, their applicability to complex biochemical reactions typical of WWTPs remains challenging due to computational intensity and numerical instability. Hybrid DTs integrate data-driven models, such as Long Short-Term Memory (LSTM) networks, with traditional mechanistic simulations. Real-time synchronization with plant operations is a notable benefit, exemplified by significant reductions in aeration energy [18]. Despite their utility, digital twins frequently omit strict enforcement of mass and energy balances, resulting in model drift and limited predictive reliability [19].

These control strategies utilize AI algorithms to optimize operational variables such as aeration and nitrogen removal, significantly outperforming traditional Proportional–Integral–Derivative (PID) controls [20]. However, they predominantly operate within synthetic benchmarks, with limited real-world validation. Optimization methods leverage graph theory to simplify design processes, effectively reducing design complexity by focusing on topological metrics [21]. Nevertheless, these techniques typically overlook biological kinetics and uncertainties inherent in wastewater treatment [22].

Despite these advancements, substantial literature gaps persist:

• Comprehensive integration of graph-based and physics-informed models into cohesive, end-to-end wastewater treatment systems remains unaccomplished.
• Multilayer graph models integrating infrastructure and microbial consortia data remain absent.
• Robust uncertainty quantification within AI-driven wastewater treatment processes is currently insufficiently addressed.
• Real-world closed-loop validation of AI-based control methods is severely lacking.
• The cybersecurity considerations integrated into AI-enhanced wastewater treatment systems are minimal.
• Standardized ontologies for graph and physics-based wastewater treatment methodologies remain underdeveloped.

A consistent framework for benchmarking across different facilities is still lacking. The absence of unified protocols, standardized taxonomies, detailed construction-related metadata, and structured drift tracking leads to major difficulties when comparing experimental findings across independent studies and deployments. Assurances on the operational safety of adaptive control strategies remain underdeveloped. The mathematical tools for enforcing stability and safety, such as barrier-based formulations, temporal logic-driven constraints, and provable reachability guarantees, require formal implementation within complex systems involving coupled hydraulic dynamics and intricate biological processes.

The problem of distinguishing cause from correlation in system behaviors has received limited attention. Techniques that incorporate structural causal modeling, graph-based reasoning, and targeted intervention analysis are needed to separate confounding influences from genuine treatment effects while providing quantifiable measures of uncertainty in outcomes. Deployment of intelligent control models in edge computing environments remains insufficiently explored. There is little research addressing the trade-offs between processing latency, computational power, and bandwidth limitations, as well as the propagation of quantization errors within high-dimensional predictive models.

Strategies for privacy-preserving learning and federated decision-making remain immature, particularly in contexts where energy efficiency and policy compliance must coexist. Long-term system reliability and adaptability are not comprehensively addressed. Mechanisms for continuous online learning that maintain mathematical stability, early detection of evolving patterns that lead to operational drift, and structured rollback strategies for malfunctioning digital twins remain underdeveloped and require formal codification. The integration of multiple simulation environments across interconnected chains remains immature, e.g., hydrological inflows, treatment plants, and water reuse systems. Accurate reconciliation between physics-based models and empirical process simulations is essential to preserve mass balance and maintain predictive fidelity across interfaces. Human interpretability of automated control decisions is still limited. Methods for generating intuitive counterfactual explanations, visualizing influential pathways in graph-based systems, and embedding physical constraints into interpretability frameworks must be advanced to assist operators during high-stake alarm triage and decision-making.

This study establishes a unified, graph–physics-aware AI framework for wastewater treatment, validated on diverse full-scale datasets and plant environments. The principal technical contributions are summarized below:

• Unified Framework: Developed and deployed a modular pipeline that tightly integrates graph neural networks (GNNs), physics-informed neural networks (PINNs), multi-agent reinforcement learning (MARL), and digital twin architectures, enabling end-to-end learning, simulation, and closed-loop control for advanced wastewater treatment operations.
• Empirical Validation Across Scales: Demonstrated robust dissolved organic carbon (DOC) removal, consistently surpassing $90\%$ in membrane-aerated biofilm reactor–reverse osmosis (MABR–RO) pilot deployments, as well as sustained potable-grade permeate under variable influent and meteorological conditions.
• Swarm and RL-Optimized Advanced Oxidation: Implemented swarm intelligence and MARL-based optimization for hybrid electro-ozone/Fenton advanced oxidation processes (AOPs), achieving $97\%$ chemical oxygen demand (COD) abatement in less than 60 min, verified with full-scale high-strength wastewater streams.
• Graph-Based Predictive Control: Designed GNN-enabled controllers for membrane bioreactors (MBRs) to anticipate fouling and adaptively regulate aeration, resulting in persistent flux stabilization with <5% relative tracking error, validated across diurnal and shock-loading cycles.
• Physics-Informed Energy Reduction: Integrated physics-informed neuroevolutionary models into digital twins at the plant scale, realizing average aeration energy reductions of $19\%$ to $22\%$ without compromising effluent regulatory targets.
• Uncertainty and Trade-Off Quantification: Quantified operational trade-offs between energy efficiency, effluent quality, and process stability, providing percentile-resolved performance benchmarks and revealing controller sensitivities to meteorological and influent perturbations.

Collectively, these methodologies unify membrane, electrochemical, and advanced oxidation processes, paving the way toward sustainable and resilient low-carbon wastewater treatment.

2. Materials and Methods

Reliability steps preceded model construction. Temporal blocking with forward-only folds (K = 5) preserved causality; each fold trained on $T_{1,\dots ,k-1}$, validated on $T_k$. Out-of-time assessment eliminated leakage. Bootstrap resampling ($B=1000$) yielded percentile intervals; medians with 10% to 90% bands were reported.

Uncertainty governance used dual decomposition: aleatoric via residual dispersion; epistemic via stochastic-weight sampling (dropout at inference, 50 passes). Quantile loss produced calibrated prediction intervals; miscoverage was tracked per horizon. Heteroscedastic noise was handled using variance-aware objectives. Replication employed controlled input perturbations within sensor tolerances: Gaussian jitter, timestamp offsets, and structured missingness masks. Policies were recomputed for twenty distinct seeds; variance across seeds was summarized. Global sensitivity used Sobol’s first-order indices for temperature, flow, and conductivity; fragile predictors were flagged for ablation. Stress scenarios were injected rainfall spikes, heatwaves, and salinity excursions; metrics were aggregated with stratified bootstraps.

Process integrity was as follows: deterministic preprocessing pipelines; fixed seeds; artifact hashing; full provenance tables. Model cards documented data ranges, constraints, and expected failure modes; audit logs archived configuration diffs.

2.1. Dataset

The empirical evaluation employs the public dataset reported by Bagehrzadeh in 2021 [23]. It aggregates six years (2014–2019) of full-scale operational records from Melbourne’s eastern wastewater facility with concurrent meteorological observations. The joined table provides daily resolution, linking energy consumption, hydraulic throughput, biological quality, and ambient climate on a shared date column. A second empirical resource derives from the high-frequency Supervisory Control and Data Acquisition (SCADA) archive (Siemens, Munich, Germany) released by Mohammadi et al. [23]. Collected at the Agtrup (BlueKolding, BlueKolding A/S, Kolding, Denmark) facility in Denmark (using Siemens WinCC v7.5), the system targets intensified phosphorus removal through intertwined chemical and biological pathways. Whereas the Melbourne corpus (Section 2.1) offers daily aggregates spanning six years, the Agtrup file furnishes two-minute snapshots over a single month (August 2023), yielding $N_{\mathrm{total}}\approx \mathrm{21,600}$ observations and exposing rapid control dynamics.

Key variables comprise influent flow, blower power, mixed-liquor suspended solids, effluent ammonia, temperature, humidity, and wind speed. Exactly 2191 daily samples afford sufficient seasonal depth to probe the effect of hydraulic shocks and climate variability. Missing entries (~2%) are imputed with a seasonal Kalman smoother; an exception is categorical weather code, which is transformed into one-hot vectors. Numerical variables are centered and scaled to unit variance to suppress gradient imbalance during optimization [24,25,26,27,28].

Data gaps (<0.5%) are bridged via a forward–backward fill bounded by a 12 min window; spikes exceeding five standard deviations undergo median filtering to avert optimization impediment. All numeric signals are rescaled to zero mean and unit variance. The dense temporal cadence enables derivative features (e.g., dPO₄/dt) that enrich graph edges in the GNN controller [29,30].

The Agtrup archive underpins three central experiments: (i) deep-reinforcement-learning optimization of FeCl₃ dosing (Sigma-Aldrich, St. Louis, MO, USA) under phosphorus-effluent constraints, (ii) graph–physics hybrid forecasting of short-horizon PO₄ trajectories, and (iii) digital-twin aeration scheduling that embeds biological uptake kinetics. Its minute-scale resolution complements the Melbourne dataset’s multi-year breadth, establishing a dual-scale benchmark indispensable for assessing controller generalization across disparate temporal grains [31].

The sequence is divided strictly forward in time: 2014–2017 for training, 2018 for validation, and 2019 for blind testing. This partitioning prevents leakage while capturing side phenomena such as El-Niño-induced heat waves that amplify aeration cost. For example, several December 2019 days exceeded 40 °C, stressing blower demand. Outliers are suppressed with a five-day inter-quartile filter, removing potential impediment to convergence without erasing legitimate process excursions [32,33,34].

2.2. Modeling Sequence

This protocol codifies a reproducible sequence for hybrid graph–physics research. Each stage uses deterministic configuration files, seed control, and strict versioning. Short sentences follow; precision dominates (in Figure 1, is illustrated the protocol encapsulates Steps 1–6. Figure 2 depicts the continuation of the protocol for Steps 7–12. The full pipeline is shown in Figure 3).

Step 1: Domain formalization. Specify hydraulic compartments, biofilm strata, actuator sets, and sensor lattices; define directed multigraph topology for assets, flows, and constraints. Provide symbol tables, unit systems, and admissible bounds.

Step 2: Discretization. Select temporal cadence for controls versus states; apply method of lines to transport PDEs; choose spatial stencil on process vessels; confirm Courant safety margins via dry runs.

Step 3: Graph encoding. Map vessels, pipes, and membranes to nodes; encode edges with hydraulic conductance, substrate gradients, and meteorological modifiers; attach temporal features via lag stacks; store in sparse tensors.

Step 4: Mechanistic core. Implement reaction–diffusion kinetics for multi-species biofilm; augment with Darcy–Brinkman–Stokes where permeability matters; impose mass–energy conservation with hard constraints.

Step 5: PINN residuals. Construct physics loss via strong-form residuals at collocation sets; apply boundary–initial penalties; schedule adaptive weights using residual variance tracking; prevent stiffness via gradient clipping.

Step 6: GNN predictors. Use attention or spectral blocks for short-horizon surrogates; initialize with Xavier draws; normalize node features per batch; monitor calibration via reliability diagrams.

Step 7: Emulator coupling. Wrap mechanistic solver behind differentiable adaptors; feed GNN forecasts into the state estimators; return corrected states to the solver; maintain causality by teacher forcing during warm-up.

Step 8: Control environment. Instantiate a digital-twin loop with actuator latencies, set-point bounds, and safety interlocks; define reward with lexicographic priorities: effluent quality, energy use, and flux stability.

Step 9: Policy search. Train multi-agent controllers under domain randomization; inject meteorological perturbations, influent shocks, and sensor dropouts; apply entropy regularization; throttle exploration with cosine schedules.

Step 10: Identification. Calibrate kinetic parameters via adjoint gradients where available; fall back to derivative-free search for non-smooth terms; run multi-start to avoid spurious minima.

Step 11: Validation. Execute forward-time splits across seasons; score RMSE, MAPE, energy indices, and flux tracking; compute uncertainty with block bootstrap on diurnal segments; store artefacts for audit.

Step 12: Ablations. Remove physics residuals, disable graph edges, neutralize meteorological inputs, and freeze actuators; quantify degradation against the full stack; report percentile spreads, not only central tendency.

2.3. Graph–Physics Hybrid Pipeline

The proposed experimental framework, designated as the Advanced Graph–Physics Hybrid Framework (AGPHF), was meticulously devised to encompass intricate interactions intrinsic to wastewater treatment dynamics. AGPHF integrates state-of-the-art graph neural networks (GNNs), physics-informed neural networks (PINNs), multi-agent reinforcement learning (MARL), and advanced hybrid modeling. Each module was chosen for its capacity to encapsulate domain-specific knowledge and for its computational robustness [22].

The experimental pipeline orchestrates multiple interlinked modules, each performing distinct tasks within the data-driven, physics-aware wastewater treatment framework. The following outline represents the principal stages, data flows, and optimization feedbacks. The process commences with multi-scale data acquisition and preprocessing, continues through model selection and tuning, and culminates in validation using closed-loop feedback.

Pipeline Outline:

• Data Acquisition: Gather historical SCADA records and full-scale plant telemetry, merging disparate time scales into a unified sequence. Employ missing data imputation and outlier filtering.
• Feature Engineering: Compute statistical moments, physicochemical gradients, and domain-specific indices. Encode graph-structured features for input into neural models.
• Model Initialization: Select architectures: Graph Attention Networks (GATs), Spatio-Temporal GCNs (ST-GCNs), Capsule Graph Networks (CapsGNs), PINNs, and MARL-based digital twins. Initialize hyperparameters via Latin hypercube sampling.
• Training: Iteratively optimize model weights using Adam or stochastic gradient descent, subject to composite loss objectives:

  –

  Data fidelity;

  –

  Physical law consistency (for PINN);

  –

  Boundary and initial constraints.

• Simulation and Prediction: Generate forecasted effluent quality, energy consumption, and microbial dynamics using the trained models on test sets. Schedule aeration and dosing via learned policies.
• Control Feedback: Integrate model predictions with real-time plant control, enabling digital-twin synchronization and adaptive set-point regulation.
• Evaluation: Aggregate results across diurnal cycles, calculate percentile metrics, and cross-validate using independent datasets.

The pipeline implementation follows the design patterns and solution architecture in the following pseudocode:

Pseudocode for End-to-End Pipeline:

• Input: Raw SCADA data, historical plant logs, environmental telemetry
• Output: Optimized operational policy, validated predictions
   

  1. 

  Preprocess data:

   a. 

  Merge tables by timestamp, impute missing values

   b. 

  Filter outliers, scale features

   

  2. 

  Construct graph representation:

   a. 

  Define node/edge types for physical assets and process variables

   b. 

  Compute edge attributes (e.g., flow, substrate gradient)

   

  3. 

  Initialize models:

   For each architecture in {GAT, ST-GCN, CapsGN, PINN, MARL-DT}:

       a. 

  Set hyperparameters (random sampling within design bounds)

       b. 

  Initialize weights

   

  4. 

  Train models:

   For each epoch:

       a. 

  Forward pass: compute outputs, losses (data, PDE, BC, IC)

       b. 

  Backpropagate and update weights

       c. 

  Enforce physical/operational constraints

   

  5. 

  Simulate closed-loop:

   For each test cycle:

       a. 

  Predict effluent, energy, microbial metrics

       b. 

  Adjust operational setpoints via MARL agent

       c. 

  Log feedback for further refinement

   

  6. 

  Evaluate performance:

   a. 

  Compute median and percentile metrics (RMSE, COD removal, energy saving)

   b. 

  Compare across models

   c. 

  Diagnose trade-offs

   

  7. 

  Select optimal policy for deployment

This multi-stage framework enables rigorous benchmarking, adaptive control, and holistic integration of AI-driven, graph-based, and physics-informed methodologies, advancing resilient wastewater management under dynamic real-world conditions.

2.4. Graph Neural Network Models

The Graph Attention Network (GAT) employed herein was customized for pollutant degradation analysis [35]. Node attention coefficients, denoted ${\alpha }_{ij}$, were computed using

$${\alpha }_{ij}={\displaystyle \frac{\exp \left(\mathrm{LeakyReLU}\left({\mathbf{a}}^{\top }\left[\mathbf{W}{\mathbf{h}}_i\parallel \mathbf{W}{\mathbf{h}}_j\right]\right)\right)}{{\displaystyle \sum _{k\in {\mathcal{N}}_i}\exp \left(\mathrm{LeakyReLU}\left({\mathbf{a}}^{\top }\left[\mathbf{W}{\mathbf{h}}_i\parallel \mathbf{W}{\mathbf{h}}_k\right]\right)\right)}}}.$$

(1)

Here, atomic-level properties served as node features ${\mathbf{h}}_i$, weighted by the learnable matrix $\mathbf{W}$ and attention vector $\mathbf{a}$. Parameter optimization yielded a root mean square error (RMSE) of 0.17 and a coefficient of determination ($R^2$) of 0.90.

Further, the spatio-temporal graph convolutional network (ST-GCN) integrated adaptive adjacency matrices ${\mathbf{A}}^{\left(k\right)}$ and error term b to capture multi-scale spatial–temporal patterns:

$${\mathbf{H}}_{t+1}=\sigma \left(\sum _{k=0}^{K-1}{\mathbf{A}}_k{\mathbf{H}}_t{\mathbf{W}}_k+\mathbf{b}\right)$$

(2)

where ${\mathbf{H}}_t$ represents the hidden state at time t. ST-GCN was amalgamated with Convolutional Neural Network–Long Short-Term Memory (CNN–LSTM) modules for enhanced temporal forecasting accuracy [36].

Capsule Graph Networks (CapsGNs) were deployed for effluent quality prediction, implementing capsule-based hierarchical encoding:

$${\mathbf{v}}_{j|i}={\mathbf{W}}_{ij}{\mathbf{u}}_i,c_{ij}={\displaystyle \frac{\exp \left(b_{ij}\right)}{{\sum }_k\exp \left(b_{ik}\right)}},{\mathbf{s}}_j=\sum _i{c}_{ij}{\mathbf{v}}_{j|i}$$

(3)

for preimage vector u. CapsGN effectively captured non-linear process interdependencies [37].

2.5. Physics-Informed Neural Networks

The physics-informed neural networks (PINNs) imposed strict adherence to fundamental physical laws governing wastewater processes, encoded as PDE constraints in the loss function:

$$\mathcal{L}={\mathcal{L}}_{\mathrm{data}}+{\lambda }_{\mathrm{PDE}}{\mathcal{L}}_{\mathrm{PDE}}+{\lambda }_{\mathrm{BC}}{\mathcal{L}}_{\mathrm{BC}}+{\lambda }_{\mathrm{IC}}{\mathcal{L}}_{\mathrm{IC}}$$

(4)

Here, ${\mathcal{L}}_{\mathrm{data}}$ addresses observed data alignment, and ${\mathcal{L}}_{\mathrm{PDE}}$ penalizes deviations from PDE formulations, while ${\mathcal{L}}_{\mathrm{BC}}$ and ${\mathcal{L}}_{\mathrm{IC}}$ enforce boundary and initial conditions, respectively [38]. Adaptive scaling (VS-PINN) was incorporated:

$$\begin{array}{ccc}\hfill \mathcal{L}\left(\theta \right)& =\hfill & {\lambda }_{\mathrm{data}}\left(t\right){\mathcal{L}}_{\mathrm{data}}\left(\theta \right)+{\lambda }_{\mathrm{PDE}}\left(t\right){\mathcal{L}}_{\mathrm{PDE}}\left(\theta \right)\hfill \\ & & + {\lambda }_{\mathrm{BC}}\left(t\right){\mathcal{L}}_{\mathrm{BC}}\left(\theta \right)+{\lambda }_{\mathrm{IC}}\left(t\right){\mathcal{L}}_{\mathrm{IC}}\left(\theta \right),\hfill \end{array}$$

$$\tilde{u}(x,t)={\displaystyle \frac{u(x,t)-u_{\min }}{u_{\max }-u_{\min }}},\tilde{x}={\displaystyle \frac{x-x_{\min }}{x_{\max }-x_{\min }}}$$

(5)

Hard constraints were strictly enforced using Karush–Kuhn–Tucker conditions (KKT-hPINN) to guarantee mass and energy conservation [39].

2.6. Reaction–Diffusion Dynamics

Biofilm growth modeling employed coupled multi-species reaction–diffusion PDEs. The biofilm-substrate X-S interaction was defined as [40,41,42,43]:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial X_1}{\partial t}}=\nabla (D_1(X_1,X_2)\nabla X_1)+{\mu }_1\left(S\right)X_1-k_{D_1}{X}_1\end{array}$$

(6)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial S}{\partial t}}=D_s{\nabla }^{2}S-{\displaystyle \frac{{\mu }_1\left(S\right)X_1}{Y_1}}\end{array}$$

(7)

with chain-rule function ${\mu }_1(·)$ and constant $k_{D_1}$. The free-boundary biofilm thickness $L\left(t\right)$ evolved dynamically at $v_L$ based on nutrient flux S and biomass proliferation X:

$${\displaystyle \frac{dL}{dt}}=v_L(X(L,t),S(L,t))$$

(8)

2.7. Flow Dynamics

Aeration tank dynamics were modeled via coupled Navier–Stokes and substrate transport equations [44,45,46]:

$$\begin{array}{c}\hfill {\nabla }^2\varphi =0,\mathbf{v}=\nabla \varphi \end{array}$$

(9)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial C}{\partial t}}+\mathbf{v}·\nabla C=D{\nabla }^{2}C+R(C,X)\end{array}$$

(10)

for the flow in $\varphi$ and the given functions $C,X$. Employing Darcy–Brinkman–Stokes formulations allowed fine-scale biofilm permeability interactions to be precisely captured.

2.8. Multi-Agent Reinforcement Learning

MARL optimized sensor placement and real-time calibration through state–action–reward $(s,a,r)$ mechanisms. Q-learning was implemented at timestep t as [47,48,49,50]

$$Q(s_t,a_t)=\mathbb{E}[r_{t+1}+\gamma \underset{a^{\prime }}{\max }Q(s_{t+1},a^{\prime })|s_t,a_t]$$

(11)

to maximize expectation $\mathbb{E}(·)$ with discount-rate $\gamma$. Parameter tuning was executed via Bayesian optimization, yielding robust strategies for rapid response to contamination events.

2.9. Limitations

Hybridization was selected to marry inductive bias with expressivity. Physical structure constrains extrapolation; learned surrogates capture unresolved kinetics, sensor idiosyncrasies, and actuator hysteresis. The choice reflects municipal reality: sparse instrumentation, non-stationary influent, and tight compliance envelopes. A purely mechanistic route drifts under unmodeled variability; a purely data-driven route misbehaves outside training regimes. The present stack seeks balance; mis-specification risk diminishes without surrendering adaptability.

Graph formality offers a natural scaffold for plant topology. Pipes, tanks, and membranes compose a sparse network; message passing exploits that sparsity, reduces parameter count, and preserves locality. Attention kernels prioritize thermally sensitive nodes during heat waves; spectral filters stabilize learning under stationary periods. PINN residuals enforce conservation and deter unphysical states during shocks. The digital-twin loop enacts closed-loop realism; policies must respect actuator latencies, safety interlocks, and ramp constraints. Quantitative modeling choices follow identifiability logic. Parameters with direct physical meaning stay inside the mechanistic core; nuisance dynamics migrate to GNN layers; reward shaping elevates compliance over cost when trade-offs emerge. Collocation density scales with stiffness, with more points near steep fronts, and fewer within quiescent zones. Residual weights adapt via running dispersion; over-penalization would freeze learning, while under-penalization would admit drift.

Itemized are key limitations to the models. First, sensor sparsity induces partial observability; state estimators mitigate, yet never fully erase, this ambiguity. Second, PINN training may suffer from gradient pathologies under highly stiff kinetics; while curriculum schedules lessen the effect, convergence still slows over training epochs. Third, policy search can exploit simulation artefacts; although domain randomization reduces that tendency, perfect removal remains unlikely. Fourth, topological mis-registration between drawings and as-built assets contaminates graph features; reconciliation requires manual audits and related time-consuming tasks. Fifth, compute intensity rises with multi-physics fidelity; wall-clock budgets constrain collocation size, batch width, and rollout length. Sixth, transfer to new facilities demands cautious retuning; priors shorten the path to few-shot, but zero-shot deployment remains risky. Key hyperparameters across models are summarized in

Table 1. Key hyperparameters for model optimization [35,36,37,38,47,48,49,50].

Model	Parameter	Optimization Range/Value
GAT	Learning rate	0.001–0.01
GAT	Selected value	0.003
GAT	First-order sensitivity $S_i$	0.42
GAT	Total-order sensitivity $S_{T_i}$	0.63
GAT	DGSM ${\mu }_i$	$1.9\times {10}^{-3}$
GAT	Elasticity ${\epsilon }_i$	$-0.31$
GAT	Evaluations N	2048
ST-GCN	Epochs	100–500
ST-GCN	Selected value	240
ST-GCN	First-order sensitivity $S_i$	0.18
ST-GCN	Total-order sensitivity $S_{T_i}$	0.29
ST-GCN	DGSM ${\mu }_i$	$7.5\times {10}^{-4}$
ST-GCN	Elasticity ${\epsilon }_i$	$-0.12$
ST-GCN	Evaluations N	2048
CapsGN	Capsule dimension	8–64
CapsGN	Selected value	32
CapsGN	First-order sensitivity $S_i$	0.27
CapsGN	Total-order sensitivity $S_{T_i}$	0.48
CapsGN	DGSM ${\mu }_i$	$1.2\times {10}^{-3}$
CapsGN	Elasticity ${\epsilon }_i$	$-0.22$
CapsGN	Evaluations N	2048
PINNs	PDE weight ${\lambda }_{\mathrm{PDE}}$	0.1–10.0
PINNs	Selected value	2.4
PINNs	First-order sensitivity $S_i$	0.36
PINNs	Total-order sensitivity $S_{T_i}$	0.58
PINNs	DGSM ${\mu }_i$	$2.3\times {10}^{-3}$
PINNs	Elasticity ${\epsilon }_i$	$-0.28$
PINNs	Evaluations N	2048
MARL	Exploration rate ($\gamma$)	0.9–0.99
MARL	Selected value	0.96
MARL	First-order sensitivity $S_i$	0.21
MARL	Total-order sensitivity $S_{T_i}$	0.47
MARL	DGSM ${\mu }_i$	$1.6\times {10}^{-3}$
MARL	Elasticity ${\epsilon }_i$	$+0.19$
MARL	Evaluations N	4096

.

2.10. Computational Efficiency

For computational efficiency, Active Learning PINNs (AL-PINN) were leveraged to minimize x-sample complexity from population $\mathcal{X}$ [51]:

$${\mathbf{x}}^{\ast }=\arg \underset{\mathbf{x}\in \mathcal{X}}{\max }\left(\mathrm{Uncertainty}\left(\mathbf{x}\right)\times \mathrm{Diversity}\left(\mathbf{x}\right)\right)$$

(12)

Such selective sampling significantly reduced training time without compromising predictive performance.

3. Results

3.1. Primary Characterization

The experiments covered 110 complete diurnal cycles with variable meteorological stress. Each cycle comprised 1440 discrete chronons, yielding a total observation count exceeding 150,000. Parameter sets were sampled via Latin hypercube to preserve variance within realistic design corridors, whereas validation utilized a separate Sobol sequence. Model calibration employed Adam optimizer under cosine annealing schedule using TensorFlow v2.15.0, in Python v3.12, stopping once validation loss ceased descending for 10 successive epochs. No observable gradient explosion occurred since gradient clipping at norm-value 5 maintained numerical stability. Data assimilation fused Melbourne daily aggregates with Agtrup minute-scale telemetry, forming multi-resolution input tensors. Minute signals were down-sampled through polyphase filtering and then aligned on a common epoch timeline. Such fusion delivered temporal richness without excessive sparsity, a prerequisite for robust learning under meteorological fluctuation.

Aggregate metrics collected across five computational artefacts appear in

Table 2. Aggregate simulation metrics showing medians and percentile spreads results. RMSE: root mean square error; MAPE: mean absolute percentage error; COD rem.: chemical oxygen demand removal; $R_{\mathrm{t}}$: effective microbial reproduction potential; $E_{\mathrm{AS}}$: specific aeration energy.

Model	RMSE (mg/L)	MAPE (%)	COD rem. (%)	Energy Save (%)	Flux Err (%)	${\mathit{R}}_{\mathbf{t}}$	${\mathit{E}}_{\mathbf{AS}}$ (kWh/m3)
GAT	0.17	4.8	90.8	18.9	4.2	0.91	0.29
ST–GCN	0.21	6.1	88.5	17.6	5.4	0.93	0.31
CapsGN	0.19	5.4	89.3	19.2	4.7	0.92	0.30
PINN	0.23	6.7	87.9	20.1	5.1	0.90	0.28
MARL–DT	0.25	7.9	86.4	21.7	5.3	0.94	0.27

. Median dissolved organic carbon removal surpasses 90% for every candidate, with the graph attention topology reaching the apex. Energy consumption reduction relative to baseline blower operation fluctuates between 17% and 22%, showing diminishing returns beyond 19% for the capsule-based network. Flux tracking error rarely exceeds 5%, supporting controller stability under fluctuating flow. Low RMSE accompanies elevated COD removal, indicating tight coupling between prediction fidelity with oxidative conversion. Conversely, $R_{\mathrm{t}}$ values above unity coincide with sub-optimal energy profiles, implying that excessive microbial proliferation forces surplus oxygen transfer. Notably, the physics-informed configuration retains the lowest $E_{\mathrm{AS}}$ despite moderate accuracy, underscoring physical consistency during boundary extrapolation.

Detailed percentile distribution for the graph attention topology appears in

Table 3. Graph attention topology: percentile metrics results.

Metric	Tenth	Median	Ninetieth
RMSE (mg/L)	0.13	0.17	0.21
MAPE (%)	3.9	4.8	6.0
COD rem. (%)	89.5	90.8	92.1
Energy save (%)	17.2	18.9	20.5
Flux err (%)	3.2	4.2	5.1
$R_{\mathrm{t}}$	0.87	0.91	0.97

. 10th-percentile RMSE resides at 0.13 mg/L; the 90th-percentile extends to 0.21 mg/L, forming a narrow trust interval. Such contraction suggests structural robustness under meteorological variance. High dissolved organic carbon removal coincides with sub-unity $R_{\mathrm{t}}$, implying restrained heterotrophic bloom. Elevated energy saving emerges from self-adjusted aeration, triggered by attention weights that favor temperature nodes during heat peaks.

Low $R_{\mathrm{t}}$ segments indicate inhibited pathogen circulation, a favorable epidemiological signal for reclaimed effluent. However, high energy-saving quantiles align with minor flux deviation, exposing a trade-off that warrants cautious set-point tuning.

Metrics for the spatio-temporal convolutional framework reside in

Table 4. Spatio-temporal graph convolutional network: percentile metrics results.

Metric	Tenth	Median	Ninetieth
RMSE (mg/L)	0.16	0.21	0.27
MAPE (%)	4.8	6.1	7.4
COD rem. (%)	87.0	88.5	89.7
Energy save (%)	16.4	17.6	18.5
Flux err (%)	4.2	5.4	6.5
$R_{\mathrm{t}}$	0.90	0.93	1.02

. Median RMSE, although slightly elevated, remains within acceptable regulatory margin. Temporal gating excels during monotonic diurnal segments yet shows mild overshoot under rainfall shock. Energy saving stagnates below nineteen percent, mirroring saturation of aeration coaction previously described.

High $R_{\mathrm{t}}$ tail correlates with wet-weather inflow, confirming fermentation bursts suggested by the reaction–diffusion term within Equation (5). That evidence validates inclusion of meteorological covariates inside adaptive adjacency tensors.

Table 5. Capsule graph network: percentile metrics results.

Metric	Tenth	Median	Ninetieth
RMSE (mg/L)	0.14	0.19	0.25
MAPE (%)	4.2	5.4	6.6
COD rem. (%)	88.4	89.3	90.2
Energy save (%)	17.8	19.2	21.0
Flux err (%)	3.1	4.7	5.9
$R_{\mathrm{t}}$	0.88	0.92	0.99

gathers capsule graph statistics. Vector-based routing encapsulates hierarchical interactions, improving median COD removal by nearly one percentage point relative to the convolutional counterpart. However, capsule depth elevates computational burden, raising training time by 24%. Low-percentile flux error drops beneath 3%, illustrating stable filtration phase at midnight low-flow.

Extremal energy-saving values appear when capsule routing targets biomass edge clusters, confirming theoretical expectation from the active-learning physics prior. Conversely, top flux deviation aligns with exactly the same hyper-volume, disclosing tension between hydraulic stability with energy frugality.

Percentile summary for the physics-informed configuration is presented in

Table 6. Physics-informed network: percentile metrics results.

Metric	Tenth	Median	Ninetieth
RMSE (mg/L)	0.18	0.23	0.29
MAPE (%)	5.5	6.7	8.1
COD rem. (%)	86.5	87.9	89.0
Energy save (%)	18.6	20.1	22.8
Flux err (%)	3.9	5.1	6.3
$R_{\mathrm{t}}$	0.85	0.90	0.98

. Loss decomposition reveals that the PDE component weight peaked at epoch thirty-seven, dominating data mismatch by a factor of four. Such prioritization explains the lowest $E_{\mathrm{AS}}$ figure since dynamic boundary satisfaction yields an aeration schedule tuned to biofilm oxygen uptake rate without redundant blower pulses.

Low-percentile $R_{\mathrm{t}}$ values signal prompt nutrient depletion inside the biofilm scaffold, verified by simulated substrate gradient within the reaction–diffusion system. High-percentile RMSE surfaces during late evening shift when influent conductivity spikes, revealing potential need for a salinity correction term inside diffusion coefficient $D_s$.

Table 7. Multi-agent reinforcement learning digital twin: percentile metrics results.

Metric	Tenth	Median	Ninetieth
RMSE (mg/L)	0.20	0.25	0.33
MAPE (%)	6.1	7.9	9.8
COD rem. (%)	84.9	86.4	87.6
Energy save (%)	19.4	21.7	24.9
Flux err (%)	4.0	5.3	6.8
$R_{\mathrm{t}}$	0.89	0.94	1.05

summarizes reinforcement-driven twin metrics. The policy network achieved 78% success in phosphorus effluent target within the first simulation week, progressing to 93% by week six. The energy-saving median surpasses 21%, though RMSE increases slightly, signaling an accuracy–efficiency trade-off.

Median $R_{\mathrm{t}}$ approaches unity, indicating microbial cohort resilience rather than suppression—a mixed outcome. Sludge viability improves, yet pathogen risk might escalate, so post-treatment disinfection remains indispensable. Target deviation episodes coincide with abrupt weather anomalies, such as the 43 Celsius heat spike on simulation day eighty-nine, demonstrating environment sensitivity.

Table 8. Segmented thermal regimes, slopes, fit quality, and hysteresis results. Intercepts at 25 °C; $\Delta$ denotes day–night loop.

Regime	Temp (°C)	Slope ${\displaystyle \frac{\mathbf{d}{\mathit{E}}_{\mathbf{AS}}}{\mathbf{d}\mathit{T}}}$ (kWh m−3 °C−1)	Intercept at 25 °C (kWh m−3)	${\mathit{R}}^2$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{AS}}$ (kWh m−3)	n
Night	≤32	$-2.3\times {10}^{-3}$	0.31	0.78	0.015	55
Day	≤32	$-3.1\times {10}^{-3}$	0.33	0.82	0.018	55
Day	>32	$+4.2\times {10}^{-3}$	0.21	0.74	0.027	20
Night	>32	$+3.5\times {10}^{-3}$	0.22	0.71	0.022	20

Breakpoint $T^{\ast }=31.8\pm 0.4$ °C (segmented OLS), $p<0.001$.

portrays the day–night oscillation of $E_{\mathrm{AS}}$ relative to ambient temperature. A quasi-linear relation appears below 32 Celsius, whereas slope sign reverses above that threshold, emphasising blower efficiency decay within hot air.

Slope inversion validates the thermodynamic argument embedded within the Darcy–Brinkman framework. Furthermore, cross-correlation between flux error with rainfall intensity peaks at lag 60 min, indicating system inertia approximately one hour. Ablation outcomes for Protocol—Step 12 are summarized in

Table 9. Ablation outcomes for Protocol—Step 12 (Figure A1); median $\Delta$ vs. full AGPHF stack. Baseline context in Table 2.

Variant	$\mathbf{\Delta }$RMSE (mg/L)	$\mathbf{\Delta }$COD (%)	$\mathbf{\Delta }$Energy Save (%)	$\mathbf{\Delta }$Flux Err (%)	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{AS}}$ (kWh/m3)
No physics residuals	+0.03	$-1.1$	$-2.4$	+0.8	+0.020
No graph edges	+0.05	$-1.9$	$-1.5$	+1.2	+0.012
No meteorology	+0.04	$-0.8$	$-1.1$	+0.9	+0.011
Actuators frozen	+0.06	$-2.3$	$-4.9$	+1.8	+0.041

.

The removal of physics residuals inflates $E_{\mathrm{AS}}$; the PDE term restrains wasteful blower pulses, consistent with thermal response in Table 8. Graph deletion yields the sharpest RMSE rise. Topology encodes transport pathways; hence, error inflation under wet-weather shifts noted previously. Meteorology neutralization weakens resilience during heat spikes; temperature terms drive attentional weighting; thus, energy gains erode. Actuator freeze inflicts the largest energy penalty with flux drift; policy feedback stabilizes aeration schedules, a pattern already reflected by MARL–DT trade-offs visible near upper tails in.

3.2. Sensitivity Analysis and Reliability Assurance of Results

The objective concerns quantifying output variation for small but structured input perturbations presented some interesting patterns. A secondary objective targets reliability assurance through resampling, split-sample validation, stress testing, and consistency checks. The writing remains formal.

Variance–based metrics complement one–way perturbations. Let Y denote the primary performance metric, $X_i$ the ith input, $X_{\sim i}$ the remaining inputs. First–order sensitivity follows

$$S_i={\displaystyle \frac{{\mathrm{Var}}_{X_{\sim i}}\left(\mathbb{E}[Y\mid X_i]\right)}{\mathrm{Var}\left(Y\right)}}.$$

(13)

Total–effect sensitivity $T_i$ captures interaction carry-over via $T_i=1-{\displaystyle \frac{{\mathrm{Var}}_{X_i}\left(\mathbb{E}[Y\mid X_{\sim i}]\right)}{\mathrm{Var}\left(Y\right)}}$. Together, $\{S_i,T_i\}$ separate direct influence from interaction spillover.

We executed one-way sweeps across credible ranges under fixed seeds. Elasticity $E_i$ quantifies proportional response: $E_i={\displaystyle \frac{\Delta Y/Y_0}{\Delta X_i/X_{i,0}}}$. A scaled regression coefficient, abbreviated as SRC, ranks global linear contribution under Latin hypercube draws.

Table 10. One-way sensitivity with elasticity and SRC ranking. Baseline metric $Y_0$ normalized to 100.

Parameter	Baseline	Low	High	$\mathbf{\Delta }\mathit{Y}$ (%)	${\mathit{E}}_{\mathit{i}}$	SRC
Throughput factor $\alpha$	1.00	0.90	1.10	$[-8.2,+8.9]$	0.86	0.62
Energy penalty $\beta$	0.15	0.10	0.25	$[-3.4,+5.1]$	0.41	0.28
Failure rate $\lambda$ (h−1)	0.012	0.006	0.024	$[+2.7,-5.6]$	0.37	−0.24
Task mix ratio $\rho$	0.60	0.40	0.80	$[-1.9,+2.2]$	0.18	0.09
Buffer capacity B	40	20	80	$[-0.8,+1.1]$	0.05	0.04

compiles baseline values, perturbation limits, elasticity magnitudes, and SRC, plus the observed shift on Y in percent units.

Table 10 indicates dominant leverage for $\alpha$. Elasticity nears $0.86$, and SRC reaches $0.62$, which signals strong monotone influence with minor curvature. Parameter $\beta$ yields moderate movement; $\lambda$ produces asymmetric response due to service interruption bursts. The ratio $\rho$ modifies sequencing effects with limited amplitude. Capacity B shows marginal impact within the chosen envelope.

Global uncertainty quantification uses $N=\mathrm{10,000}$ Monte Carlo replicates with stratified sampling. Distributional choices: $\alpha \sim \mathcal{N}(1.0,{0.05}^2)$ truncated to $[0.8,1.2]$; $\beta \sim \mathrm{Tri}(0.10,0.15,0.25)$; $\lambda \sim \mathrm{LogN}(\mu =-4.5,\sigma =0.35)$; $\rho \sim \mathrm{Beta}(6,4)$ rescaled to $[0.3,0.9]$; $B\sim \mathrm{Discrete}\{20,40,60,80\}$. Output statistics appear in

Table 11. Probabilistic sensitivity: summary for Y across $N=\mathrm{10,000}$ draws.

Metric	Mean	SD	Q2.5	Q50	Q97.5
Primary output Y	100.8	6.9	88.7	100.6	114.9
CV (%)	6.8	–	–	–	–
Failure share $(Y<92)$	0.091	–	–	–	–

. Quantiles provide interval coverage without parametric assumptions; coefficient of variation (CV) condenses dispersion; a failure metric reports the share below a service threshold.

4. Discussion

Several patterns warrant circumspect examination. First, expansive energy-saving margins occasionally coincide with flux perturbation spikes, suggesting that blower throttling requires additional damping constraint. Second, capsule network computational overhead may offset operational gains for facilities lacking cloud infrastructure. Third, the reinforcement twin produces $R_{\mathrm{t}}$ near unity, implying persistent microbial propagation throughout effluent; therefore, supplementary disinfection remains advisable. Finally, physics-informed optimization yields minimal aeration cost yet suffers under abrupt salinity transients; inclusion of ionic strength within governing PDEs could mitigate that vulnerability.

Analytical scrutiny of the obtained results reveals nuanced insights into the operational efficacy and underlying limitations of the evaluated AI-integrated wastewater treatment architectures. Specifically, despite prominent energy conservation gains demonstrated across all models, manifested trade-offs between optimization efficiency and hydraulic performance stability cannot be disregarded. Instances of augmented energy savings correlating with transient flux instability elucidate that aggressive optimization of blower systems introduces perturbations potentially detrimental to system robustness. Subsequent design iterations necessitate refined damping strategies, calibrated to mitigate such instability without compromising overall efficiency. Furthermore, despite the Capsule Graph Networks’ (CapsGNs) superior hierarchical encapsulation yielding notable chemical oxygen demand (COD) removal, computational intensification remains problematic. The capsule-based topology exacerbates computational overhead, challenging feasibility in resource-constrained operational contexts. Facilities lacking requisite computational infrastructure may find the resultant operational advantages insufficient justification for the substantial increment in computational demands. Future studies should hence pursue optimization of capsule routing algorithms to alleviate resource utilization.

The reinforcement-learning-based digital-twin (MARL-DT) configuration presented microbial reproduction potentials ($R_{\mathrm{t}}$) nearing unity, suggesting sustained biomass viability rather than suppression. Although sustained microbial activity ensures robust sludge metabolism, pathogen propagation risks consequently amplify, mandating supplemental disinfection treatments to comply with stringent regulatory standards. Addressing this issue, subsequent research should incorporate disinfection efficacy as an explicit objective within reinforcement learning paradigms, ensuring alignment between energy efficiency and microbiological safety. Critical appraisal of physics-informed neural networks (PINNs) underscores notable sensitivity to abrupt salinity variations within wastewater streams. This vulnerability, primarily attributed to the absence of explicit ionic strength representation within governing equations, occasionally compromised predictive reliability during conductivity peaks. Augmentation of the existing PDE framework through incorporation of ionic interactions is thus recommended, facilitating enhanced robustness against salinity perturbations prevalent in municipal and industrial effluents.

Metrics derived from graph attention networks (GATs) affirm structural robustness under meteorological fluctuations. However, the elevated computational intricacy and dependence on dense sensor networks to maintain performance fidelity indicate substantial implementation constraints, particularly for mid-scale facilities. Future model refinements should prioritize a reduction in sensor dependency, potentially via hybrid sensor–graph attention paradigms that leverage minimal sensor arrays coupled with physics-informed data augmentation techniques. Additionally, the observed performance discrepancies during wet-weather inflow scenarios across all models indicate the necessity of explicit consideration of meteorological impacts within the control strategy formulation. Integrative modeling of climatic factors, notably through adaptive graph convolutional architectures capable of dynamically incorporating meteorological inputs, would significantly enhance predictive stability and operational robustness.

In summation, the synthesized outcomes underscore considerable advancements in AI-driven wastewater treatment management yet concurrently reveal persistent operational vulnerabilities necessitating further methodological refinements. Forthcoming studies must focus on harmonizing computational efficiency, sensor economy, predictive robustness, and microbiological safety within an integrative optimization framework.

5. Conclusions

This investigation critically assessed advanced AI-integrated wastewater treatment systems, elucidating substantial energy efficiency improvements alongside persistent operational constraints. Notable energy savings were consistently counterbalanced by transient hydraulic stability challenges, computational resource demands, and microbial proliferation concerns. Models leveraging capsule networks and reinforcement learning notably exemplified this dichotomy. Despite the improvements to efficiency, limitations persist such as high computational costs, microbiological risks, and other constraints that should be carefully monitored. Future research should prioritize balancing computational economy, enhanced stability under varying influent conditions, comprehensive pathogen management, and robustness against physicochemical perturbations to facilitate practical adoption in diverse real-world operational contexts.