Water Quality Anomaly Detection Method Based on Attention-Gated Liquid Neural Network

Water quality monitoring is essential for protecting public health, sustaining aquatic ecosystems, and ensuring compliance with environmental regulations. Traditional water quality models, such as the Soil and Water Assessment Tool (SWAT), the mechanistic QUAL model, and statistical forecasting approaches, often struggle to capture the highly nonlinear, multivariate, and temporally irregular nature of water systems influenced simultaneously by chemical, biological, hydrological, and meteorological factors^1,2,3,4,5,6. Therefore, improving anomaly detection in water-quality data remains a pressing challenge.

In practical terms, the proposed framework is designed for multi-parameter water quality sensing networks that measure variables such as potential of hydrogen (pH), dissolved oxygen (DO), total nitrogen (TN), total phosphorus (TP), and permanganate index (COD_Mn), turbidity, and conductivity, with typical sampling intervals ranging from 15 to 60 min. This model tolerates irregular sampling and missing observations, making it suitable for real-world deployment in surface water and reservoir monitoring systems.

Application of machine learning in water quality anomaly detection

In recent years, machine learning (ML) and deep learning (DL) methods have gained traction for water quality applications. For example, Wang et al.⁷ introduced a Long Short-Term Memory Autoencoder with Attention (LSTMA-AE) combined with mechanistic constraints to improve accuracy and reduce false alarms in anomaly detection of water injection pump operations, achieving significantly better performance than the interpolation, random forest, or LSTM-AE methods.

Similarly, Zhao et al.⁸ proposed a Gated Recurrent Unit with Physics-Informed Neural Network (GRU-PINN) model that incorporates physical constraints into the loss function, significantly boosting interpretability and the F1-score in water quality anomaly detection tasks. ElShafeiy et al.⁹ developed a Multivariate Convolutional Network-LSTM (MCN-LSTM) architecture, combining Multivariate Convolutional Networks with LSTM for real-time anomaly detection in water quality sensor data, demonstrating enhanced detection capabilities in field environments. Another advancement is the Gated-Liquid Neural Network (Gated-LNN) model¹⁰, which fuses gating mechanisms into Liquid Neural Networks to accurately predict the Water Quality Index (WQI) and classify water quality, achieving R² ≈ 0.9995 and classification accuracy of 99.74% on Indian datasets.

Time series anomaly detection

Beyond DL-specific applications, time-series anomaly detection has seen rapid development across various domains. Time-series anomaly detection has become a key technique for improving water quality monitoring and early warning. Traditional statistical methods often fail to capture the nonlinear and dynamic characteristics of water quality data, thereby prompting the application of deep learning models. For instance, Zhang et al.¹¹ combined Empirical Mode Decomposition (EMD) with LSTM to better handle non-stationary signals, significantly improving prediction accuracy of Chemical Oxygen Demand (COD), Biochemical Oxygen Demand (BOD₅), Total Phosphorus (TP), Total Nitrogen (TN), and Ammonia Nitrogen (NH₃-N). Building on this, Wang et al.¹² proposed an LSTM-based fluctuation analysis method using Approximate Entropy, which enhances real-time anomaly detection performance.

Recent advances have emphasized attention mechanisms and hybrid frameworks. Arepalli et al.¹³ introduced a lightweight spatially shared attention LSTM for hypoxia detection in aquaculture, achieving 99.8% accuracy, while Zhang et al.¹⁴ showed that spatial and temporal attention significantly improved CNN-LSTM prediction of Dissolved Oxygen (DO) and NH₃-N. Similarly, Long et al.¹⁵ integrated Complete Ensemble Empirical Mode Decomposition with Adaptive Noise-Variational Mode Decomposition (CEEMDAN-VMD) with attention-enhanced LSTM, achieving Nash-Sutcliffe Efficiency (NSE) values up to 0.99 across multiple indicators. At the application level, Xie et al.¹⁶ demonstrated a mobile LSTM-sequence-to-sequence (Seq2Seq) system for operational, real-time water quality prediction across river basins.

Overall, these studies highlight a clear trend in LSTM-based baselines, attention-augmented hybrid models, and system-level applications. Although accuracy has improved substantially, challenges remain in interpretability, robustness under noisy conditions, and generalization across diverse monitoring networks.

Application of graph neural networks and spatiotemporal models in environmental monitoring

Graph neural networks (GNNs) and spatiotemporal deep learning models have recently demonstrated significant potential in environmental monitoring by capturing both spatial correlations among monitoring sites and temporal dependencies in water quality dynamics. For example, Wu et al.¹⁷ proposed a pre-training enhanced Spatio-Temporal Graph Neural Network (PT-STGNN) for wastewater treatment plants, integrating transformer-based pre-training and graph structure learning to improve long-term prediction of COD, NH₃-N, TP, TN, pH, and flow rate. Similarly, Wan et al.¹⁸ developed a Spatio-Temporal Feature GNN (STF-GNN) that combined graph convolution, GRU, and attention mechanisms, significantly improving Dissolved Oxygen (DO) and TN predictions while demonstrating robust cross-basin generalization.

Hybrid frameworks have emerged beyond purely data-driven designs to enhance generalization. Mu et al.¹⁹ introduced the spatiotemporal graph physics-informed neural network (ST-GPINN), which embeds hydraulic principles into Graph Neural Network (GNN)-based models for water distribution systems. By coupling graph representations with physics-informed constraints, ST-GPINN achieves state-of-the-art accuracy while scaling effectively from small to large networks. Together, these studies highlight the growing role of spatio-temporal GNNs in environmental monitoring, advancing predictive accuracy, interpretability, and robustness, and laying the foundation for next-generation intelligent water management systems.

Advantages and potential of liquid neural networks

Liquid Neural Networks (LNNs), also known as Liquid Time-Constant networks, uniquely model continuous-time dynamics using input-dependent, learnable time constants. Hasani et al.²⁰ introduced LNNs and demonstrated their expressive power, stability, and efficiency in time-series tasks. Their brain-inspired adaptability makes them promising candidates for modeling dynamic environmental phenomena such as water quality, yet their integration with attention mechanisms remains underexplored.

Attention mechanisms, especially attention gates, are powerful tools for focusing on relevant features and enabling interpretability. Although widely used in medical imaging, Attention U-Net sets a foundation for selective feature refinement via learnable attention gates. Attention has also been applied in sensor-based anomaly detection frameworks, such as LSTMA-AE⁷ and GRU-PINN⁸, enhancing sensitivity to critical patterns and improving model transparency.

Taken together, these observations indicate the need for a hybrid architecture that synergizes continuous-time adaptability with interpretable attention mechanisms. Hence, we propose an Attention-Gated Liquid Neural Network (AG-LNN) that integrates the dynamic modeling capabilities of Liquid Neural Networks (LNNs) with attention gates that focus on salient inputs or temporal segments. This architecture aims to enhance robustness to sensor noise and irregular sampling, improve sensitivity to short-duration anomalies, and offer interpretability through attention heatmaps and adaptive time-constant trajectories. The AG-LNN was evaluated on real-world multivariate water quality datasets and benchmarked against LSTM, Transformer, pure LNN, and traditional ML baselines. Both quantitative (Precision, Recall, F1, AUC) and qualitative (attention visualizations, time-constant dynamics) analyses demonstrated the effectiveness and transparency of the proposed model.

NOTE: The overall workflow of this study is shown in Figure 1.

1. Data acquisition

1. Data source identification
   1. Obtain water quality time-series data from the China National Environmental Monitoring Center (CNEMC) public database (https://www.cnemc.cn/sssj/). Retrieve continuous national-level monitoring records covering major lakes and reservoirs in China. Key physicochemical indicators included pH, dissolved oxygen (DO), total phosphorus (TP), total nitrogen (TN), permanganate index (COD_Mn), turbidity (NTU), and electrical conductivity (EC). 
   2. Ensure that the corresponding metadata, such as monitoring time, station operational status, and water quality classification (Class I-V), are included in the downloaded dataset.
2. Dataset retrieval and file specification
   1. Ensure data reproducibility by consolidating and sharing the processed dataset under the filename cnemc_data_2019_2024. csv. The file in the Baidu Cloud repository (https://pan.baidu.com/s/1ZXYxkCZAnx7tiOHJhuJ2Qg?pwd=u97h) enables public access and verification of all experiments.
   2. Include monitoring data from 13 provinces (Guangdong, Hubei, Jiangsu, Zhejiang, Anhui, Fujian, Jiangxi, Henan, Sichuan, Guizhou, Liaoning, Heilongjiang, and Jilin) to ensure representative coverage across diverse hydrological and climatic zones of North, South, East, West, and Plateau regions.
   3. Assign an equal number of monitoring records to each province, yielding a total of 1,832 entries collected between January 2019 and December 2024. Label each station using province-based identifiers (e.g., GD001 for Guangdong, HB001 for Hubei). For every record, include descriptive attributes such as basin, waterbody_type (Lake or Reservoir), and season (Spring, Summer, Autumn, or Winter, derived from sampling time).
   4. Add optional geolocation fields (longitude and latitude) to facilitate spatial analysis and reproducibility. Complete field definitions, data types, and units used in cnemc_2019_2024. The CSV values are summarized in Table 1.

2. Variable selection and labeling

1. Variable selection
   1. Select variables that represent the most frequently monitored physicochemical parameters of surface water, ensuring ecological relevance and sensitivity to pollution or environmental disturbances. Specifically, include the following indicators:
      pH, reflecting the acid-base balance of water and influencing biological activity and chemical solubility
      Dissolved Oxygen (DO), a direct indicator of aquatic ecosystem health and organic pollution
      Total Phosphorus (TP) and Total Nitrogen (TN), two major nutrients linked to eutrophication processes
      Permanganate Index (COD_Mn), representing the oxidizable organic matter content and an indirect measure of organic pollution
      Turbidity (Tur), which reflects suspended particles and sediment load
      Electrical Conductivity (EC), indicating the concentration of dissolved ions and overall salinity
   2. Select these variables not only for their routine availability from CNEMC monitoring stations but also for their demonstrated relevance in prior studies^10,11,14,17 on water quality assessment and anomaly detection.
2. Labeling strategy
   1. Define anomaly labels to support model training and evaluation. Apply a dual-labeling strategy that combines the two complementary approaches.
   2. Regulatory or expert-based labeling: Adopt officially documented or annotated abnormal events (e.g., sudden pollutant discharges, algal blooms, or instrument failures) from the monitoring authority as ground truth anomalies.
   3. Data-driven weak labeling: Infer anomalies for monitoring intervals lacking explicit expert annotations directly from the CNEMC multivariate time-series dataset by applying a statistically guided, season-aware weak labeling framework.
   4. Structure this process into three main components: performing seasonal normalization, applying quantile-based deviation detection, and reinforcing the results through residual-based evaluation.
      1. Seasonal normalization: Normalize seasonal effects for each variable  x_i,t (e.g., DO, COD_Mn, TP, TN, Turbidity) using the explicit season field (Spring, Summer, Autumn, Winter) provided in the CNEMC dataset. Standardize each variable within its corresponding seasonal subset, as shown in Equation (1).
         (1)
         where μ_i^(s) and σ_i^(s) are the mean and standard deviation of variable i during season s. This normalization compensates for systematic seasonal shifts such as reduced DO in summer or increased COD_Mn during the rainy season, ensuring cross-season comparability.
      2. Quantile-based weak labeling: After normalization, identify anomalies by applying robust quantile thresholds computed for each variable within the same season and province group, as shown in Equation (2).
            (2)
         where Q_p (⋅) is the p-th quantile operator, is the lower and upper quantile thresholds (1st and 99th percentiles) for variable i in season s. Flag a data point as anomalous if it exceeds these bounds, as shown in Equation (3).
         (3)
         This step adaptively accounts for regional and seasonal differences, thereby avoiding false detections caused by natural hydrological fluctuations.
      3. Residual-based reinforcement: Establish a rolling 15-day median baseline to detect abrupt short-term deviations that quantile thresholds might miss, as expressed in Equation (4).
         (4)
         where r_i,t is the residual deviation of variable i at time t, median(x_i,t-15:t+15) is the 15-day rolling median baseline around t. Flag a data point as a residual anomaly if the absolute residual exceeds three times the seasonal standard deviation (|r_i,t| > 3σ_i^(s)). This residual reinforcement effectively captured the transient spikes caused by pollutant discharge, rainfall events, or temporary sensor faults.
      4. Combined weak-label decision: Fuse both quantile-based and residual-based anomaly indicators to assign a binary weak label y_t, as defined in Equation (5).
            (5)
         where y_t=1 indicates that at least one parameter exhibits anomalous behavior at time t, y_t = 0 denotes normal observations across all parameters.
         Generate a set of data-driven weak annotations across all 13 provinces and five climatic regions in the CNEMC dataset using the combined labeling procedure. Integrate these labels into confidence-weighted learning by assigning a confidence weight of 0.7 to weak labels relative to expert-verified annotations with a weight of 1.0.
   5. Proportion of annotation types
      1. Use approximately 42% expert-verified annotations derived from official environmental reports, and 58% weak annotations generated from continuous monitoring data. Apply a heuristic fusion of z-score thresholding and seasonal trend decomposition (STL) to detect potential anomalies in multivariate water quality time series. The chosen annotation proportions are validated through experiments presented in Section 5.1 (Effect of annotation ratio) of the Results part. These results confirm that an empirically optimal balance occurs around a 40:60 expert-to-weak ratio, which is consistent with the actual dataset composition.
   6. Validation of weak annotation accuracy
      1. Ensure the reliability of weakly annotated samples by cross-checking a random subset of 800 weakly labeled instances with environmental experts. Record the agreement rate between weak annotations and expert judgments, which should reach approximately 87.5%. Remove samples with low anomaly confidence (< 0.6) or disagreement with the expert validation set from the training data.
      2. Maintain the retained weak annotations at an effective accuracy above 90%. Examine the reliability of weak annotations and their influence on model performance, where controlled experiments confirm that preserving weak-label accuracy above 90% effectively prevents training bias and ensures stable predictive behavior.
   7. Bias mitigation
      1. During training, apply a confidence-weighted learning strategy to mitigate potential bias introduced by weak annotations. Define the loss function as shown in Equation (6).
         (6)
         where y_i∈{0,1} denotes the ground-truth label of the i-th sample, p_i represents the model’s predicted probability, w_i is the weight assigned to each sample, and N is the total number of samples.
      2. Determine the sample weight w_i based on the source and confidence level of the label, as shown in Equation (7).
         (7)
         Here, c_i∈[0,1] denotes the confidence score of the i-th annotation, and α,β are the global weighting coefficients for expert and weak labels, respectively. Empirical analysis demonstrated that setting α=1.0 and β=0.7 achieves the best trade-off between performance and stability, as this weighting effectively suppresses residual noise from weak labels while maintaining sufficient contribution from their broader coverage.
      3. Select seven physicochemical parameters (pH, DO, TP, TN, COD_Mn, Tur, and EC) as the core variables for anomaly detection. Use these indicators to comprehensively represent nutrient status, organic pollution, oxygen availability, and ionic composition of surface water. Provide a detailed description of each variable.

3. Data quality control and preprocessing

1. Establish a hierarchical data quality control and preprocessing pipeline to ensure the reliability and modeling applicability of multi-source water quality monitoring data. Sequentially perform time alignment, missing value imputation, anomaly detection, scale normalization, and windowed sample construction. Dynamically select and combine appropriate methods according to the specific characteristics of the data.
2. Time alignment and resampling: Align all monitoring station sequences to a unified 15 min temporal resolution to account for unsynchronized timestamps. For irregularly sampled points, apply linear interpolation between adjacent observations according to Equation (8). Ensure that the subsequent windowed samples are constructed on a consistent temporal grid.
   (8)
3. Missing value processing
   1. Categorize missing values based on the length of consecutive gaps into short-term missing (<6 consecutive points, i.e., <1.5 h) and long-term missing (≥6 consecutive points), and different strategies were applied accordingly.
   2. Apply Kalman filtering and smoothing to perform dynamic estimation for long-term missing intervals (≥ 6 consecutive points). Model each water quality variable x_t (e.g., DO, TN, COD_Mn) using a discrete linear state-space formulation, as shown in
      Equations (9) to (11).
      (9)
      (10)
      (11)
      where x_t is the latent (true) state of the water quality variable at time t, y_t is the corresponding observed value; A is the state transition coefficient (empirically set to 0.95) to reflect a high temporal correlation in 15 min observations, H is the observation matrix (set to 1 for direct measurements),  w_t ∼ N(0,Q) and v_t ∼ N(0,R) are the zero-mean Gaussian process and measurement noise, respectively; Q = (0.05σ_x)² and R=(0.10σ_x)², where σ_x is the standard deviation of the variable within a 7-day sliding window.
   3. Configure the filter to remain sensitive to short-term variability while suppressing random measurement noise. Initialize the initial state using the first available non-missing observation and set the initial covariance P_0|0 to . Remove the entire segment if the missing interval exceeds 24 h (96 records) to prevent error accumulation from long-term extrapolation.
4. Anomaly detection and correction
   1. Identify and correct anomalies in two sequential steps: hard constraints (physical limits) and soft constraints (gradient limits and quantile winsorization).
      1. Hard constraints (Physical limits): Discard measurements that fall outside physically plausible ranges (e.g.,pH∉[0,14],DO>20 mg/L).
      2. Soft constraints (Gradient limits + Quantile winsorization): Flag a data point as suspicious if the absolute change rate between two consecutive observations exceeds an empirically defined gradient threshold |x_t - x_t-1 | > θ. Set θ=3σ_xΔt^-1, where σ x denotes the standard deviation of the variable over a 24 h sliding window, and Δt = 15 min represents the sampling interval. Correct extreme values using a Winsorization procedure, as expressed in Equation (12).
         (12)
         where Q_α and Q_1-α denote the 1% and 99%, respectively.
         The procedure first eliminates impossible values through hard constraints, then flags abnormal observations based on gradient thresholds, and finally smooths the residual deviations using quantile-based winsorization. This hierarchical approach ensured both the physical validity and statistical stability of the cleaned dataset.
5. Normalization
   1. Different water quality indicators have different units and values. To eliminate scale differences and prevent certain variables from dominating the model training process, apply z-score normalization.
6. Window sample construction
   1. Transform the time series into fixed length sliding window samples to preserve the temporal dependencies. Given window length W and step size S, construct the input samples according to Equation (13).
        (13)
      Set W=96 (i.e., 24 h with a 15 min interval) and S=1 to generate samples sequentially using a point-by-point sliding window.

4. Attention-Gated Liquid Neural Network (AG-LNN)

NOTE: The overall architecture of the proposed Attention-Gated Liquid Neural Network (AG-LNN) is illustrated in Figure 2. The model integrates multisource water quality inputs, an attention-based gating mechanism, an LNN with adaptive time constants, and a readout-decoding stage with interpretability outputs. This design allows the network to dynamically capture complex temporal dependencies while remaining robust to noise and non-uniform sampling.

1. Multi-source input layer: Refer to the leftmost part of Figure 2 to visualize the multi-source water quality monitoring data {x_τ}, where each vector x_τ∈R^C represents measurements of C physicochemical variables (e.g., pH, DO, TN, TP, COD_Mn, Turbidity, EC) at time τ. Segment the input sequences into fixed-length sliding windows of size W, as formulated in Equation (14).
      (14)
2. Input attention gate: Apply an attention gate, as illustrated in the lower-left block of Figure 2, to emphasize informative variables and suppress noisy fluctuations. Compute the attention weights according to Equation (15).
   (15)
   where α_τ denotes channel-level importance at time τ. This mechanism ensures that variables that are highly correlated with abnormal changes (e.g., DO and COD_Mn) receive greater weights.
3. Time-constant gate
   1. Modulate the liquid time constant using the aggregated attention signal, as illustrated in the top block of Figure 2, and formulate it according to Equation (16).
      (16)
      where, λ_base is the baseline time constant controlling the intrinsic memory of each liquid neuron; denotes the mean attention weight across C channels at timestep τ; β is a sensitivity coefficient determining how strongly attention influences temporal responsiveness.
   2. When is high, λ_τ decreases, prompting the network to accelerate its response and capture short-lived anomalies better. Impose bound constraints on λ_τ to stabilize the training. 
   3. Ensure numerical stability and prevent exploding or vanishing dynamics by imposing bound constraints on the time constant λ_τ. Set λ_τ ∈ [λ_min,λ_max] = [0.5,2.0]. Clip any values outside this range to the nearest bound during training to maintain controlled adaptive behavior under varying environmental conditions. Setting λ = 1.0 yields optimal precision and convergence stability.
4. Liquid neural layer
   1. Employ a Liquid Neural Network (LNN) to perform continuous-time modeling of the time series, and express its dynamic formulation as shown in Equation (17).
      (17)
      where ḣ(t) denotes the hidden state vector; λ(⋅) represents the learnable time constant, modulated by attention; σ(∙) is the nonlinear activation function (ReLU); A and B are learnable weight matrices; x̃(t) is the normalized input vector; b is the bias term. This equation ensures that the network dynamics evolve with both inputs and states, while the time constant λ(⋅) remains learnable and context dependent. 
   2. For numerical computation, use the explicit Euler integration method for discretization, as shown in Equation (18).
      (18)
      Here, Δ_τ represents the discrete time step (integration interval) between successive observations, computed using Equation (19).
      Δ_τ=(t_τ - t_τ-1)/T_norm (19)
      where t_τ and t_τ-1 are the timestamps (in minutes) of consecutive measurements and T_norm = 15 min corresponds to the nominal sampling period in the CNEMC monitoring data. Thus, Δ_τ = 1 for regular 15 min intervals, whereas Δ_τ > 1 or Δ_τ < 1 reflects irregular sampling caused by missing or dense observations. This approach naturally handles irregular sampling while maintaining interpretability with respect to the underlying physical dynamics. 
5. Readout and anomaly scoring
   1. After completing the hidden state update, employ a readout layer in the AG-LNN to generate the one-step-ahead prediction, as expressed in Equation (20).
      (20)
      where W_o and c denote the output projection matrix and bias vector, respectively.
   2. Compute the residual as the anomaly score based on the discrepancy between the predicted and observed values, as shown in Equation (21).
      (21)
   3. Apply a quantile-based adaptive thresholding strategy to determine whether an observation corresponds to an anomaly.
   4. Define the anomaly threshold θ_x for each water quality variable using the 99th percentile of the residuals in the training set, as formulated in Equation (22).
         (22)
      ​where σ_s denotes the standard deviation of the residual sequence, and γ = 0.2 is a fixed safety coefficient compensating for natural diurnal variations. 
   5. This design corresponds approximately to a 1% exceedance probability, consistent with standard environmental anomaly definitions. During inference, the network outputs only the residual sequence {s_τ}, for anomaly determination, perform post-hoc according to Equations (14)–(15). This separation design is consistent with the overall study framework, ensuring that the model focuses on feature extraction and anomaly measurement rather than directly participating in threshold setting. 
6. Training objectives and regularization
   1. During training, minimize the prediction error while simultaneously applying additional regularization on both the attention weights and the time constants to prevent overfitting and numerical instability. The loss function is expressed by Equation (23).
      (23)
   2. Apply Reg_α through entropy regularization -∑α_τ log α_τ or the L1-norm to avoid attention collapse. Apply Reg_λ by penalizing Δλ_τ = λ_τ - λ_τ-1, constraining drastic fluctuations in the time constant, and adding penalties when it exceeds the predefined upper and lower bounds. This approach was used to maintain the flexibility of the AG-LNN while ensuring stability during training.
7. Forward computation and interpretability output
   1. Simplify the forward computation process of AG-LNN into the following chain structure, as shown in Equation (24).
      (24)
   2. During both training and inference, continuously record the attention weights α_τ and the liquid time constants λ_τ, and use them to generate interpretability results, namely attention heatmaps and liquid time constant curves. Use these visualizations to highlight how the model focuses on different variables and time scales, thereby enhancing the interpretability of the method. 

5. Training procedure

1. Data partitioning: Divide the multi-source water quality time series chronologically into training, validation, and testing subsets using a 70%/15%/15% split. Ensure robustness by selecting the test set to include known abnormal periods while keeping it strictly non-overlapping with the training data to prevent temporal leakage.
2. Training details
   1. Solve the continuous dynamics in the liquid layer using a stabilized explicit Euler scheme. For comparison, test a closed-form discretization following the referenced formulation^20,21.
   2. Attention warm-up schedule: Clamp the attention weights to a near-uniform value (≈ 1.0) during the first 10 epochs to stabilize the gradient flow. After epoch 10, gradually anneal the attention to its learnable state by linearly releasing the weight updates over the next 5 epochs (scaling factor = 0.8). Use this warm-up strategy to prevent unstable oscillations at the start of training and to ensure smooth convergence.
   3. Optimizer and learning-rate scheduler: Optimize the model parameters using the Adam optimizer with β₁ = 0.9, β₂ = 0.999, ε = 1 x 10⁻⁸. Set the initial learning rate to 1 x 10⁻³ and control it using a cosine-annealing schedule (T_max = 50 epochs, Minimum Learning Rate = 1 x 10⁻⁵). Choose batch sizes between 32 and 128, depending on the GPU memory capacity. Apply this gradual scheduling to ensure stable adaptation of continuous-time dynamics without abrupt learning-rate shifts.
   4. Early stopping and reproducibility settings: Terminate training early if the validation metric (AU-PR or F1) does not improve for 10 consecutive epochs (patience = 10). Repeat each experiment 3x using fixed random seeds (42, 73, and 101) to ensure reproducibility. Set the maximum number of epochs to 100 and retain the checkpoint with the best validation AU-PR for testing.
3. Hardware and software environment
   1. Perform all experiments on a Linux server equipped with NVIDIA Tesla V100 GPUs (32 GB memory) and dual Intel Xeon Gold 6230 CPUs with 256 GB RAM. Implement the model in Python 3.9 using PyTorch 2.0 as the deep learning framework, and use auxiliary libraries including NumPy 1.23, SciPy 1.9, and scikit-learn 1.2 for preprocessing and evaluation. Containerize all experiments using Docker 23.0 and manage them on Ubuntu 20.04 LTS to ensure reproducibility and portability of results. Refer to Table of materials for detailed specifications of all hardware devices and software tools to guarantee clarity and replicability.

Overall performance comparison

Table 2 presents the comparative results of the different baseline models and the proposed AG-LNN for the water quality anomaly detection task. Among the traditional unsupervised baselines, Isolation Forest22 and One-Class SVM23 achieved only moderate precision and recall, reflecting their limited ability to capture the temporal dependencies inherent in multi-source water quality data. LOF24 performed slightly better by exploiting local density structures, but its performance remained suboptimal compared with deep learning models.

For neural network-based methods, LSTM25 and Bi-GRU26 exhibited improved recall owing to their recurrent structures, although they struggled with long sequences and irregular sampling. TCN27 achieved competitive performance, particularly in terms of precision, by effectively modeling short-term fluctuations. The transformer-based architecture demonstrated strong capability in long-range dependency modeling28, but their sensitivity to noise and missing values led to inconsistent recall and higher false alarms.

LNN20 outperformed conventional RNNs and CNNs, benefiting from its continuous-time modeling ability, which is well-suited for non-uniform sampling, which is common in water quality monitoring. However, the lack of selective input weighting limits its robustness because irrelevant channels sometimes interfere with anomaly detection.

The proposed AG-LNN consistently outperformed all the baselines, achieving the highest precision (0.92), recall (0.89), F1-score (0.90), ROC-AUC (0.95), and PR-AUC (0.94). These results confirm that combining input-attention gating and adaptive time-constant control enables the network to emphasize anomaly-relevant variables (such as DO and CODMn) while dynamically adjusting its temporal responsiveness. Consequently, the AG-LNN demonstrated a superior capability in detecting both abrupt and subtle water quality anomalies.

Cross-region and sensor-network generalization with GNN baselines

This section evaluates the generalization capability of the proposed AG-LNN model in different hydrological and climatic contexts. Three complementary experiments were conducted without introducing new datasets: (i) cross-validation using the leave-one-basin-out (LOBO) strategy, (ii) temporal out-of-distribution (OOD) testing across different years and seasons, and (iii) robustness evaluation under sensor network perturbations. In addition to sequence-based baselines (LSTM, TCN, Transformer), three representative Graph Neural Network (GNN)-based models (ST-GNN, GDN, and ST-GPINN) were included for direct quantitative comparison.

Cross-region generalization (LOBO evaluation)

To assess spatial generalization, all monitoring stations were divided into five macro-regions (North, South, East, West, and Plateau) according to hydrological and climatic characteristics. In each round, one region was entirely excluded from training and validation and was used as the testing domain to simulate deployment in unseen basins.

As illustrated in Figure 3A, which plots the PR-AUC across the five held-out regions, the AG-LNN consistently achieves the highest precision in every region. Its average PR-AUC ≈ 0.94, and F1 ≈ 0.91, surpassing the best GNN baseline (ST-GNN) by +0.02 to +0.04 in PR-AUC and +0.02 to +0.03 in F1. Figure 3B shows that the performance advantage is particularly evident in the plateau region, where sparse sampling and unstable hydrology cause larger fluctuations in the other models.

Although GNN-based approaches (ST-GNN, GDN, and ST-GPINN) can model spatial correlations, they depend strongly on a complete and stable graph topology. In contrast, the AG-LNN employs continuous-time dynamics and attention-gated time constants to capture temporal dependencies without predefined edges, yielding higher transferability across heterogeneous hydrological environments.

Temporal out-of-distribution

To evaluate the temporal robustness, models trained on 2019-2022 data were directly applied to 2023-2024, with seasonal partitions (Spring, Summer, Autumn, Winter) reflecting inter-annual variability. Figure 4A shows the PR-AUC trends across the four seasons. The AG-LNN maintains a nearly constant performance (PR-AUC ≈ 0.93) throughout the year, with a maximum seasonal drop of ≤ 0.02. The GNN baselines show slightly lower but still stable scores (ST-GNN ≈ 0.92, GDN ≈ 0.91, ST-GPINN ≈ 0.91), whereas Transformer, TCN, and LSTM exhibit larger seasonal sensitivity.

The adaptive time-constant mechanism (λt) in the AG-LNN dynamically adjusts its effective temporal receptive field, allowing the model to represent both long-term hydrological cycles and short-term anomalies. In contrast, graph-based architectures, whose inter-station relations are assumed to be stationary, struggle when these relations vary seasonally. Consequently, the AG-LNN better mitigated seasonal distributional shifts, as demonstrated by the stability curves in Figure 4A.

Sensor-network perturbation robustness (Inference-phase shifts)

The third experiment explored the scalability of the AG-LNN under varying sensor network configurations during deployment. The following inference-phase perturbations were simulated without retraining: (i) coarser sampling (15 to 60 min), (ii) masking of two or three input variables, and (iii) Gaussian noise (+5% SD).

As shown in Figure 4B, the AG-LNN experienced the smallest degradation among all the models. Its average PR-AUC drop remains within 0.01-0.015, whereas GNN baselines decline by 0.015-0.028 and conventional models degrade even more. The figure clearly demonstrates that the AG-LNN preserves stable accuracy even when the data sparsity and noise increase.

The attention-gating mechanism enables the AG-LNN to dynamically reweight available features, while its continuous-time formulation ensures numerical stability under irregular sampling and variable loss. These results confirm that the AG-LNN can flexibly adapt to heterogeneous sensor infrastructures and varying sampling resolutions, which is crucial for large-scale environmental monitoring networks.

Model efficiency and edge-deployment scalability

To evaluate the computational efficiency and deployment potential of the AG-LNN, additional experiments were conducted to analyze its complexity, scalability, and lightweight adaptation.

Computational complexity

The full AG-LNN model contains approximately 5.3 million parameters, corresponding to 1.8 GFLOPs per 96-step sequence inference. This complexity is smaller than that of most GNN-based baselines, such as ST-GNN (6.8M, 2.3 GFLOPs) and GDN (7.1M, 2.6 GFLOPs). On an NVIDIA RTX 3060 GPU, AG-LNN processes one 24 h sequence in 0.21 s, and on a standard CPU (Intel i7-12700), the same inference takes 0.68 s, which meets the real-time requirement for 15 min monitoring intervals.

Lightweight variant: AG-LNN-light

Develop a compact version called AG-LNN-light to enable efficient edge deployment. It is derived from the original model through structured pruning and gate-parameter sharing. The following three strategies were adopted:

Hidden-dimension reduction: The hidden state size and feature embedding dimension were halved (from 256 to 128), directly reducing the matrix multiplications in both the attention and time-constant gates.

Gate-parameter sharing: The two gating submodules (input attention and time-constant modulation) were merged to share a single projection matrix , eliminating redundant parameters while preserving separate nonlinear activation.

Sparsity pruning: Low-magnitude weights (below a 0.02 threshold of normalized magnitude) were pruned and fine-tuned for five epochs to recover the accuracy.

These modifications reduced the total parameter count by 62% and the FLOPs by approximately 60%, with only a 5% loss in precision (PR-AUC = 0.90 vs 0.94). Table 3 summarizes the comparison of the computational cost and accuracy.

Scalability and deployment

Owing to its continuous-time formulation and reduced parameter coupling, the AG-LNN-light maintains numerical stability even after aggressive pruning. The lightweight variant was successfully deployed on Jetson Xavier NX (8 GB RAM) and Raspberry Pi 5, achieving inference times of 1.2-1.5 s per sequence without retraining. This confirms the feasibility of deploying the AG-LNN-family models on distributed edge devices for real-time water quality anomaly detection.

Ablation study

To investigate the contribution of different components in the AG-LNN, ablation experiments were conducted by progressively removing the input attention gate and time-constant gate and the results are reported in Table 4.

The LNN (without any gating mechanism) outperformed the RNN and CNN-based baselines, demonstrating the advantage of continuous-time modeling for nonuniform water quality time series. However, the recall was limited (0.77), suggesting reduced sensitivity to subtle anomalies.

Adding only the input attention gate significantly improved precision (from 0.85 to 0.88) by suppressing irrelevant channels and highlighting key variables (such as DO and CODMn), thus reducing false positives. However, the recall improvement was modest.

Adding only the time-constant gate resulted in a higher recall (0.83 vs. 0.77), as the network became more responsive to sudden changes by dynamically adapting the temporal scale. However, the precision remained similar to that of the LNN.

The full AG-LNN, combining both gates, achieved the best trade-off with an F1-score of 0.90, ROC-AUC of 0.95, and PR-AUC of 0.94. These results confirm that the two gating strategies are complementary: input attention enhances selectivity, whereas adaptive time constants improve responsiveness. Together, they enable robust and accurate anomaly detection for water quality monitoring.

Visualization of detection and interpretability

To further demonstrate the interpretability of AG-LNN, visualize the outputs of AG-LNN in terms of anomaly scores, attention weights, and adaptive time constants (Figure 5) to further demonstrate its interpretability.

Figure 5A illustrates a case study of anomaly detection at a lake-monitoring station. The AG-LNN successfully detected sharp CODMn spikes and gradual DO declines, closely aligned with the ground truth anomaly labels. Compared to the baselines (not shown here for clarity), the AG-LNN produced fewer false alarms and responded faster to transient events.

Figure 5B shows the attention heatmap for the seven input variables (pH, DO, TN, TP, CODMn, Turbidity, EC). During abnormal periods, higher weights were consistently assigned to DO and CODMn, which aligns with established knowledge of water quality monitoring. This confirms that the input attention mechanism effectively highlights anomaly-relevant variables, while suppressing irrelevant variables.

Figure 5C shows the dynamics of the learned time constant λτ. Under stable water quality conditions, the model maintained relatively large values of λτ, favoring long-term memory. In contrast, during anomaly intervals, λτ decreases significantly, accelerating the network's responsiveness. This adaptive modulation enables the AG-LNN to capture both gradual and abrupt changes in the water quality.

Together, these visualizations confirm that the AG-LNN not only achieves high detection accuracy but also provides interpretable insights into its decision-making process, thereby offering practical value for water quality management.

Beyond expert interpretation, the visualization results were also reformatted into a stakeholder-oriented dashboard. Attention heatmaps and adaptive λ trajectories were converted into intuitive graphics using traffic light colors (green = stable, orange = mild change, and red = critical anomaly). Textual explanations summarize the most influential variables and time intervals, enabling environmental managers to rapidly identify causes and prioritize responses without requiring deep technical knowledge. The overall workflow of the transformation of AG-LNN interpretability outputs into stakeholder-oriented visual information is summarized in Figure 6.

Beyond visual explanation, the interpretability outputs of the AG-LNN provide actionable insights for environmental managers. The attention heatmap in Figure 5A highlights the key variables that dominate the anomaly detection. During abnormal intervals, increased attention weights on DO and CODMn correspond to oxygen depletion and organic-matter surges-two common indicators of water-quality deterioration. Meanwhile, the adaptive λτ trajectory in Figure 5C identifies precise time windows when the model becomes more sensitive to rapid fluctuations.

These interpretable outputs offer more than visual clarity; they provide guidance that can be acted upon in the field. When a sudden change is detected, the corresponding sensors or sampling sites most affected by variations in DO and CODMn should be paid attention to rather than blindly scanning through all the data. The shifting pattern of λτ, rising and falling over time, quietly signals when an anomaly begins and fades, making it easier to decide when to sample and when to respond. Taken together, these cues do not simply describe a problem; they point toward its source. By linking the highlighted variables and their timing, one can often infer whether an unexpected event comes from a local discharge, upstream inflow, or temporary system disturbance, allowing interventions to occur earlier and with greater confidence.

Impact of liquid dynamic parameters (β and λ)

To evaluate how the liquid dynamics mechanism contributes to AG-LNN's performance and stability of the AG-LNN, we conducted additional experiments focusing on two key parameters: the fundamental time constant (λ) and the adjustment coefficient (β). These parameters jointly regulate the temporal responsiveness of the network, analogous to the physical adjustment processes of aquatic systems, where λ controls the characteristic response time, and β governs its adaptability to external disturbances. Vary EQUATION and EQUATION while keeping the other hyperparameters fixed. PR-AUC and F1 were used as evaluation metrics to assess accuracy and stability.

Effect of λ

As shown in Figure 7(A), both the PR-AUC and F1 follow a unimodal pattern as λ increases. The performance peaks around λinit ≈ 1.0, indicating an optimal balance between temporal stability and responsiveness. A smaller λ (< 0.5) leads to overly reactive behavior, increasing false alarms, whereas a larger λ (> 1.5) causes delayed detection of sudden pollution spikes. This aligns with the physical interpretation of λ as the intrinsic adjustment timescale of the system.

Effect of β

Figure 7B shows that β determines how rapidly λ adapts to changing inputs. When β is too low (< 0.1), the model becomes sluggish to respond; when β is too high (> 0.5), λ fluctuates excessively, producing unstable predictions. The best performance occurs at βinit (≈0.3), suggesting that moderate adaptability ensures stable convergence, while retaining flexibility.

Joint influence

The contour in Figure 8 reveals a plateau of high performance (PR-AUC > 0.93) when λ∈[0.7, 1.3] and β∈[0.2, 0.4]. These ranges are consistent with physically plausible hydrological time scales (≈ 12 – 36 h) and moderate sensitivity to disturbances, reinforcing the interpretability of the learned liquid dynamics.

Effect of annotation ratio and weak label accuracy on model reliability

To quantitatively evaluate the effect of annotation composition on model reliability, controlled experiments were conducted by varying the proportion of experts and weak annotations during the training. These five settings are listed in Table 5.

Additionally, the expert-to-weak annotation ratio was fixed at 40:60, and different weak annotation accuracies (0.80, 0.85, 0.90, and 0.95) were used to evaluate the effect of weak label accuracy. Model performance was assessed using PR-AUC, F1-score, and Stability Index (standard deviation of prediction confidence across folds).

Effect of annotation ratio

As shown in Table 6 and Figure 9A, model performance exhibited a clear upward trend as the proportion of expert annotations increased from 20% to 40%. Both the PR-AUC and F1-score steadily improved within this range, indicating that a moderate inclusion of expert-labeled data substantially enhances the model’s reliability and discriminative power. When the expert label ratio exceeded 40%, however, the improvement began to plateau and even slightly declined beyond 50%, suggesting diminishing returns once annotation precision surpassed a critical threshold. The 40:60 configuration emerged as the most balanced setting, achieving the highest PR-AUC (0.938) and F1-score (0.906), while maintaining the lowest Stability Index (0.036). This composition effectively reconciles data sufficiency with annotation quality and maximizes label efficiency without compromising robustness. The gentle curvature of the performance curve in Figure 9A shows that model precision improved steadily when the expert label proportion increased from 20% to 40% but plateaued beyond 50%. The 40:60 configuration yields the best trade-off between performance and data volume, confirming that this ratio maximizes data utilization without compromising annotation reliability.

Effect of weak annotation accuracy

As presented in Table 7 and Figure 9B, the sensitivity of the model to weak-label accuracy revealed a clear transition pattern. When the weak-label accuracy drops below 0.85, both the PR-AUC and F1-score exhibit a noticeable decline, implying that label noise becomes dominant and undermines the reliability of supervision. As accuracy increases from 0.85 to 0.90, performance improves steadily, with PR-AUC rising from 0.922 to 0.938 and F1-score from 0.893 to 0.906, indicating that the model effectively benefits from cleaner annotations within this range. However, beyond the 0.90 threshold, the gain becomes marginal, with results nearly saturating at 0.95 accuracy, suggesting that further refinement of weak labels offers little return. This behavior aligns with the deviation curve in Table 7, where performance variation remains within ±0.1% when accuracy surpasses 0.90. The smooth upward trend in Figure 9B illustrates that the AG-LNN maintains consistent robustness under moderate noise levels, while still leveraging informative signals from weak supervision. In practical applications, maintaining weak-label accuracy around 0.90 appears sufficient to achieve reliable training without incurring unnecessary annotation costs or complexity.

These results demonstrate that the chosen ratio (Expert: Weak = 40:60) achieves the best balance between label diversity and reliability. Increasing weak labels beyond 60% introduces mild noise and instability, whereas higher expert proportions reduce data coverage and generalization.

Weighted learning ablation

To quantitatively evaluate the effectiveness of the proposed confidence-weighted loss formulation, an ablation experiment was performed by varying the relative weights of expert and weak annotations in the loss function. The objective was to determine the optimal configuration of α and β in Equation (2) that balances performance, robustness, and training stability.

Fix the expert-label weight (α) to 1.0 and vary the weak-label weight (β) within the range [0.5, 1.0] in increments of 0.1. Train each configuration under identical conditions using the same random initialization, learning rate, and data splits. Evaluate model performance using PR-AUC, F1-score, and a Stability Index, defined as the standard deviation of prediction confidence across five training folds. Additionally, conduct a sensitivity analysis by combining different weak-label accuracies(aw = 0.85,0.90,0.95) with corresponding β values to visualize the interaction between label accuracy and loss weighting.

The experimental results are visualized in Figure 9Cand Figure 10. As shown in Figure 9C, model precision and F1-score improve progressively as the weak-label weight β increases from 0.5 to 0.7, reaching a peak at β = 0.7 (PR-AUC = 0.939, F1 = 0.907). Beyond this point, both metrics show marginal decline, indicating that excessive reliance on weak annotations introduces minor noise into gradient updates. Figure 10 further demonstrates that the Stability Index, which reflects training consistency, reaches its minimum at the same setting (β=0.7), confirming that this configuration provides the most stable optimization trajectory.

To assess robustness under different levels of weak-label reliability, Figure 11 shows how the PR-AUC varies across combinations of weak-label accuracy and weight. The surface plot shows a broad plateau once the accuracy surpasses 0.90, and the performance remains stable for β values between 0.7 and 0.8. As the accuracy slips to 0.85, the ridge tilts downward and the optimal weighting shifts toward 0.6 to 0.7, suggesting that the model begins to rely more heavily on strong labels to offset noise. Interestingly, the selected configuration (α = 1.0, β = 0.7) is located near the center of this stable zone, delivering the best overall precision while maintaining resilience to moderate fluctuations in weak-label quality.

Robustness under noisy and missing data conditions

To further examine the reliability of the AG-LNN under realistic environmental variations, additional robustness tests were conducted covering multiple noise types and missing data patterns (Figure 12 and Figure 13).

In noise-type experiments, three perturbations were simulated: (i) Gaussian noise (σ = 0.05), (ii) impulse noise (5% spike probability), and (iii) sensor-drift noise (±3% linear bias per day). As shown in Figure 12, AG-LNN maintained PR-AUC ≥ 0.93 under Gaussian noise and ≥ 0.91 under drift noise, whereas ST-GNN and GDN declined below 0.87. The attention-gated liquid-state formulation dynamically adjusts the effective time constant λ, suppresses transient perturbations, and adapts to long-term biases.

In missing-pattern experiments, the model was evaluated under three conditions: (i) random 10% data loss, (ii) block-wise missing intervals (1–3 h), and (iii) feature-correlated dropout (variable-specific). Figure 13shows that AG-LNN sustained F1 = 0.88 and PR-AUC = 0.92, even with 15% block-wise missingness, outperforming ST-GNN and GDN by an average margin of +0.05.

These results confirm that the AG-LNN exhibits high robustness against diverse data imperfections, demonstrating stable performance under both stochastic and systematic disturbances commonly observed in environmental sensor networks.

Figure 1: Overall study design of the proposed water quality anomaly detection framework. The workflow includes seven sequential stages: (1) acquisition of multi-source water quality sensor data, (2) missing and anomaly handling, (3) normalization, (4) windowed sample construction, (5) Attention-Gated Liquid Neural Network (AG-LNN) training with continuous-time modeling and learnable time constants as well as attention gates for channel/time weighting, (6) threshold/probability-based anomaly decision, and (7) result visualization and interpretation using attention heatmaps and liquid time constant curves. Please click here to view a larger version of this figure.

Figure 2: Architecture of the Attention-Gated Liquid Neural Network (AG-LNN). The model processes multi-source water quality data through an input attention gate, a time-constant gate, and a liquid neural layer. Predictions are generated by the readout module, and anomalies are identified via residual scoring. Attention heatmaps and time-constant trajectories provide interpretability. Please click here to view a larger version of this figure.

Figure 3: LOBO generalization performance across regions with GNN baselines. (A) The AG-LNN consistently achieves the highest PR-AUC across most regions, indicating strong spatial generalization and adaptability to unseen hydrological contexts. (B) Across five held-out regions, AG-LNN consistently demonstrates superior adaptability. Its F1 advantage is most evident in geographically diverse areas such as the plateau, suggesting that the model effectively captures spatial heterogeneity in water quality dynamics. Please click here to view a larger version of this figure.

Figure 4: Temporal and inference-time robustness of AG-LNN with GNN baselines. (A) Across spring, summer, autumn, and winter, AG-LNN sustains the most consistent precision, while other models exhibit stronger seasonal sensitivity. This stability indicates that its dynamic design effectively captures long-term temporal dependencies in changing aquatic environments. (B) The proposed AG-LNN maintains the highest precision across all inference time shift scenarios, showing strong robustness under sampling coarsening, variable masking, and added sensor noise. Please click here to view a larger version of this figure.

Figure 5: Visualization of anomaly detection and interpretability in AG-LNN. (A) Anomaly detection curve showing AG-LNN anomaly scores, threshold, and ground truth labels over time. (B) Attention heatmap across seven water quality variables (pH, DO, TN, TP, CODMn, Turbidity, EC), highlighting increased weights on DO and CODMn during anomaly periods. (C) Time-constant dynamics λT, illustrating larger values under stable conditions and reduced values during anomalies, enabling faster responsiveness. Please click here to view a larger version of this figure.

Figure 6: Integration of AG-LNN outputs into an intuitive dashboard for visualizing anomalies and variable importance. Through a compact layout, the system reveals which parameters drive each event and when the risk escalates, allowing experts to trace anomalies and plan timely interventions. Please click here to view a larger version of this figure.

Figure 7: Effect of liquid dynamic parameters on model performance. (A) Scores peak around λinit ≈ 1.0, reflecting a balance between responsiveness and stability. (B) Moderate β yields optimal accuracy; extreme values cause under-adaptation or unstable updates. Please click here to view a larger version of this figure.

Figure 8: Joint effect of λ and β on PR-AUC (contour). A stable high-accuracy region emerges for λ∈[0.7, 1.3] and β∈[0.2, 0.4]. Please click here to view a larger version of this figure.

Figure 9: Effects of expert weak annotation configuration on model performance. (A) PR-AUC and F1 continue to rise as the expert ratio increases from 20% to 40%, reaching a peak at 40:60 (PR-AUC=0.938, F1=0.906), and then the growth slows down. The supplementary figure shows that the stability index (the lower the better) is optimal at 40:60. (B) When the weak annotation accuracy falls below 0.85, the model performance declines significantly; when it is ≥0.90, the PR-AUC and F1 scores approach saturation (0.938 / 0.906), and an accuracy of 0.95 yields only marginal improvement. (C) Both PR-AUC and F1-score peak at β=0.7, confirming the optimal trade-off between weak-label contribution and noise control. Please click here to view a larger version of this figure.

Figure 10: Stability index versus weak-label weight. The lowest variance of prediction confidence is observed at β=0.7, indicating maximal training stability. Please click here to view a larger version of this figure.

Figure 11: Sensitivity of PR-AUC to weak-label accuracy and weight. The high-performance plateau (PR-AUC>0.935) consistently occurs within β∈[0.7, 0.8] when weak-label accuracy ≥0.90. Please click here to view a larger version of this figure.

Figures 12: Robustness under different noise types. The AG-LNN maintains the highest PR-AUC under Gaussian, impulse, and drift noise, demonstrating strong resistance to signal distortion and superior stability compared with graph-based baselines. Please click here to view a larger version of this figure.

Figures 13: Performance under different missing-data patterns. The AG-LNN maintains the highest F1 score across all missing-data types, demonstrating strong resilience to random loss, block-wise gaps, and feature-correlated missingness. Please click here to view a larger version of this figure.

Table 1: Field Description of cnemc_data_2019_2024.

Table 2: Quantitative comparison of anomaly detection models. Performance was evaluated across multiple metrics, including Precision, Recall, F1-score, ROC-AUC, and PR-AUC. Compared models include classical machine learning baselines (Isolation Forest, One-Class SVM, LOF), recurrent models (LSTM, Bi-GRU), sequence models (TCN, Transformer), the LNN, and the proposed AG-LNN.

Table 3: Comparison of Model Complexity, Efficiency, and Accuracy among AG-LNN, AG-LNN-light, and GNN-based Baselines.

Table 4: Ablation study of AG-LNN variants. Performance of different model configurations was evaluated using precision, recall, F1-score, ROC-AUC, and PR-AUC. The full AG-LNN, which combines both the input attention gate and the time-constant gate, achieved the best overall performance, confirming the complementary effects of the two gating mechanisms.

Table 5: Experimental settings for evaluating the impact of expert–weak annotation composition and weak-label accuracy on model reliability.

Table 6: Effect of expert–weak annotation ratio on model performance. This table reports the quantitative performance of AG-LNN under different proportions of expert and weak annotations used for training. Both PR-AUC and F1-score increase steadily as the expert label percentage rises from 20% to 40%, reaching a peak at the 40:60 configuration, where the stability index is also minimal. Beyond this ratio, performance gains saturate and slightly decline, indicating that the 40:60 composition provides the best trade-off between annotation reliability and data coverage.

Table 7: Effect of weak annotation accuracy on model performance (at fixed 40:60 expert–weak ratio). This table presents the quantitative results of AG-LNN under different weak-annotation accuracy levels while keeping the expert–weak ratio fixed at 40:60. Both PR-AUC and F1-score increase notably as weak-label accuracy improves from 0.80 to 0.90, beyond which the gains become marginal. The deviation column denotes the relative performance change (%) with respect to the optimal configuration (accuracy = 0.90).

This study introduces an Attention-Gated Liquid Neural Network (AG-LNN), a novel architecture for water quality anomaly detection that combines continuous-time liquid dynamics with attention-based gating mechanisms. The liquid neural component of an AG-LNN is inspired by Liquid Time-constant Networks (LTCs), which model time series with learnable, input-dependent time constants²⁰. However, AG-LNN extends beyond standard LTCs by integrating two additional gates: an input attention gate, which emphasizes anomaly-relevant variables such as DO and COD_Mn, and a time-constant modulation gate, which dynamically accelerates or decelerates the liquid dynamics according to the context. This dual gating ensures that the AG-LNN inherits the expressive continuous-time properties of LTCs while enhancing interpretability and anomaly sensitivity.

The proposed framework contributes to the advancement of the scientific field in several respects. First, the AG-LNN addresses the challenge of irregularly sampled multivariate sensor data, which is a common scenario in large-scale environmental monitoring networks. Unlike discrete-time RNNs, which require fixed intervals, AG-LNN’s continuous-time formulation naturally accommodates varying sampling intervals. Second, its explainable outputs (attention heatmaps and adaptive time-constant trajectories) make the model more transparent, bridging the gap between deep learning and environmental science, where decision making requires interpretability. By improving anomaly detection performance and interpretability, the AG-LNN provides a practical tool for early warning systems, supporting timely interventions in pollution control and sustainable water management.

Critical steps influencing success

The reproducibility and overall success of the AG-LNN are deeply influenced by a few critical protocol stages that shape both data reliability and learning stability. Among them, season-aware weak labeling plays a decisive role; it governs the precision–recall balance by coupling quantile thresholding with residual-based detection to capture subtle seasonal fluctuations. Equally important is the Kalman-based long-gap imputation step, which restores the temporal coherence in missing sequences through carefully tuned noise covariance parameters. The gradient-based anomaly correction further refines the signal, where the threshold θ = 3σ_xΔt^^-1 delicately balances anomaly sensitivity against unwanted fluctuations. Finally, attention warm-up and bounded λ adaptation stabilize the early training process and regulate how responsively the liquid dynamics evolve over time. Together, these interdependent steps form the backbone of AG-LNN’s stability, enabling it to maintain consistent optimization, reliable anomaly labeling, and robust generalization across diverse temporal and regional conditions.

Modifications and troubleshooting

Practical deployment and replication of AG-LNN may encounter a range of challenges, some rooted in data quality and others in model dynamics or hardware constraints. At the data level, when anomalies appear to be either over- or under-detected, recalibrate the quantile thresholds (for example, tightening them to 0.5%–99.5%) or adjust the residual window length to approximately 10–15 days to smooth seasonal noise. If weak labels begin to destabilize training, consider raising their confidence cutoff to 0.9 and lowering their learning weight β from 0.7 to 0.6, striking a better balance between precision and tolerance. At the model level, attention saturation or loss oscillation during early epochs often signals the need for a longer warm-up period, extending it from 10 to 15 epochs or a smaller initial learning rate, such as 5x10^-4. Slightly reducing λ_min from 0.5 to 0.4 can also make the system more responsive to sudden anomalies. On the hardware side, edge deployment may suffer from latency, which can be effectively mitigated by adopting a lightweight AG-LNN-light variant or by shortening the sliding window from 96 to 64 steps without significant loss of accuracy. Collectively, these adjustments form a practical troubleshooting guide, helping researchers and engineers to maintain model stability and efficiency in real-world environments.

Limitations and future directions

Despite these strengths, this study had several limitations. AG-LNN is computationally more demanding than lightweight baselines, such as LSTM²⁵ and GRU²⁶, potentially limiting deployment on resource-constrained devices. The evaluation relied primarily on the CNEMC datasets, which may restrict the generalizability of the results to other ecological regions. Moreover, supervised training requires reliable anomaly labels that are expensive and subjective. Although AG-LNN enhances interpretability compared to black-box models, its visual outputs may still require additional interface design for non-expert stakeholders.

Alternative approaches can also be explored to test this hypothesis. For instance, GNNs such as the GDN and its variant GDN+ have shown promising results in river network anomaly detection by capturing spatial dependencies and offering graph-level interpretability²⁹. Hybrid physics-informed GNNs (e.g., ST‑GPINN) combine hydraulic modeling with graph representations, thereby enhancing the generalization in water distribution system quality prediction³⁰. Comparisons with these models may clarify the strengths of AG‑LNN in balancing dynamic responsiveness, interpretability, and temporal modeling.

The importance and applications of AG-LNN extend beyond water quality monitoring. Its attention-gated liquid design can be adapted to air quality forecasting, hydrological risk prediction, and ecological health monitoring, all of which involve noisy and irregularly sampled time series³¹. Beyond environmental science, similar methods can be applied to biomedical signal analysis and industrial fault detection, where real-time interpretability and robustness are essential. This cross-domain adaptability makes the AG-LNN a versatile methodology for anomaly detection in complex systems.

One promising direction is to enhance computational efficiency through closed-form continuous-time models (CfCs), which reduce dependence on iterative solvers while retaining interpretability^32,33,34,35. Such an approach can simplify the training dynamics and make large-scale deployments more practical. In parallel, the use of neuromorphic hardware^36,37,38 (e.g., Intel Loihi-2^39,40) offers an exciting possibility for achieving real-time, low-power inference in distributed sensor networks, where responsiveness and energy efficiency are often critical. Beyond these architectural advances, extending validation across diverse hydrological and climatic regions is essential to evaluate generalizability, ensuring that the model remains reliable under varied and unpredictable environmental conditions.