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

Summary

This study proposes an Attention-Gated Liquid Neural Network for water quality anomaly detection that achieves superior accuracy and interpretability through continuous-time modeling and attention-based gating, thereby enhancing the reliability of environmental monitoring and supporting sustainable water management.

Abstract

With the rapid expansion of water monitoring networks, time-series anomaly detection has become increasingly crucial for safeguarding aquatic environments and ensuring their sustainable management. However, conventional models often struggle with irregular sampling intervals, multivariate correlations, and interpretability in practical applications. To address these challenges, this study proposes an Attention Gated-Liquid Neural Network (AG-LNN) that integrates the dynamic modeling capability of the Liquid Neural Network (LNN) with attention-based gating mechanisms. The model introduces an input-attention gate to emphasize anomaly-relevant variables such as Dissolved Oxygen (DO) and the Permanganate Index (COD_Mn), and a time-constant gate that adaptively adjusts the model's temporal memory. Using data from the China National Environmental Monitoring Center (CNEMC) collected between 2019 and 2024 across 13 provinces, AG-LNN demonstrated superior performance over baseline models, including Long Short-Term Memory (LSTM), Temporal Convolutional Network (TCN), Transformer, and Graph Neural Network (GNN) architectures. It achieves a Precision-Recall Area Under Curve (PR-AUC) of 0.95 and an F1-score of 0.90, while maintaining stability under cross-region and temporal evaluations. A compact version, AG-LNN-light, reduces parameters by 62% with minimal accuracy loss, enabling efficient edge deployment. The results confirmed that attention-gated continuous-time modeling provides a robust and interpretable approach for large-scale water quality anomaly detection.

Introduction

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.

Protocol

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.

Results

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 da...

Discussion

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 empha...

Materials

List of materials used in this article

Name	Company	Catalog Number	Comments
256 GB DDR4 RAM	Samsung	M393A4K40DB3	High-capacity memory for handling large multivariate time-series datasets
CUDA 11.6	NVIDIA	N/A	GPU acceleration toolkit for PyTorch
Intel Xeon Gold 6330 CPU (2.0GHz, 28 cores)	Intel	BX80708-6330	Used as the main computation server for model training
Matplotlib 3.4, Seaborn 0.11	Open Source	N/A	Visualization of experimental results
NumPy 1.21, Pandas 1.3	Open Source	N/A	Data preprocessing and feature engineering
NVIDIA A100 GPU (40GB)	NVIDIA	900-21001-0000-001	Accelerated training of AG-LNN with CUDA support
Python 3.9	Python Software Foundation	N/A	Main programming language for implementation
PyTorch 1.12	Meta AI	N/A	Deep learning framework used for building AG-LNN
Scikit-learn 0.24	Open Source	N/A	Evaluation metrics and baseline models
Ubuntu 20.04 LTS OS	Canonical	N/A	Operating system for the computational environment