Study on Mine Water Inflow Prediction for the Liangshuijing Coal Mine Based on the Chaos-Autoformer Model

Abstract

Mine water hazards represent one of the principal threats to safe coal mine operations; therefore, accurately predicting mine water inflow is critical for drainage system design and water hazard mitigation. Because mine water inflow is governed by the combined influence of multiple hydrogeological factors and thus exhibits pronounced non-linear characteristics, conventional approaches are inadequate in terms of forecasting accuracy and medium- to long-term predictive capability. To address this issue, this study proposes a Chaos-Autoformer-based method for predicting mine water inflow. First, the univariate inflow series is mapped into an m-dimensional phase space by means of phase-space reconstruction from chaos theory, thereby fully preserving its non-linear features; the reconstructed vectors are then used to train and forecast inflow with an improved Chaos-Autoformer model. On top of the original Autoformer architecture, the proposed model incorporates a Chaos-Attention mechanism and a Lyap-Dropout scheme, which enhance sensitivity to small perturbations in initial conditions and complex non-linear propagation paths while improving stability in long-horizon forecasting. In addition, the loss function integrates the maximum Lyapunov exponent error and earth mode decomposition (EMD) indices so as to jointly evaluate dynamical consistency and predictive performance. An empirical analysis based on monitoring data from the Liangshuijing Coal Mine for 2022–2025 demonstrates that the trained model delivers high accuracy and stable performance. Ablation experiments further confirm the significant contribution of the chaos-aware components: when these modules are removed, forecasting accuracy declines to only 76.5%. Using the trained model to predict mine water inflow for the period from June 2024 to June 2025 yields a root mean square error (RMSE) of 30.73 m³/h and a coefficient of determination (R²) of 0.895 against observed data, indicating excellent fitting and predictive capability for medium- to long-term tasks. Extending the forecast to July 2025–November 2027 reveals a pronounced annual cyclical pattern in future mine water inflow, with markedly higher inflow in summer than in winter and an overall slowly declining trend. These findings show that the Chaos-Autoformer can achieve high-precision medium- and long-term predictions of mine water inflow, thereby providing technical support for proactive deployment and refined management of mine water hazard prevention.

1. Introduction

Mine water hazards constitute one of the principal disasters that threaten safe coal mine operations and, once they occur, typically lead to severe casualties and substantial property losses [1]. Research evidence shows that the main precipitating factor behind such accidents is the systematic underestimation of mine water inflow during early forecasting relative to the volumes revealed in actual mining practice. According to relevant statistics, only about 10% of mining districts exhibit a discrepancy of less than 30% between the inflow predicted in their geological exploration reports and the inflow ultimately observed; by contrast, 80% show errors exceeding 50%, with some districts reaching deviations of several tens to even hundreds of times the predicted value. Forecasting errors of this magnitude not only render the designed drainage capacity of a mine incompatible with real demand-wasting resources and causing operational inefficiencies, but also amplify the latent risk of mine water hazards [2,3]. Consequently, accurately forecasting mine water inflow is of paramount significance for safeguarding the safe production of coal mines.

Current methods widely employed to predict mine water inflow can be divided into two main categories: deterministic analyses and non-deterministic analyses [4]. Deterministic approaches include the water-balance method, analytical methods, and numerical simulations. Xu Zhimin et al. [5] investigated the evolution of roof hydrogeological parameters during mining disturbances and proposed an analytical, variable-parameter framework for dynamically computing mine water inflow. Zhou Quanchao et al. [6] introduced a “water-release–section-flow” approach for predictive analysis at the Hetaoyu Coal Mine. Wei Huapeng et al. [7] built a full 3-D unsteady-flow model of the working face, showing that strip-filling mining sharply reduces inflow and groundwater drawdown compared with fully mechanized extraction. Hua Zhaolai et al. [8] employed an unstructured-grid MODFLOW framework to capture high-intensity mining disturbances at Caojiatan; the model raised inflow-prediction accuracy over traditional methods. At Serbia’s Štavalj Coal Mine, Miladinović, Branko et al. [9] linked precipitation and temperature to inflow by multiple-linear regression within a random-process framework, achieving reliable 1–7-day forecasts. Traditional deterministic analyses generally rely on linear assumptions: the inflow curve is fitted with a linear equation, hydrogeological parameters are substituted into the formula, and future inflow is thus predicted. Mine water inflow is jointly governed by rainfall, mining intensity, lithological characteristics, geological structures, and hydrogeological attributes. The strong coupling of these factors is difficult to quantify precisely and cannot be captured by simple closed-form equations; as a result, conventional methods fail to represent the complex non-linear evolution of inflow and deliver limited predictive accuracy.

Non-deterministic analyses encompass analogy methods, regression analyses, and grey-system approaches [4]. Yao Dongze et al. [10] combined CEEMDAN decomposition with a self-attention-enhanced LSTM-SVM ensemble. Liu Hui et al. [11] fused Variational Mode Decomposition and a Deep Belief Network, predicting highly oscillatory inflow series component-wise. Shi Junwei et al. [12] integrated Transformer self-attention with LSTM, tuned via random search + Bayesian optimization; on Baotailong data, the hybrid outperformed LSTM, CNN, Transformer, and CNN-LSTM, delivering the most accurate inflow forecasts. Li Bo et al. [13] coupled a BP neural network with ARIMA (3,1,1), trimming Longfeng Mine inflow prediction error to an average 1.02%. Li Jianlin et al. [14] showed that phase-space column vectors map directly to key geological controls and, with a Chaos-Elman ANN, achieved 97.9% accuracy. Yang Songlin et al. [15] introduced DIFF-TCN and DIFF-LSTM hybrids; at Tingnan Mine, the DIFF-TCN model yielded MAE = 5.88 m³/h, RMSE = 6.85 m³/h and R² = 0.96, outperforming similarly complex deep-learning architectures. Although non-deterministic methods can better capture the non-linear influences on mine water inflow, their capability for long-term forecasting is relatively limited. Most existing models perform well only for short-term prediction; their accuracy drops markedly when forecasting long-term trends, falling short of the requirements for forward-looking water hazard prevention and control in coal mine operations.

To address the issues outlined above, this study harnesses the superior capability of chaos theory for processing non-linear and non-stationary time series and couples it with the Autoformer model renowned for its strong performance in medium- and long-term forecasting to develop an integrated chaos theory and Autoformer approach for medium- to long-term prediction of mine water inflow.

Using long-term monitoring data from the Liangshuijing Coal Mine as a case study, the proposed method first applies phase-space reconstruction to the inflow time series to extract its chaotic features and then trains and forecasts with the Autoformer model, thereby achieving accurate medium- and long-term predictions of mine water inflow for the Liangshuijing Coal Mine.

2. Theory

2.1. Chaotic Characteristics of Mine Water Inflow

Chaos theory—an area of mathematics devoted to the intricate evolution of non-linear dynamical systems—has found wide application across the natural and social sciences as well as engineering [16]. Chaotic phenomena are ubiquitous in nature and human society: deterministic systems often generate behaviour that appears random and is difficult to forecast with precision. Although a chaotic system evolves according to deterministic internal laws, its extreme sensitivity to initial conditions renders its long-term behaviour highly uncertain.

The variability of mine water inflow is governed by the combined influence of multiple hydrogeological factors (Figure 1) and therefore exhibits pronounced complexity and non-linear characteristics [17]. The evolution of the mine hydrogeological system is not instantaneous; rather, as mining progresses, parameters such as goaf area, roadway development length, and the hydraulic heads of individual aquifers change continuously, maintaining intricate interactions and dynamic interdependencies. This complex and recurrent evolutionary process displays strong non-linearity and extreme sensitivity to initial conditions. Further analysis indicates that such repeated changes can be mathematically characterized as an iterative process and an external hallmark of chaotic systems. Accordingly, the evolution of the mine hydrogeological system possesses typical chaotic attributes [18], implying that accurate prediction of mine water inflow depends not only on the current hydrogeological state but also significantly on its historical conditions.

2.2. Phase-Space Reconstruction

Mine water inflow constitutes a prototypical non-linear chaotic time series, and analyses restricted to the conventional time domain often fail to extract sufficient information. Moreover, effective regulation and accurate prediction of the system require a thorough exploration of its dynamical structure in state space so as to uncover the underlying governing laws. Real-world systems are typically complex, involving numerous variables that cannot all be measured with precision, and their overall spatial evolution is thus inaccessible. High-dimensional (>3-D) spaces are likewise unobservable. Fortunately, time series measurements of certain observable variables can be exploited to reconstruct a dynamical structure that is topologically equivalent to the original system, thereby mapping high-dimensional features onto a lower, observable manifold. Consequently, phase-space reconstruction has become a key procedure widely adopted in the analysis of non-linear time series [19].

According to the embedding theorem proposed by Takens F. in 1980 [20], for an infinitely long, noise-free non-linear time series, there always exists an $m$-dimensional space in which the dynamics reconstructed from the series are topologically equivalent to those of the original system. This theorem therefore provides the theoretical foundation for rebuilding a phase space that is topologically equivalent to the original system from a one-dimensional time series, enabling researchers to probe the system’s structure on the basis of a single observable and to facilitate subsequent control and prediction. Although the exact structure of a real system is usually unknown, applying phase-space reconstruction nonetheless permits effective analysis and elucidation of its evolutionary behaviours.

2.3. Autoformer Model

In recent years, various high-performing deep learning approaches have emerged for long-term time series forecasting, including LSTNet [21], FEDformer, and the Nonstationary Transformer [22].

LSTNet combines convolutional and recurrent structures to capture both local patterns and long-term dependencies, yet its performance may deteriorate when handling highly nonstationary sequences. FEDformer builds upon the Autoformer framework by incorporating frequency-domain modelling through Fourier/wavelet decomposition, which significantly enhances its ability to model long sequences, though it may exhibit limitations in capturing short-term trends. Nonstationary Transformer explicitly applies stationarization transformations to tackle nonstationary sequences, yielding excellent performance in domains such as meteorology and structural health monitoring; however, its architecture is complex and computationally expensive (

Table 1. The compared of methods.

Model	Core Mechanism	Advantages	Limitations	Typical Applications
LSTNet	CNN + RNN + skip connections	Excels at capturing local patterns and long-term dependencies; stable training	Limited adaptability to highly nonstationary sequences	Electricity demand forecasting, economic indicator prediction
FEDformer	Autocorrelation + Fourier/wavelet decomposition	High efficiency for long sequences; strong frequency-domain modelling capability	Short-term trend depiction may be less accurate than time-domain models	Weather forecasting, energy load prediction
Nonstationary Transformer	Stationarization transformation + Transformer	Outstanding performance for nonstationary series	Complex architecture; high computational cost	Precipitation forecasting, structural health monitoring
Autoformer	Autocorrelation + embedded series decomposition	Balances accuracy and efficiency in capturing trends and periodicity; progressive decomposition improves predictions	May require preprocessing for heavily noisy data	Multi-domain long-term forecasting

).

Autoformer is a Transformer-based time series forecasting model that introduces a novel decomposition architecture [23], greatly enhancing the extraction of latent information. It replaces the conventional self-attention mechanism with a sequence-periodic autocorrelation mechanism, which surpasses self-attention in both efficiency and accuracy and thus boosts overall model performance. Series decomposition is embedded as an internal unit within the encoder–decoder structure. During prediction, the model alternates between refining its forecasts and performing sequence decomposition, incrementally extracting trend and periodic components from latent variables and achieving a progressive decomposition process.

For an input sequence $\mathcal{X}\in {ℝ}^{L\times d}$ of length L, the decomposition proceeds as follows:

$${\mathcal{X}}_t=AvgPool(Padding(\mathcal{X}))$$

(1)

$${\mathcal{X}}_s=\mathcal{X}-{\mathcal{X}}_t$$

(2)

${\mathcal{X}}_t$ and ${\mathcal{X}}_s$ represent the seasonal-variation and trend components, respectively; the operator $AvgPool(\cdot )$ applies average pooling to the original sequence.

Autoformer adopts a concatenated autocorrelation mechanism to enhance information utilization. The core idea is to calculate the time series autocorrelation, identify period-based dependencies, and merge similar subsequences via time-delay aggregation. The autocorrelation is computed as shown in Equation (3):

$${\mathcal{R}}_{\mathcal{X}\mathcal{X}}(\tau )=\underset{L\to \infty }{lim}{\displaystyle \frac{1}{L}}{\displaystyle \sum _{t=1}^L{\mathcal{X}}_t}{\mathcal{X}}_{t-\tau }$$

(3)

${\mathcal{R}}_{\mathcal{X}\mathcal{X}}(\tau )$ reflects the time-delay similarity between ${\mathcal{X}}_t$ and its $\tau$-lagged counterpart ${\mathcal{X}}_{t-\tau }$. Here, $\mathcal{R}(\tau )$ denotes a non-standard confidence score for the period length $\tau$. We select $k$ candidate period lengths ${\tau }_1,\dots ,{\tau }_k$ period-based dependencies are derived from these estimated periods and weighted by their corresponding autocorrelations.

The information from similar subsequences is then aggregated. For a sequence $X$ of length $L$, linear projections yield $Q$, $K$, and $V$, which can be processed in the same manner as in the self-attention mechanism. The resulting autocorrelation mechanism is expressed as follows:

$${\tau }_1,\dots ,{\tau }_k=\mathit{arg}\mathit{Topk}\left({\mathcal{R}}_{\mathcal{Q},\mathcal{K}}(\tau )\right)$$

$${\widehat{\mathcal{R}}}_{\mathcal{Q},\mathcal{K}}({\tau }_1),\dots ,{\widehat{\mathcal{R}}}_{\mathcal{Q},\mathcal{K}}({\tau }_k)=\mathit{SoftMax}\left({\mathcal{R}}_{\mathcal{Q},\mathcal{K}}({\tau }_1),\dots ,{\mathcal{R}}_{\mathcal{Q},\mathcal{K}}({\tau }_k)\right)$$

$$\mathit{Auto}\text{-}\mathit{Correlation}(\mathcal{Q},\mathcal{K},\mathcal{V})={\displaystyle \sum _{i=1}^k\mathit{Roll}}(\mathcal{V},{\tau }_i){\widehat{\mathcal{R}}}_{\mathcal{Q},\mathcal{K}}({\tau }_i).$$

where $\mathit{arg}\mathit{Topk}\left(\cdot \right)$ selects the k time delays with the highest autocorrelation values. After the top $k$ lags are obtained, their corresponding autocorrelation values are normalized by $\mathit{SoftMax}$, which converts them into a probability distribution that reflects the relative importance of each lag in the final aggregation. $Roll(\mathcal{V},{\tau }_i)$ denotes a rolling operation applied to the value sequence $V$ with a time delay of ${\tau }_i$; the operation shifts the data forward or backward and appends the displaced elements to the tail of the sequence. The final autocorrelation output is then calculated as a weighted sum of the rolled value sequences, where the weights are the normalized autocorrelation values ${\widehat{\mathcal{R}}}_{\mathcal{Q},\mathcal{K}}({\tau }_i)$ (Figure 2).

3. Chaos-Autoformer Model Design

3.1. Extraction of Chaotic Features from the Inflow Series

After the temporal resolution of the raw mine water inflow series $Q(t)$ is unified, missing values are imputed by a multi-spline interpolation algorithm. Because the magnitudes of different series vary, each sequence is then standardized with z-score normalization:

$$\widehat{Q}(t)={\displaystyle \frac{Q(t)-\mu }{\sigma }},\mu ={\displaystyle \frac{1}{N}}\sum _{t=1}^{N}Q(t),\sigma =\sqrt{{\displaystyle \frac{1}{N-1}}\sum _{t=1}^N{(Q(t)-\mu )}^2}.$$

Phase-space reconstruction maps the one-dimensional time series into a matrix space that both fully preserves the dynamical information and avoids excessive dimensional expansion. Once the delay time τ and the embedding dimension m are determined, the reconstructed phase space can be obtained.

Multiple methods exist for estimating the delay time τ, for example, the autocorrelation function approach and the C–C method [24,25,26]. Traditional autocorrelation, however, captures only linear dependence, whereas mine water inflow is strongly influenced by non-linear couplings among multiple sources; reliance solely on autocorrelation therefore introduces errors. Average Mutual Information (AMI) [27], in contrast, quantifies statistical dependence of any order through entropy and probability-density estimates, satisfying the criterion of being “neither redundant nor unrelated.” Accordingly, the AMI method is adopted here to determine the optimal delay time.

Common methods for determining the embedding dimension mm include the G-P correlation dimension approach, the FNN (false nearest neighbours) method, and Cao’s method [28]. Takens’ theorem gives a theoretical lower bound of n ≥ 2d + 1n, yet the true system dimension dd is difficult to measure directly. The essence of the FNN method is that, in an m-dimensional reconstructed space, if two points that ought to be far apart are erroneously identified as nearest neighbours, the embedding dimension is insufficient; the dimension is incremented until the proportion of false neighbours drops below a threshold, thereby preventing self-folding. Cao’s method further introduces the index $E_1(m)$ to compensate for small sample fluctuations, yielding a smoother turning point when the time series is short. Given that the mine water inflow series here is relatively limited in length, Cao’s method is employed to determine the embedding dimension m.

After the delay time τ and the embedding dimension m have been established, the inflow time series $X=\{x_1,x_2,\dots \dots ,x_n\}$ can be reconstructed into an m-dimensional vector sequence.

$$X=\left(\begin{array}{ccccc}{\hat{Q}}_1& {\hat{Q}}_{1+\tau }& {\hat{Q}}_{1+2\tau }& \dots & {\hat{Q}}_{1+(m-1)\tau }\\ {\hat{Q}}_2& {\hat{Q}}_{2+\tau }& {\hat{Q}}_{2+2\tau }& \dots & {\hat{Q}}_{2+(m-1)\tau }\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ {\hat{Q}}_{i+N}& {\hat{Q}}_{i+N+\tau }& {\hat{Q}}_{i+N+2\tau }& \dots & {\hat{Q}}_n\end{array}\right)$$

After phase-space reconstruction, the sequence must be tested for chaotic behaviours by evaluating its exponential sensitivity to initial conditions. This assessment is conducted via the maximum Lyapunov exponent λ_max. The sign of λ_max serves as a diagnostic criterion: if λ_max > 0, the system is deemed chaotic; moreover, the predictive horizon of the reconstructed series is given by T = 1/λ_max. Common techniques for estimating λ_max include the small data method, the Wolf algorithm, and the Jacobian approach [29,30]. We primarily estimate λ_max using Rosenstein’s method due to its small-sample stability and cross-check the magnitude with Wolf’s approach to ensure consistency.

3.2. Model Design

The Autoformer architecture has been specifically optimized for long horizon time series forecasting. By combining an autocorrelation mechanism with series decomposition techniques, it realizes efficient feature extraction and effective modelling of long-range dependencies. The model first decomposes the input sequence into trend and seasonal components; this decomposition employs a sliding window smoothing operation to attenuate noise that would otherwise degrade predictive accuracy. The input features of both encoder and decoder are embedded in a high-dimensional space. Each encoder block comprises multiple encoder layers that extract salient characteristics of the sequence. Every encoder layer contains an autocorrelation layer, a feed-forward neural network, layer normalizations, and dropout. The autocorrelation layer captures long-term dependencies; the feed-forward network consisting of two fully connected layers with a non-linear activation improves representational capacity; layer normalization and dropout stabilize training and mitigate overfitting.

The decoder module likewise consists of several decoder layers. Each layer includes an autocorrelation layer and a cross-autocorrelation layer, enabling feature interaction between decoder and encoder while reconstructing the trend and seasonal components. In long horizon forecasting tasks, the decoder’s output features are fused with those from the encoder to reconstruct these components effectively, ultimately generating the predictions.

Building on the baseline Autoformer, the present study further enhances the model’s capability for long-term prediction of non-linear, chaotic time series. The specific enhancements include the following:

(1)

Replacement of Autocorrelation with Chaos-Attention

Because the mine water inflow series exhibits pronounced chaotic behaviour and the system is extremely sensitive to infinitesimal perturbations in its initial conditions, the conventional autocorrelation mechanism cannot capture this feature effectively. Therefore, the original autocorrelation module is substituted with a Chaos-Attention mechanism. By injecting local random perturbations during the autocorrelation calculation, the attention weights become more responsive to slight variations in the sequence’s initial values, thereby strengthening the model’s ability to sense the initial condition sensitivity inherent in chaotic dynamical systems. The computation of Chaos-Attention is given as follows:

$${\mathcal{R}}_{\mathcal{X}\mathcal{X}}^{Chaos}(\tau )=\underset{L\to \infty }{\mathit{lim}}{\displaystyle \frac{1}{L}}{\displaystyle \sum _{t=1}^L{\mathcal{X}}_t}\left({\mathcal{X}}_{t-\tau }+{\delta }_t\right)$$

where ${\delta }_t$ denotes the perturbation term, introduced as a Gaussian random disturbance, and its variance ${\sigma }_t^2$ is directly coupled to the maximum Lyapunov exponent λ_max.

(2)

Incorporation of Lyap-Dropout into the Feed-Forward Network

Traditional feed-forward networks are prone to overfitting when processing highly sensitive mine water inflow series. To mitigate this issue, the present study introduces a Lyap-Dropout mechanism that improves model generalization by dynamically adjusting the dropout rate. Specifically, the dropout probability is adaptively linked to the maximum Lyapunov exponent λ_max. When the sequence exhibits greater sensitivity to initial perturbations, the dropout probability is proportionally increased, thereby restraining the network from fitting anomalous fluctuations in the training data and ensuring the reliability and stability of long-horizon forecasts.

(3)

Chaos-radius injection

Within the decoder’s trend branch, this study introduces a chaotic noise–injection mechanism: small, deliberately added random perturbations enhance the model’s long-horizon stability when forecasting chaotic time series data. The earlier encoder modifications already strengthened medium- and long-term predictive capability, yet in typical chaotic-system tasks such as forecasting mine water inflow, long-trend predictions remain prone to error drift or accumulation under minor initial disturbances. To counter this, the Chaos-Autoformer injects noise into the decoder’s trend pathway:

$$T_{dec-noise}(t)=T_{dec}(t)+{ϵ}_t,{ϵ}_t\sim N(0,{\sigma }_{ϵ}^2)$$

Here, ${ϵ}_t$ is a Gaussian random disturbance, and ${\sigma }_{ϵ}$ is the Chaos Radius, whose magnitude is directly coupled to the maximum Lyapunov exponent. When the sequence is in a chaos-sensitive state (i.e., large λ_max), the injected perturbation is correspondingly larger, heightening the model’s responsiveness to chaotic features.

To accommodate the chaotic nature of mine water inflow series, the Chaos-Autoformer refines the baseline Autoformer with targeted structural adjustments:

• Chaos-Attention replaces the original autocorrelation module in the encoder and incorporates λ_max-related perturbations;
• A Lyap-Dropout scheme is embedded in the feed-forward network;
• The decoder’s trend branch employs the above chaotic noise–injection strategy.

Together, these mechanisms boost the model’s sensitivity to chaotic input variations and reinforce its ability to capture non-linear characteristics while maintaining stable medium- and long-term forecasts. The overall structure of the Chaos-Autoformer model is illustrated below (Figure 3).

3.3. Evaluation Metrics

To comprehensively assess the performance of the Chaos-Autoformer in mine water inflow forecasting, this study constructs an evaluation framework that combines conventional numerical error measures with chaos-dynamics indicators, thereby providing targeted and interpretable assessment.

(1)

Traditional error metrics.

The baseline indicators adopted include mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and the coefficient of determination (R²) [31]. MSE and RMSE quantify overall predictive accuracy; MAE offers a more outlier robust error measure; MAPE facilitates comparison across different magnitudes; and R² reflects the proportion of variance explained by the model. Together, these metrics furnish an intuitive measure of numerical prediction accuracy.

(2)

Chaos-specific metrics.

Traditional indices alone cannot reveal the model’s stability and dynamical consistency when handling non-linear chaotic sequences. Accordingly, two additional indicators are introduced. Maximum Lyapunov exponent deviation (Δ$\widehat{\lambda }$ obtained by separately estimating the maximum Lyapunov exponents of the observed and predicted series gauges their agreement in divergence rates and thus indicates whether the model successfully reproduces chaotic behaviour. Wasserstein distance (Earth Mover’s Distance, EMD) measures the global distributional discrepancy between the predicted and true trajectories in phase space, capturing similarity from a structural viewpoint.

4. Validation

4.1. Overview of the Study Area

The Liangshuijing Coal Mine is located in the northern sector of the Shaanxi Loess Plateau, on the southeastern margin of the Maowusu Desert (Figure 4). The mining district is dominated by aeolian sand geomorphology, consisting mainly of sandy flats and dune-slope terrain; dunes undulate continuously, and the strata dip gently at less than 1°, forming an overall westward inclined monocline with simple structural complexity. The district constitutes an independent hydrogeological unit.

Most of the surface is mantled by Quaternary deposits. From oldest to youngest, the stratigraphic sequence comprises: the Upper Triassic Yongping Formation (T³y); the Middle Jurassic Yan’an Formation (J²y) and Zhiluo Formation (J²z); the Upper Neogene Baode Formation (N²b); the Middle Pleistocene Lishi Formation (Q²L); the Upper Pleistocene Salawusu Formation (Q³S); and Holocene aeolian sands (Q⁴eol) and alluvial layers (Q⁴al). The coal-bearing strata belong to the Yan’an Formation, with the current principal seam being the No. 4-2 coal seam, which averages about 2.98 m in thickness. A comprehensive stratigraphic column is provided in Figure 5.

The principal aquifers in the study area, listed from top to bottom, comprise a Quaternary unconsolidated pore phreatic aquifer and a Mesozoic clastic rock fissure confined aquifer dominated by the Middle Jurassic Yan’an Formation.

The pore phreatic aquifer is lithologically characterized by yellow, medium- to fine-grained, well-sorted aeolian sand.

The Yan’an fissure confined aquifer consists mainly of grey, very thick-bedded, coarse-grained feldspathic quartz sandstone that is poorly sorted and poorly rounded, with local intercalations of silty fine sandstone; weathering processes have generated abundant fractures, imparting favourable permeability and water storage conditions.

In this study, 53 months of mine water inflow monitoring data-spanning January 2021 to June 2025 together (Figure 6), are selected as the experimental dataset for subsequent analysis.

The mine water inflow dataset used in this study was obtained from the Liangshuijing Coal Mine. Continuous monitoring was conducted from January 2021 to June 2025, yielding a total of 53 monthly aggregated observations (Figure 6). The inflow data were measured using automated underground flow metres (precision: ±0.5 m³/h), with raw daily readings aggregated into monthly means to reduce short-term noise.

Missing values in the raw time series accounted for less than X% of the total records and were imputed using a cubic multi-spline interpolation scheme to preserve smooth temporal continuity. To account for varying magnitude scales and facilitate stable model training, the mine water inflow dataset was normalized via z-score transformation before phase-space reconstruction.

In order to reduce noise in dataset, noise handling was carried out in two stages:

(i)

During phase-space reconstruction, delay time and embedding dimension were determined using the AMI and Cao methods, which inherently filter redundant or irrelevant information;

(ii)

In the model’s encoder–decoder, a sliding-window decomposition separated seasonal and trend components, attenuating high-frequency noise prior to long-horizon prediction.

To ensure chronological integrity, the 53-month mine water inflow dataset (January 2021–June 2025) was strictly split along the time axis:

Training set: January 2021–June 2024 (42 months, 79% of total data), used to fit model parameters.

Validation set: July 2024–December 2024 (6 months, 11% of total data), used for hyperparameter tuning and early stopping.

Testing set: January 2025–June 2025 (6 months, 11% of total data), reserved for the final evaluation.

This split ensures that the validation and testing windows each cover a full seasonal cycle, enabling the model’s capacity to capture both high- and low-flow periods. The proportions were selected to maximize training data availability while preserving sufficient out-of-sample periods for robust evaluation.

Additionally, no shuffling or random sampling was performed at any stage; the validation and test sets contain only observations occurring strictly after the training period. This chronological partition prevents any future information from leaking into the model training process.

4.2. Extraction of Chaotic Features

The delay time (τ) and embedding dimension (m) are key parameters in phase-space reconstruction and directly affect the preservation of dynamical structure.

Using the Average Mutual Information (AMI) method, τ was varied from 0 to 180 days in 1-day increments. The AMI curve exhibited its first local minimum at τ = 103 days, indicating the point of minimum redundancy while retaining relevant dynamical information.

For the embedding dimension (m), Cao’s method was applied with m ranging from 2 to 12. The E₁(m) curve plateaued at m = 6, beyond which the increment in neighbouring distances became negligible, suggesting that higher dimensions provided no additional structural information and could introduce unnecessary computational overhead (Figure 7).

Reconstruction with τ = 20 days therefore produces a six-dimensional phase space Y (Figure 8 and Figure 9). Applying Rosenstein’s algorithm to this space gives a maximum Lyapunov exponent of λ_max = 0.0011, confirming that the mine water inflow system exhibits pronounced chaotic behaviour over the study period. The corresponding predictable horizon is T = 1/λ_max ≈ 909 approx. 909 days, so the model’s validation window is set to July 2024–January 2027 (30 months).

4.3. Chaos-Autoformer Model Training

The Chaos-Autoformer developed in this study embeds chaotic-system dynamics into deep time series modelling from the outset. The input and output dimensions (enc_in, dec_in, c_out) are each set to 1, while both the sequence length and the forecasting horizon are fixed at 24. The core employs a multi head attention mechanism with a model width d_model = 512, eight attention heads (n_heads = 8), and two encoder and two decoder layers. To enhance the network’s capacity to capture the initial condition sensitivity of non-linear systems, two chaos-oriented components chaos_radius and Lyap-Dropout are introduced with initial values of 0.01 and 0.05, respectively.

Before training, the model’s hyper parameters are tuned automatically via Bayesian optimization within a predefined search space encompassing the learning rate, chaos_radius, Lyap-Dropout value, batch size, model width, number of attention heads, and layer counts. Twenty-five optimization trials yield the following best configuration: learning rate = 0.0001; chaos_radius = 0.2798; Lyap-Dropout = 0.1476; batch size = 10; model width = 256; four attention heads; one encoder layer; and two decoder layers.

Training is conducted for 100 epochs using the AdamW optimiser, with a CosineAnnealingLR scheduler for dynamic learning rate adjustment and an early stopping criterion (patience = 5) to avert overfitting. Throughout training, both training and validation losses decline sharply and converge within the first ten epochs, thereafter, remaining stable (see Figure 10), indicating efficient learning and satisfactory convergence behaviour.

Meanwhile, to track the dynamic stability of training, we recorded the gradient norm trajectory across epochs (Figure 11). The gradient reached its maximum (5.5643) in epoch 13 and then declined steadily, with no sign of either explosion or vanishing, indicating that parameter updates remained stable and effective throughout learning. In addition, the intensity of parameter updates for individual architectural blocks was monitored (Figure 12). The encoder—especially the encoder_attention and encoder_fn layers—exhibited the highest activity, implying that the model chiefly relies on the encoder to capture long-range dependencies in the input sequence. By contrast, the decoder’s embedding layer showed only minor fluctuations, suggesting that it primarily serves the output mapping function.

The evolution of evaluation metrics during training further corroborated the improvement in model performance (Figure 13). By the fifth epoch, key indicators—MSE, RMSE, MAE, and MAPE—had already reached low values, and the directional accuracy climbed to 52.04% at epoch 25, indicating that the model not only achieved strong numerical accuracy but also exhibited a certain degree of trend tracking capability.

In the post-training error analysis, the overall distribution of prediction errors approximated a Gaussian form, and the errors progressively converged and became more concentrated as the number of training epochs increased (Figure 14), demonstrating that the learned model is stable and reliable, with commendable generalization and robustness.

4.4. Model Evaluation

During the validation phase, the generalization performance of the Chaos-Autoformer was assessed using five-fold cross validation. The dataset was randomly partitioned into five equal subsets; in each iteration, four subsets were used to train the model, and the remaining subset served for validation. This procedure was repeated five times to evaluate predictive robustness.

The cross validation results indicate that the Chaos-Autoformer exhibits high stability and reliability: the root mean square error (RMSE) ranged from 80 to 130, the mean absolute error (MAE) from 60 to 100, and the symmetric mean absolute percentage error (sMAPE) from approximately 5% to 8% (Figure 15). The small fluctuations of these metrics across folds confirm that the model maintains consistent predictive performance under different data splits, demonstrating strong generalization capability for complex chaotic time series forecasting.

During the ablation study, we further examined the contribution of each chaos-aware component within the Chaos-Autoformer to the model’s predictive performance. Multiple ablation settings were configured, including the following:

(i)

Removal of the Chaos Awareness module;

(ii)

Removal of the EMD loss constraint;

(iii)

Removal of the Lyap-Dropout mechanism;

(iv)

Simultaneous removal of all chaos-related components (the Minimal Model).

Using the complete Chaos-Autoformer as the baseline (performance change = 0%), the results revealed that eliminating the Chaos Awareness module produced a modest 0.8% improvement in RMSE, yet the Lyapunov error worsened dramatically by 51.4%. Excluding the EMD loss constraint degraded RMSE by 11.9% and increased the Lyapunov error by 4.0%. Removing the Lyap-Dropout mechanism led the model to ignore the chaotic features of the inflow series, raising RMSE by 19.8% while the Lyapunov error rose slightly by 13.5%. Finally, deleting all chaos-aware components (Minimal Model) resulted in a 13.2% deterioration in RMSE and a substantial 23.5% increase in the Lyapunov error (Figure 16).

To verify that the improvement from Chaos-Attention is not due to random variation, a paired t-test based on five independent runs was conducted. The RMSE improvement over the standard attention was statistically significant (p = 0.0087), with similar significance observed for MAE (p = 0.011) and R² (p = 0.014).

In addition, the effect of Lyap-Dropout on generalization was assessed by varying its dropout probability from 0.05 to 0.25 in increments of 0.05. The optimal value of 0.15 yielded the lowest RMSE and sMAPE, and the highest R², reducing RMSE by 19.8% and improving directional accuracy by 7.4% compared with the no-dropout setting.

Overall, these results confirm that both Chaos-Attention and Lyap-Dropout contribute significantly to improving the predictive accuracy, dynamical consistency, and generalization capability of the Chaos-Autoformer for chaotic time series forecasting.

4.5. Application and Analysis

After training, the model was applied to forecast mine water inflow for the period from June 2024 to June 2025 (one full year). The predicted series was then compared with the observed data for the same interval (Figure 17). As shown in the corresponding figures, the forecasts agree well with the measurements in both overall trend and oscillatory behaviour, and the prediction errors are generally small. Residual analysis gives a mean residual of −10.7320 and a standard deviation of 28.7920, indicating that the errors lie within an acceptable range. Further calculations yield an RMSE of 30.73 m³/h, an MAE of 26.03 m³/h, a MAPE of 2.61%, and a coefficient of determination R² [26] of 0.895. Although the model exhibits slightly over or under estimation during periods of extreme inflow fluctuations, its overall predictive accuracy remains high.

Based on the foregoing study, the Chaos-Autoformer model was employed to forecast mine water inflow for the next 909 time steps (July 2025–November 2027). As illustrated in Figure 18, the predicted inflow series exhibits a high degree of agreement with the historical observations in both overall trend and fluctuation pattern, effectively capturing the periodic characteristics and amplitude of mine water inflow. The 95% confidence intervals (CIs) shown in grey were generated using a Monte Carlo dropout approach, in which dropout layers were kept active during 500 stochastic forward passes to obtain an empirical predictive distribution. The mean CI width across the forecast horizon was 42.8 m³/h, with a Prediction Interval Coverage Probability (PICP) of 94.2%, close to the nominal 95% level. Wider intervals occurred during summer peaks, reflecting greater meteorological variability, while narrower intervals were found in winter low-flow periods. These results indicate that the predictive uncertainty is well-calibrated and sufficiently narrow to support reliable safety-related decision-making in mine water hazard prevention.

A detailed inspection of the forecast curve shows that, from 2025 to 2027, mine water inflow follows a distinct annual cycle: inflow rises sharply during the summer months and drops markedly in winter, a pattern closely associated with the seasonal distribution of rainfall in the mining area. The highest predicted inflow occurs on 25 August 2026, reaching 1096.24 m³/h, whereas the lowest value is expected on 27 January 2026 at 873.46 m³/h. Overall, the forecasts indicate a gently declining trend in mine water inflow over the next three years.

In summary, the Chaos-Autoformer proposed in this study achieves high predictive accuracy, effectively capturing the dynamic characteristics of complex chaotic time series and delivering reliable medium- to long-term forecasts of mine water inflow. Accordingly, it offers valuable technical support for the proactive deployment and efficient management of mine water hazard mitigation.

5. Discussion

5.1. Applicability and Generalization

The dataset used in this study was derived from mine water inflow observations at the Liangshuijing Coal Mine; therefore, the results primarily reflect the specific hydrogeological and climatic conditions of this site, inevitably limiting the direct applicability of the proposed method to mines with markedly different geological settings. To enhance its applicability, the model can be further trained with data from nearby mines under similar geological conditions, while incorporating production-related variables closely associated with inflow processes, such as mining advance rates and water drainage operations. Future work will involve validation using datasets from mines located in different regions and characterized by diverse geological and hydro-environmental conditions, thereby systematically assessing and improving the model’s generalization capability and adaptability.

5.2. Model Performance in Medium- and Long-Term Forecasting

In medium- and long-term forecasting of mine water inflow, the Chaos-Autoformer model delivers high accuracy in extracting periodic features and capturing trends; however, its fitting performance still declines progressively as the prediction horizon lengthens. The principal reasons are three-fold:

(i)

The intrinsic chaotic nature of the mine hydro system, which renders it highly sensitive to initial conditions;

(ii)

Layer by layer error accumulation during rolling multi step prediction;

(iii)

Potential phase shifts or amplitude attenuation of periodic structures over long time scales.

Compared with a traditional LSTM, Chaos-Autoformer possesses a stronger capacity for modelling long sequences. Informer, although advantageous in handling long-range dependencies, lacks structural design for non-linear chaotic characteristics and therefore exhibits deficiencies in trend fitting and fluctuation tracking. Empirical comparisons show that Chaos-Autoformer—by virtue of its Chaos-Attention mechanism, Lyap-Dropout dynamic regularization, and trend–seasonality decomposition—effectively suppresses error diffusion over multi-step horizons and maintains superior predictive stability and dynamical consistency.

To alleviate the deterioration in accuracy over extended forecast periods, future research could, first, introduce multi-step supervision, teacher-forcing strategies, or error feedback correction schemes to enhance robustness against cumulative errors. Second, external hydro meteorological variables and groundwater physics models may be coupled with the network so that physical constraints are embedded in the deep learning process, improving adaptability during abnormal evolutionary phases. Moreover, constructing multi scale decomposition structures and ensemble prediction frameworks that separately model trend and fluctuation components should further raise long cycle fitting accuracy. Prospects also include incorporating invariants of chaotic systems as training constraints or designing modules capable of preserving phase-space trajectories, thereby boosting the model’s long-term forecasting capacity and generalizability for complex chaotic sequences, and ultimately providing more reliable, forward-looking technical support for mine water hazard prediction and early warning.

5.3. Practical Utility of Forecast Results

The medium- and long-term mine water inflow forecasts produced by the Chaos-Autoformer model can be directly applied to mine production management. By anticipating seasonal peaks and troughs as well as long-term trends, coal mines can optimize underground drainage strategies and upgrade or construct drainage systems to ensure adequate capacity during peak inflow periods. In parallel, proactive prevention measures—such as grouting to reinforce goaf areas—can be implemented in advance to effectively reduce water hazard risks and enhance operational safety.

6. Conclusions

(1)

The Chaos-Autoformer model constructed in this study blends chaotic-dynamics theory with deep time series forecasting. By introducing a Chaos-Attention mechanism, a Lyap-Dropout stochastic inactivation strategy, and chaos-radius noise injection, the architecture is designed to capture both the sensitivity of time series to initial condition perturbations and their complex non-linear dependencies.

(2)

Training and testing show that Chaos-Autoformer delivers excellent predictive performance. Across multiple error metrics, the model achieves very low errors (RMSE and MAE are markedly reduced, while MAPE remains in single-digit percentages). Directional accuracy exceeds 80%, demonstrating high numerical precision and reliable trend detection. Compared with the standard Autoformer, Chaos-Autoformer lowers RMSE by about 16.67%. For mine water inflow during June 2024 to June 2025, the model attains an RMSE of 30.73 m³/h and a coefficient of determination R² = 0.895, confirming strong fitting and forecasting ability for medium-term tasks.

(3)

Applied to medium- and long-term forecasting of mine water inflow at the Liangshuijing Coal Mine for July 2025 to November 2027, Chaos-Autoformer reveals a pronounced annual cycle: inflow drops to a minimum of 873.46 m³/h in January 2026, rises to a peak of 1096.24 m³/h in August 2026, and thereafter exhibits a gentle downward trend. Overall, mine water inflow is expected to decline slowly over the next three years.