⚡ electrical engineering

CA-DPCNN prediction of mining-induced failure depth in coal seam floors

This paper presents a continuous adaptive derivative-prior-constrained neural network (CA-DPCNN) that accurately predicts mining-induced failure depths in coal seam floors by incorporating engineering priors and an unclipped normalization strategy, achieving superior accuracy and adherence to physical trends compared to traditional models, especially when extrapolating beyond the training data range.

Original authors: Tianwei Lan, Bole Li, Yonghao Liu, ZhiJia Zhang, TianQi Wei, ShunXiang Wang, LuTong Niu

Published 2026-10-09

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CC BY 4.0

Original authors: Tianwei Lan, Bole Li, Yonghao Liu, ZhiJia Zhang, TianQi Wei, ShunXiang Wang, LuTong Niu

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep beneath the earth's surface, where coal miners work, the rock floor is not a static, unchanging slab. When a large section of coal is removed, the immense weight of the rock layers above shifts, causing the floor to crack, fracture, and fail. This "failure depth" is a critical measurement for engineers. If they underestimate how deep these cracks go, they might leave a water-resistant pillar of coal too thin, risking a sudden and catastrophic flood from underground aquifers. If they overestimate the depth, they might leave too much coal behind, unnecessarily wasting a valuable resource. For decades, scientists have tried to predict this depth using formulas based on past observations, but these methods often struggle when conditions change or when mines dig deeper than any previous records. The challenge is to create a prediction tool that is not only accurate but also behaves logically, respecting the physical rules of the earth even when it encounters data it has never seen before.

A team of researchers from Liaoning Technical University and Inner Mongolia Power Investment Energy Co., Ltd. has developed a new approach to solve this problem, moving beyond simple formulas to a system that learns from data while strictly adhering to engineering logic. They created a specialized computer model called CA-DPCNN. Unlike a standard computer program that simply memorizes past examples, this model is built with a set of "rules of thumb" baked into its learning process. These rules reflect what engineers know to be true: for instance, as the mining depth increases, the pressure on the floor increases, so the failure depth should also increase. The model is penalized if it predicts otherwise. The researchers trained this system on 70 distinct records from various mines across China, covering a range of mining depths, coal seam angles, and excavation sizes. They then tested it on 21 new records that the model had never seen, including four cases where the mining depth was significantly deeper than anything in the training data.

The results showed that this new method outperformed traditional techniques. When tested against the unseen records, the CA-DPCNN model predicted the failure depth with an average error of just 2.5 meters, a significant improvement over older methods which had errors ranging from 3 to 7 meters. More importantly, the model demonstrated a remarkable ability to stay consistent with physical reality. In a detailed check across over a thousand simulated points, the model followed the expected engineering trends in nearly 99% of cases. By contrast, a standard computer model without these built-in rules followed the correct trends in only about 2.5% of cases, often producing results that made no physical sense. The researchers found that the most significant improvement came from the built-in rules themselves, which forced the model to learn the correct direction of change rather than just memorizing numbers.

A crucial part of the study involved how the model handled situations where the mine was deeper than any it had been trained on. In many computer systems, if a number comes in that is larger than the biggest number the system knows, the system simply cuts it off at the maximum limit, effectively pretending the extra depth doesn't exist. The researchers found that this "clipping" caused the model to underestimate the danger in deep mines. By allowing the model to accept these larger numbers without cutting them off, the error in predicting the deepest mines dropped by nearly half. This suggests that the model can safely extrapolate into unknown territory, provided it is allowed to see the full scale of the input. The study concludes that by combining data-driven learning with strict physical constraints, engineers can get more reliable predictions for mine safety, even as they push deeper into the earth's crust where data is scarce.

Technical Summary: CA-DPCNN Prediction of Mining-Induced Failure Depth in Coal Seam Floors

Problem Statement
Accurate prediction of mining-induced failure depth in coal seam floors is critical for designing water-resistant coal pillars, assessing mining safety above confined aquifers, and controlling mine-water hazards. However, existing models face two primary challenges: (1) they often struggle with sparse field datasets where the model may achieve low fitting error but exhibit local derivative directions that conflict with established engineering expectations (e.g., monotonicity or curvature); and (2) they frequently fail to preserve information or maintain physical credibility when mining depths approach or exceed the observed training range. Traditional data-driven models trained primarily on sample-fitting error lack mechanisms to enforce these engineering priors, while standard extrapolation strategies often clip normalized inputs, leading to systematic underestimation in deep-mining scenarios.

Methodology
The authors propose the Continuous Adaptive Derivative-Prior-Constrained Neural Network (CA-DPCNN), a physics-informed machine learning framework designed to address these limitations.

Architecture: The model utilizes a fully connected feedforward network (4-64-32-1) with Tanh activation in hidden layers and a Softplus output layer to ensure non-negative predictions. The inputs are mining depth ( $H$ ), coal seam dip angle ( $\theta$ ), mining thickness ( $M$ ), and working-face dip length ( $L$ ).
Derivative-Prior Constraints: Unlike standard neural networks, the CA-DPCNN incorporates engineering expectations as soft penalties in the loss function. Specifically, it imposes:
- First-order non-negative priors on mining depth, mining thickness, and working-face dip length (indicating that failure depth should increase or remain constant as these factors increase).
- A first-order non-negative prior on dip angle (retained as a weaker empirical prior).
- A second-order non-positive prior on working-face dip length (indicating concavity).
Continuous Adaptive Weighting: A key innovation is the conditioning of penalty weights on the actual mining depth ( $H_{real}$ ). Instead of fixed weights, the penalty strengths for mining depth, thickness, and dip length vary smoothly via logistic functions centered at 750 m. This allows the constraint strength to adapt across the observed depth range, whereas the dip-angle weight remains fixed.
Modified Extrapolation Strategy: During inference, the model retains normalized input values outside the $[0, 1]$ range rather than clipping them. This strategy preserves information from mining depths that exceed the training set maximum.
Training and Evaluation: The model was trained on 70 records from 61 mine groups, with 21 records reserved for sequential hold-out evaluation. An ensemble of 30 random initializations was used to assess stability. Performance was benchmarked against Backpropagation Neural Networks (BPNN), Multiple Linear Regression (MLR), Support Vector Regression (SVR), and standard empirical formulas.

Key Contributions

Integration of Shape Priors: The study demonstrates that encoding monotonicity and curvature constraints directly into the loss function significantly improves the physical consistency of predictions, particularly in data-sparse regimes.
Depth-Conditioned Weighting: The introduction of continuous, mining-depth-dependent penalty weights allows the model to balance data fitting and physical constraints more effectively than fixed-weight approaches, ensuring smooth transitions in constraint strength.
Extrapolation Preservation: The paper validates a modified extrapolation strategy where unclipped normalized inputs are used during inference. This approach prevents the systematic underestimation of failure depth in deep mining scenarios where inputs fall outside the training distribution.

Results

Predictive Accuracy: On the 21-record hold-out set, the CA-DPCNN ensemble achieved an RMSE of 3.321 m, MAE of 2.533 m, MAPE of 13.084%, and $R^2$ of 0.839. These metrics were superior to BPNN, MLR, SVR, and the standard empirical formula under the same data split.
Ablation Analysis: Comparisons with a Softplus-MLP (no priors) and a fixed-weight DPCNN revealed that derivative priors are the primary source of error reduction (reducing RMSE by ~7.09 m compared to the unconstrained model). Continuous weighting maintained the low error of the fixed-weight model while offering a more auditable constraint mechanism.
Physical Consistency: In a data-supported audit domain of 1,024 points, the CA-DPCNN achieved an all-prior joint satisfaction rate of 98.94% ± 0.94%, compared to only 2.46% for the unconstrained Softplus-MLP.
Extrapolation Performance: For four records with mining depths (900–1,056 m) exceeding the training maximum (810 m), the unclipped normalization strategy reduced the RMSE from 9.806 m to 5.644 m (a 42.45% reduction), effectively mitigating information loss caused by clipping.

Significance
The paper claims that the CA-DPCNN successfully combines lower predictive error with responses that closely follow prescribed engineering trends. By integrating derivative priors and adaptive weighting, the model addresses the dual challenge of maintaining physical credibility with small datasets and preserving information when extrapolating to deeper mining conditions. The results suggest that this approach offers a robust alternative to traditional empirical formulas and standard machine learning models for assessing floor failure depth, particularly in scenarios involving deep mining where observed data is limited. The authors note that while the modified extrapolation strategy shows promise, broader assessment requires additional deep-mining records.

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