Study on slurry mass concentration prediction method driven by multi-source data fusion

Abbreviations and acronyms coal preparation process

In the coal preparation process, slurry mass concentration serves as a core control variable that permeates the entire flowsheet and exerts a decisive influence on key unit operations such as dense-medium cyclone separation, magnetic recovery, flotation, and thickening/filtration. As illustrated in Fig. 1, from raw coal pretreatment through dense-medium separation, magnetic-medium recovery, flotation, thickening, filtration, and water recycling, the operational stability and product quality of each stage depend directly on the accurate regulation of slurry concentration. In dense-medium separation, the separation precision is governed by the medium density, which is itself determined by the feed slurry mass concentration; maintaining a stable concentration is therefore essential for stabilizing the separation interface and achieving the desired clean-coal ash content. During magnetic recovery, excessively high feed concentration increases slurry viscosity and promotes particle agglomeration, thereby elevating tailings grade and constraining processing capacity, whereas excessively low concentration reduces recovery per unit volume and may lead to losses of magnetic medium (Lv et al., 2022). Thus, the feed concentration must be maintained within an optimal range to balance recovery rate, product yield, and throughput.

Within the flotation circuit, slurry concentration directly affects collector dispersion, hydrophobic agglomeration, and the probability of effective bubble–particle collisions, thereby determining flotation selectivity and clean-coal recovery. In the subsequent thickening and filtration stages, slurry mass concentration becomes the dominant factor controlling dewatering performance: insufficient concentration results in slow settling and turbid overflow, while overly high concentration impairs flocculant dispersion and deteriorates thickening efficiency; in filtration, low feed concentration prolongs filter cake formation and decreases dewatering efficiency. Overall, slurry mass concentration governs medium density, separation stability, flotation response, and dewatering performance, and is tightly coupled with pump speed, dilution water addition, and reagent dosing. It therefore represents a fundamental parameter for coordinating product quality and energy efficiency and for ensuring the stable and efficient operation of the entire coal preparation process.

Problem description of slurry mass concentration measurement in coal preparation

At coal preparation plants, real-time monitoring of feed slurry mass concentration is essential to ensure stable operation of key process stages. However, the commonly utilized manual measurement methods typically involve multiple steps such as settling, filtering, drying, and weighing, which suffer from poor sampling representativeness, long detection cycles, and delayed experimental outcomes (Bamberger & Greenwood, 2004).

For example, in the dense medium cyclone and magnetic separator feed stages, manually determining slurry mass concentration cannot promptly capture fluctuations, making it difficult to detect and correct issues such as unstable separation density limits and medium losses in time. In the flotation feed process, the low frequency of manual sampling cannot dynamically guide the appropriate dosing and distribution of flotation reagents, negatively impacting dispersion performance and clean coal recovery. For thickeners, manual measurement of feed concentration is susceptible to disturbances such as slurry stratification, leading to imprecise control of flocculant and coagulant dosing, which affects settling and separation efficiency. Similarly, fluctuations in filter press feed concentration are difficult to detect and adjust promptly through manual inspection, reducing filter cake formation efficiency and dewatering performance.

As shown in Fig. 2, the current status of slurry mass-concentration monitoring and control reveals that conventional grab sampling and offline assays suffer from delayed feedback and operator-induced variability. Field adjustments rely on experience-based rapid actions and assay-driven lagged corrections. Under fluctuating operating conditions, these approaches fail to achieve timely process stabilization, highlighting the necessity of continuous online characterization and automated, coordinated control.

Overall, manual testing not only entails high labor intensity but also fails to provide real-time, online monitoring and closed-loop control, making it inadequate for meeting the modern coal preparation plant’s requirements for efficient, stable, and intelligent production management.

Bimodal collaborative measurement approach for slurry mass concentration

In the measurement of slurry mass concentration, both infrared transmitted light intensity and pressure difference can characterize changes in slurry properties, and there is a certain correlation between the two. The transmitted light intensity is affected by factors such as slurry volumetric concentration, true particle density, and particle size distribution, while the pressure difference depends on parameters including slurry volumetric concentration, fluid dynamic characteristics, and the resistance of the pipeline system (Jiang et al., 2022). After optimization measures, these two measurement methods are primarily constrained by slurry volumetric concentration and true particle density, thus providing complementary advantages in slurry concentration monitoring. Meanwhile, given known volumetric concentration and true particle density, the mass concentration of the system can be further calculated. Mass concentration is defined as the mass of solid particles contained in a unit volume of the suspension and can be directly obtained as the product of the volumetric fraction and the true particle density. enabling estimation of slurry mass concentration without additional measurement conditions.

Infrared transmitted light intensity measurement is a widely utilized method for investigating the physical and chemical properties of materials such as slurries, liquids, or gases (Zhao et al., 2024). The fundamental principle of the method is that an infrared laser emitted by the infrared light source penetrates the sample under test, and the characteristics of the sample are inferred by measuring variations in the transmitted light intensity. In infrared transmission intensity measurements, the infrared light source emits an infrared laser, typically within a wavelength range of 700 nm to 1 mm. Depending on the properties of the substance, the infrared light may be absorbed, scattered, or transmitted during propagation. In this study, slurry mass concentration is reported in g L⁻¹, slurry volumetric concentration in %, and particle true density in g cm⁻³.

In infrared transmitted light intensity measurement, the Lambert–Beer law provides the theoretical basis for describing the relationship between light intensity and sample concentration. According to the Lambert–Beer law, there is an exponential relationship between the transmitted light intensity and the concentration and thickness of the sample. Specifically, the transmitted light intensity decreases as the sample concentration increases, as expressed in Eqs. (1) and (2):

(1) $$T\left(l,r_i\right)=T_0\exp \left(-{\displaystyle \frac{2r_i}{l}}\right)$$

(2) $$l(d,\varphi )={\displaystyle \frac{2d}{3\varphi Q}}$$where $l$ is the mean free path of photons, with higher $l$ values indicating more photons passing through the sample cell; $r_i$ is the inner diameter of the sample cell; where $T_0$ denotes the incident light intensity and $T$ the transmitted light intensity; $d$ is the particle diameter; $Q$ is the absorbance; and $\varphi$ is the slurry volumetric concentration.

By combining Eqs. (1) and (2), we obtain

(3) $$\varphi =-{\displaystyle \frac{d}{3rQ}(\ln T-\ln T_0).}$$

Under ideal conditions, parameters $d$ and $r_i$ are considered constants, while $Q$ is influenced by the absorbance of the dispersed phase. In the slurry system, ash content affects the slurry absorbance, and there is a strong correlation between particle ash content and density. Therefore, in this study, $Q$ is considered a function of the true particle density, i.e., $Q={\rho }_{dis}$.

The principle of pressure difference measurement is based on the pressure differential in fluid mechanics. When slurry flows through a pipeline, the presence of solid particles introduces resistance, resulting in a pressure difference (Wang et al., 2012). By precisely measuring the pressure difference between the two ends of the pipeline, the slurry volumetric concentration can be inferred. In this study, since the measurement of slurry volumetric concentration is performed in a vertical pipe, the effect of local resistance losses on the overall pressure difference can be neglected. Therefore, this work focuses on the influence of frictional (along-the-length) resistance losses on pressure difference measurement.

In practical applications, coal-slurry flow is influenced by pipe-wall friction, which generates distributed (along-the-pipe) resistance that cannot be neglected during transport. The corresponding frictional loss is calculated as follows:

(4) $$h_f=\gamma {\displaystyle \frac{l}{d}{\displaystyle \frac{v^2}{2g}}}$$where $\gamma$ denotes the along-the-pipe friction factor, $l$ is the pipe length, $d$ the internal pipe diameter, $v$ the cross-section–averaged velocity, and $g$ the gravitational acceleration.

Accordingly, the differential pressure measured by the laboratory instrument comprises two contributions—the ideal hydrostatic head and the distributed frictional loss. The total differential pressure can therefore be expressed as $\mathrm{\Delta }p=\rho gh-h_f$. Hence, Eq. (5) is obtained.

(5) $$\rho ={\displaystyle \frac{\mathrm{\Delta }p+h_f}{gh}}$$where $\rho$ is the slurry density; $\mathrm{\Delta }p$ is the pressure difference; $h_f$ is the frictional (along-the-pipe) resistance loss; $g$ is the acceleration due to gravity and $h$ is the liquid level difference;

The slurry density is determined by the densities of both solid particles and the liquid phase (water). In the slurry, the proportion of solid particles and water varies with the volumetric concentration, resulting in changes in slurry density. Specifically, the slurry density can be calculated as the weighted average of the true density of solid particles and the density of water, where the proportion of solid particles is determined by the slurry volumetric concentration $\varphi$. Therefore, the slurry density can be calculated by the following relation:

(6) $${\rho }_{slurry}=\varphi \cdot {\rho }_{dis}+\left(1-\varphi \right)\cdot {\rho }_{water}$$where ${\rho }_{slurry}$ is the slurry density; ${\rho }_{water}$ is the density of water; ${\rho }_{dis}$ is the true density of particles and $\varphi$ is the slurry volumetric concentration.

Substituting the slurry density ${\rho }_{slurry}$ into the above pressure difference formula (Eq. (5)), we obtain:

(7) $$\varphi ={\displaystyle \frac{\mathrm{\Delta }p+h_f-gh}{gh\left({\rho }_{dis}-1\right)}.}$$

Based on the foregoing analyses of the transmission-optics and pressure-drop theories, simultaneous solution of the governing relations allows the volumetric concentration and the true particle density to be expressed as functions of the transmitted light intensity and the differential pressure. In Eq. (9), the parameter $Q$ denotes the correlation function of the true particle density; its physical meaning and analytical form have been detailed earlier in the manuscript:

(8) $$\varphi ={\displaystyle \frac{\mathrm{\Delta }p+h_f-gh}{gh\left({\rho }_{dis}-1\right)}}$$

(9) $$\varphi =-{\displaystyle \frac{d}{3rQ}(\ln T-\ln T_0).}$$

By equating the right-hand sides of Eqs. (8) and (9)—i.e., enforcing that, under the theoretical assumptions, the differential-pressure and transmission-based formulations provide a consistent description of the volumetric concentration—we obtain:

(10) $$\left(\mathrm{\Delta }p+h_f-gh\right)\cdot {\rho }_{dis}=-{\displaystyle \frac{dgh}{3r}\left(\ln T-\ln T_0\right)\left({\rho }_{dis}-1\right).}$$

Since several physical quantities in the equations are constant under the experimental conditions, we simplify the expressions for ease of derivation. Let:

(11) $$-{\displaystyle \frac{dgh}{3r}=k_1}$$

(12) $$h_f-gh=k_2$$

(13) $$\ln T_0=k_3.$$Under these definitions, the original expression simplifies to:

(14) $$\left(\mathrm{\Delta }p+k_2\right)\cdot {\rho }_{dis}=k_1\left(\ln T-k_3\right)\left({\rho }_{dis}-1\right).$$Hence, the true particle density ${\rho }_{dis}$ is obtained as:

(15) $${\rho }_{dis}={\displaystyle \frac{k_1\ln T-k_1{k}_3}{k_1\ln T-k_1{k}_3-\mathrm{\Delta }p-k_2}.}$$

Based on the foregoing analyses of transmission-optics and pressure-drop theories, simultaneous solution of the governing relations yields the volumetric concentration and the true particle density as functions of the transmitted light intensity and the differential pressure. Substituting the expression for true particle density into the volumetric concentration formula (Eq. (9)), and combining and simplifying the constant terms, the volumetric concentration can be expressed as:

(16) $$\varphi ={\displaystyle \frac{k_1\ln T-k_1{k}_3-\mathrm{\Delta }p-k_2}{gh}.}$$

Meanwhile, given the known volumetric concentration and true particle density, the mass concentration of the system can be further calculated. Mass concentration is defined as the mass of solid particles contained in a unit volume of the suspension, and it can be directly obtained as the product of the volumetric fraction and the true particle density:

(17) $$c={\rho }_{dis}\cdot \varphi .$$

Thereby enabling the estimation of slurry mass concentration without requiring additional measurement conditions.

Bimodal cooperative detection system for slurry mass concentration

In the coal preparation process, slurry mass concentration data originate from different sources and have different types. In this study, a bimodal cooperative detection system is utilized for feature-level fusion, combining process data and acquired data based on measurements from multi-power infrared laser emitters and differential pressure sensors to obtain more comprehensive and accurate information, thereby improving the precision of slurry mass concentration measurement (Liu et al., 2025). The experimental system consists of three main components: a multifunctional infrared detection module, a pressure difference measurement module, and a data acquisition system. The multifunctional infrared detection module is utilized to acquire slurry transmitted light signals; the pressure difference measurement module monitors slurry flow characteristics in real time; and the data acquisition system is responsible for signal acquisition and storage. As shown in Fig. 3, the experimental setup for bimodal slurry mass-concentration measurement integrates three infrared emitters of different power levels within the same rectangular channel, and incorporates pressure taps on the vertical pipe section to connect a differential-pressure transmitter. All signals are synchronously sampled by a unified data acquisition unit, achieving spatial co-location and temporal synchronization of the optical and differential-pressure sensing modalities. This configuration ensures measurement consistency and repeatability, providing a reliable experimental foundation for subsequent multi-source fusion modeling and the ARI-ICA prediction framework.

The variation in transmitted light intensity is primarily determined by the slurry volumetric concentration and true particle density, while factors such as particle size distribution and sample cell diameter also play a role. As the slurry concentration increases, the number of particles grows, enhancing the absorption and scattering effects of light, which leads to a decrease in transmitted light intensity. The true particle density also significantly influences the transmitted light intensity, as different particle densities alter the slurry’s absorbance characteristics, thereby affecting the transmitted light intensity (Zeng, Zhu & Chen, 2023). To improve the accuracy of transmission light measurements, precise control of slurry volumetric concentration and true particle density is required to reduce measurement errors. By appropriately adjusting the slurry concentration or diluting the sample, the influence of concentration fluctuations on transmitted light intensity can be effectively mitigated. Moreover, rationally defining the range of true particle density values and employing accurate physical measurement methods contribute to enhancing measurement precision.

The expression for pressure difference during slurry flow encompasses the combined effects of slurry volumetric concentration, true particle density, flow velocity, and pipeline dimensions. These variables interact synergistically to determine the accuracy of pressure difference measurements. To enhance the precision of pressure difference data for slurry concentration prediction, optimization and control of key parameters such as slurry flow rate, particle distribution, and pipeline diameter are necessary. In this study, the pressure difference measurement module integrates the aforementioned components through the careful selection of differential pressure sensors, sampling tubes, and connecting fittings, forming an efficient and stable measurement apparatus. Specifically, the sampling tube is connected via two three-way joints to ensure stable and unobstructed slurry flow. The differential pressure sensor is installed outside the three-way joint, continuously monitoring the pressure difference during flow and converting it into an electrical signal transmitted to the data acquisition system for storage, thereby guaranteeing the accuracy and reliability of the measured data (Zheng et al., 2023).

A photograph of the overall infrared and pressure difference detection system is shown in Fig. 4, clearly illustrating the layout and function of the major components. The entire system can stably acquire transmitted light intensity data required for slurry concentration measurement and accurately provide high-quality pressure difference data for slurry concentration. This setup lays a solid experimental foundation for subsequent data analysis and the establishment of slurry mass concentration prediction models.

Architecture of the ARI-ICA model with regularization decay

In soft sensing of slurry mass concentration, infrared transmission and differential-pressure signals characterize the slurry from complementary perspectives—infrared captures particle-scale absorption and scattering, whereas differential pressure reflects macroscopic hydrodynamic resistance. Fusing these two modalities under physics-based constraints enables simultaneous extraction of particle size and concentration as well as flow velocity and solids content, thereby improving noise immunity and fault tolerance while ensuring physical consistency and interpretability of the predictions.

In traditional Random Weight Neural Networks (RWNN), the weights and biases of hidden nodes are typically generated randomly. However, this approach fails to exploit the structure of the current residual, which may lead to inefficient learning. Given the data scale and feature dimensionality in this study (160 samples; inputs comprising three infrared intensity channels and one differential-pressure signal), we prioritized a lightweight, interpretable ARI-ICA model over deep networks or large ensemble methods. Backpropagation Neural Network (BP) are prone to local optima and are data-hungry; Extreme Learning Machine (ELM) shows weak adaptability in complex scenarios with poorly constrained generalization; Support Vector Regression (SVR) exhibits limited robustness and interpretability. By contrast, ARI-ICA employs incremental construction under a geometric-correlation constraint and residual-scaled dynamic sampling, and introduces a decaying ridge regularization at the output layer, thereby maintaining structural sparsity and suppressing overfitting in the small-sample regime. Moreover, the model first predicts the volumetric concentration $\varphi$ and the true particle density ${\rho }_{dis}$ from multi-source signals, and then computes the mass concentration via $c=\varphi \cdot {\rho }_{dis}$, ensuring physical consistency and interpretability. On this basis, we adopt a nonlinear mapping model, ARI-ICA, endowed with adaptive node generation and progressively decaying regularization to efficiently learn complex relationships in the multi-source input space, resulting in markedly enhanced accuracy, robustness, and stability for mass-concentration prediction.

In order to select new hidden nodes, ICA incorporates geometric constraints to ensure that newly added hidden nodes exhibit sufficient correlation with the current residual error (Nan et al., 2024). Anti-overfitting and complexity: A new hidden node is accepted only if its output passes a dynamic cosine-squared threshold against the residual; candidate weights are scaled to the residual norm, and the output layer uses decaying ridge regression. The alignment between the output of a candidate node $g$ and the residual error $E$ is quantified by the squared cosine of the angle between them, defined as

(18) $${\cos }^2(g,E)={\displaystyle \frac{{(E^{T}g)}^2}{{\Vert E\Vert }^2{\Vert g\Vert }^2}.}$$

A candidate node is accepted and added to the network if and only if ${\cos }^2(g,E)$ is greater than or equal to the adjusted threshold ${\gamma }_L$, thereby avoiding redundancy and maintaining a sparse structure. Synergy with regularization: Residual-scaled sampling anneals the candidate magnitude, while the decaying ridge readout, enforces strong-then-weak shrinkage. Together with early stopping ( $\Vert E\Vert \le tol$ or no candidate passes), this keeps the network compact and reduces overfitting on small datasets.

The threshold ${\gamma }_L$ is defined as

(19) $${\gamma }_L=1-{\displaystyle \frac{1+\sigma L}{1+L+1}}$$where, $\sigma$ is a hyperparameter that controls the convergence rate and exploration range, and $L$ denotes the number of currently selected hidden nodes. This design enables dynamic adjustment of the strictness of node selection during the stepwise construction process, ensuring that each newly introduced node effectively reduces the current residual error and enhances the model’s representational capacity.

To address the issue of uneven weight distribution in ICA and improve the model’s generalization performance, the original algorithm is extended by introducing a dynamic weight sampling range and a decaying regularization term. This modification reduces network complexity while allowing new nodes to be added either individually or in batches, thereby improving learning efficiency (Meng, Tang & Qiao, 2022). The ICA architecture for adding multiple nodes and handling multidimensional regression is illustrated in Fig. 5. Here, $X$ denotes the input variables and $Y$ the tarobtain output variables. $a_n$ and $b_n$ represent the input weight vector and bias of the $n-th$ hidden node, respectively. ${\beta }_n$ denotes the output weight obtained after incorporating the regularization term for the $n-th$ single hidden node. ${\sigma }_n$ is the iteration coefficient, and $H_n\left(a_n,X_n,b_n\right)$ represents the activation value of the $n-th$ hidden node.

ARI-ICA learning procedure with decaying regularization

After completing the transmission-light and differential-pressure experiments, the acquired raw signals—recorded as voltage time series over finite intervals—were processed to mitigate contamination from noise, sensor errors, and other external disturbances. Signal processing employed the fast Fourier transform (FFT) and frequency-domain filtering. Specifically, the FFT module in Origin (OriginLab) was used to analyze the spectral characteristics of the raw signals; appropriate filters were then applied in the frequency domain to suppress noise components and smooth the data, thereby improving the signal-to-noise ratio. The denoised signals were subsequently averaged over the analysis window, and these mean values were taken as the representative transmitted-light intensity and differential pressure for each slurry volumetric concentration.

In the model constructed in this study, the input variables include three infrared light intensity signals (5, 50, and 200 mW) and one differential pressure signal. These inputs are processed through a regression-based prediction model trained on experimental data, producing two output variables: slurry volume concentration and true particle density. Since the ultimate prediction target is the slurry mass concentration, and there exists a clear mathematical relationship among the three parameters, a specific physical equation can be utilized to convert the predicted volume concentration and particle density into mass concentration (Geng et al., 2022). Based on this relationship, the mass concentration can be calculated as: $c=\varphi \cdot {\rho }_{dis}$, where ${\rho }_{dis}$ represents the slurry volume concentration and $\varphi$ denotes the true particle density. The slurry mass concentration is obtained as the product of the volume concentration and the particle density.

To achieve the nonlinear mapping from the four input variables to the two output variables, this work adopts an Incremental Constructive Algorithm (ICA) framework. On top of this, two improvements are introduced: a dynamic weight sampling range and a dynamic decay regularization strategy, both of which enhance the model’s expressiveness and robustness (Yuan, Li & Wang, 2020). In contrast to traditional methods that utilize a fixed interval for random weight sampling, this work proposes an improved dynamic sampling range strategy, which is applied to the random initialization of hidden node weights and biases (LeCun, Bengio & Hinton, 2015).

Uniform random sampling is performed within a specified range $\left[{\lambda }_{min},{\lambda }_{max}\right]$. This method adaptively adjusts the sampling scale based on the current residual error, enabling a transition from coarse global search to fine local search. Such an approach increases the probability of satisfying geometric constraints at different training stages, facilitating to avoid the “flat region” saturation problem caused by overly small weights in early training, as well as oscillations due to excessively large weights in later stages. To prevent the sampling space from being too narrow when residuals are large early in training, or too wide during later convergence phases, a dynamic weight sampling range strategy is introduced. In the experiments, three residual-norm scaling intervals—(0.05, 1.0), (0.1, 5.0), and (0.5, 10.0)—were compared. The results show that the predicted RMSE values of ARI-ICA were 0.099, 0.139, and 0.123 for these intervals, respectively. Among them, the interval (0.05, 1.0) yielded more stable convergence behavior across different stages of the analysis.

Let ${\lambda }_{min}=0.05\times \Vert E\Vert$, ${\lambda }_{max}=1.0\times \Vert E\Vert$. Within this range, weights $w$ and biases $b$ are uniformly sampled, enabling the network to adaptively adjust the search scale of node parameters at different training stages.

The improvement of dynamic decay regularization is implemented by solving a regularized least squares problem at each step of the output layer regression.

This study proposes a dynamic decay strategy for the regularization term (Pani & Mohanta, 2014), formulated as

(20) $${\lambda }_{L+1}={\lambda }_L\cdot decay$$which allows the model to maintain strong regularization in the early stages to prevent overfitting while gradually reducing regularization strength during later fine-tuning phases. This approach aims to achieve a balance between global fitting capability and local precision. Initially, a large regularization coefficient $\lambda$ is employed to avoid overfitting; as the residual error decreases progressively, the regularization parameter is decayed at each iteration.

If the decay parameter is set too small, the regularization term will rapidly decay toward zero, potentially leading to overfitting in the later stages of training. Conversely, if it is set too large, the decay will be too slow, resulting in excessively strong regularization and underfitting during the later training phases. To verify the rationality of this parameter setting, a sensitivity analysis was conducted in the experimental section for different decay values (0.90, 0.95, and 0.99).

The results show that the RMSE values of ARI-ICA under these decay coefficients were 0.129, 0.098, and 0.087, respectively. These findings indicate that a decay value of 0.99 achieved relatively superior error performance on most datasets. Therefore, the decay parameter in this study was set to 0.99, corresponding to a 1% reduction in the regularization coefficient per iteration. This setting has been demonstrated in both the literature and practical experience to represent a balance point between convergence speed and fitting flexibility, enabling improved fitting capability in later training stages.

Specifically, let the input sample matrix be denoted by $X\in R^{n\times 4}$ and the output by $Y\in R^{n\times 2}$. The objective is to learn a nonlinear function approximation.

(21) $$Y\approx \hat{Y}=H\beta .$$

Specifically, $H$ is the hidden layer output matrix constructed in a stepwise manner, while $\beta$ denotes the output layer weights. At each iteration, a new hidden node is generated through the following procedure. First, the $Sigmoid$ activation function is defined as:

(22) $$\sigma \left(z\right)={\displaystyle \frac{1}{1+e^{-z}}.}$$

The response of the hidden node is given by:

(23) $$g=\sigma \left(Xw+b\right).$$

To select the optimal node from the candidate pool in each training iteration and ensure that the newly added hidden node contributes to the current residual, a geometric constraint is introduced:

(24) $${\cos }^2\left(g,E\right)={\displaystyle \frac{{\Vert E^{T}g\Vert }^2}{{\Vert E\Vert }^2{\Vert g\Vert }^2}}$$and set a dynamic threshold

(25) $${\gamma }_L=1-{\displaystyle \frac{1+\sigma L}{1+L+1}.}$$

A hidden node is accepted only under condition ${\cos }^2\left(g,E\right)\ge {\gamma }_L$. After $L$ hidden nodes have successfully passed this selection criterion, they are assembled to form the hidden layer output matrix (Zhang, 2012).

(26) $$H=\left[g_1,g_2,\dots ,g_L\right].$$

To improve the search efficiency, a residual-driven sampling interval is employed:

(27) $${\lambda }_{min}=0.05\times \Vert E\Vert$$

(28) $${\lambda }_{max}=1.0\times \Vert E\Vert .$$

To facilitate adaptive adjustment of the search scale for node parameters at different training stages, the weights $w$ and biases $b$ are uniformly sampled within the defined range.

The output layer weights $b$ are determined by minimizing a regularized least squares objective:

(29) $$\beta ={\left(H^{T}H+\lambda I\right)}^{-1}{H}^{T}Y.$$

In this context, $\lambda$ represents the regularization parameter that is progressively decayed throughout the training, ${\lambda }_L={\lambda }_{L+1}\cdot decay$.

The final model is represented by $\hat{Y}=H\beta$, defined as $H=\left[\sigma \left(X{w}_1+b_1\right),...,\sigma \left(X{w}_L+b_L\right)\right]$. Each column corresponds to the response of a $Sigmoid$ hidden node selected through dynamic sampling and geometric constraints, enabling a nonlinear mapping from a four-dimensional input space to a two-dimensional output space. This approach effectively enhances the model’s interpretability while improving its representational capacity and convergence performance in complex nonlinear tasks.