Predictive Modelling and Analysis of Filtration Performance for Drip Irrigation Filters Using Sediment-Laden Water Based on the Differential Evolution Optimized Random Forest (DE/RFR)

Abstract

Filtration systems are essential for drip irrigation using sediment-laden water sources such as the Yellow River. This study focused on a sand filter (filtration accuracy: 150 μm), a disc filter (filtration accuracy: 125 μm), and their combined multi-stage filtration system (flow rate: 30–50 m³/h). In situ tests were conducted under Yellow River water conditions in the Hetao Irrigation District, Inner Mongolia, China, to evaluate the response of filtration performance to sediment characteristics, flow rate, and operating time. On this basis, Differential Evolution-optimized Random Forest Regression (DE/RFR) was further established to predict filtration performance. The results showed that: (1) Under sediment concentrations of 0.62–3.6 g/L and median particle sizes of 4.70–16.03 μm, the head loss of the sand filter (ΔH_si) remained stable over the operating time. Conversely, the head loss of the disc filter (ΔH_di) increased with the operating time; the magnitude of this increase grew with higher flow rates, sediment concentrations, and median particle sizes, reaching 0.07 MPa after 16–235 min of operation. The head loss of the multi-stage filtration system (ΔH_i) was primarily generated by the disc filter. (2) The filtration efficiency of the filters and the filtration system was 2.5–6.4%. The outlet sediment concentration and particle size distribution were linearly correlated with the inlet values, and the outlet sediment particle size distribution remained below the clogging risk threshold for emitters. (3) Prediction models for ΔH_si, ΔH_di, and ΔH_i were developed based on MLR, RFR, and DE/RFR. Among these, DE/RFR exhibited the highest accuracy in predicting these variables, with R² values ranging from 0.71 to 0.93 and RMSE values from 0.0017 to 0.0104 MPa. (4) Results from Pearson correlation and feature importance analysis indicated that ΔH_si, ΔH_di, and ΔH_i were primarily influenced by flow rate, sediment concentration and operating time, and flow rate and operating time, respectively. (5) Building upon the DE/RFR model, a Filtration Cycle Prediction Model (FCPM) was developed to determine the operational duration required for the head loss across both the filters and the filtration system to reach 0.07 MPa. The two models developed in this study provide technical support for the configuration and operation of drip irrigation filtration systems using sediment-laden water.

1. Introduction

The mismatch between water and land resources exists in China’s Yellow River Basin, and drought and water scarcity have become rigid constraints on the sustainable agricultural development of irrigation districts along the Yellow River [1]. Using Yellow River water for drip irrigation is the fundamental solution to solve the water shortage in the Yellow River Basin [2]. From 2026 to 2030, the Hetao Irrigation District in Inner Mongolia plans to add a Yellow River water drip irrigation area of 147,000 hm² [3], but the high sediment concentration (reaching 3.7–26.5 kg/m³) [4] and fine particle size (<63 μm) of the water easily lead to emitter clogging problems [5]. Currently, most drip irrigation systems using Yellow River water in the Hetao Irrigation District are equipped with multi-stage filtration systems, employing sand filters for primary filtration and disc filters for secondary filtration [6]. However, investigations reveal that operational multi-stage filtration systems generally suffer from problems such as excessive head loss, high susceptibility to clogging, frequent backflushing, poor filtration accuracy, and clogging of drip irrigation systems. Furthermore, fertilizer loss caused by the initiation of backflushing during fertigation is a prevalent issue. Due to the complex spatiotemporal variations in the sediment characteristics of Yellow River water (e.g., sediment concentration and particle size distribution), determining filter specifications, system configurations, and operational modes based solely on conventional engineering design experience often fails to meet actual field requirements. The timing of backflushing initiation (defined as the duration required for the pressure drop across the filter to reach 0.07 MPa) is dependent on the sediment characteristics of Yellow River water, yet this timing remains indeterminate across different operational periods of the irrigation season. Furthermore, the declines in outlet pressure and flow rate induced by backflushing compromise the irrigation and fertigation uniformity of the drip irrigation system. Therefore, developing a head loss prediction model for Yellow River water drip irrigation filtration systems to predict backflushing initiation times under various sediment characteristics and operating conditions is essential for establishing the fertigation regime and improving the uniformity of water and fertilizer application.

Under clean water conditions, researchers have primarily investigated the influence of internal components on the hydraulic performance of sand filters through experimental tests. Compared to the water distributor and filter media layer, the filter nozzle is the primary component responsible for head loss [7]. In light of this, Arbat et al. [8] modified the Ergun equation proposed by McCabe et al. [9] by introducing a filter nozzle porosity parameter, significantly improving the calculation accuracy of head loss in sand filters at flow velocities exceeding 70 m/h. Building on this work, Bové et al. [10] simplified the Ergun equation using stepwise regression analysis, achieving high calculation accuracy for head loss. Compared to sand media filters, disc filters exhibit a more complex structural configuration; existing research has primarily focused on the calculation of head loss under clean water conditions and the structural optimization of the discs. Researchers have developed empirical equations relating head loss to flow velocity, disc structural parameters, inlet/outlet diameters, and kinematic viscosity of water, based on hydraulic performance experiments and dimensional analysis [11,12]. Based on Computational Fluid Dynamics (CFD) numerical simulations, modifying the disc channel from a linear to a discrete design reduced head loss by 20.8% [13]. Based on CFD, Zeng et al. [14] determined the optimal combination of fractal channel parameters by introducing a buffer slot and using neural network algorithms, thereby reducing the average clean pressure drop of the filter from 0.024 MPa to 0.015 MPa.

Under sediment-laden water conditions, the variation patterns of filter performance are mainly obtained through laboratory experiments. For sand filters, when the sediment concentration is 0.2–1.2 g/L, within certain ranges, increasing the filter bed thickness, reducing the media particle size, and lowering the filtration velocity are effective ways to improve filtration efficiency [15,16]. For disc filters, whether under conditions of wastewater with a low suspended solid concentration (10–60 mg/L) [17] or Yellow River water with a high sediment concentration (0.8–2.8 g/L) [18], their filtration efficiency is consistently higher than that of screen filters with the same filtration accuracy. When the sediment concentration is 0.2–1.2 g/L, the head loss of disc filters exhibits a trend of initial stability followed by a sharp increase over operating time, and is significantly affected by sediment concentration, where for every 0.1 g/L increase in concentration, the filtration cycle is shortened by 40–60% [19,20]. Structural optimization of filter discs can reduce head loss and enhance filtration efficiency, as demonstrated by [21], whose introduction of a fractal flow channel design increased efficiency by 19.6–44.7% under equivalent head loss at a sediment concentration of 0.3 g/L. Based on extensive experimental data, Li et al. [22] developed multiple linear regression models relating head loss after 5 min of operation to sediment concentration, sediment median particle size (D₅₀), and flow rate, with R² values of 0.73–0.81. In recent years, to determine appropriate filter selection schemes under sediment-laden water conditions, Yuan et al. [23] developed a multi-objective evaluation model considering sediment-laden water, disc filters, and drip emitters based on the fuzzy comprehensive evaluation method following laboratory experiments. They subsequently identified appropriate filtration accuracies for disc filters under a sediment concentration of 0.2 g/L and a D₅₀ range of 0–150 μm. The majority of these studies focus on filtration performance experiments and head loss simulation of single-stage filters under artificially prepared sediment-laden water conditions. Consequently, there is a critical need to investigate the performance response patterns and numerical simulation of filters and filtration systems under the complex spatiotemporal variations in Yellow River sediment-laden water, as well as to determine suitable specifications, configuration strategies, and operational modes.

Furthermore, in the field of reclaimed water irrigation, researchers have combined dimensional analysis with regression analysis to develop regression models relating the head loss of screen, sand, and disc filters to inlet suspended solid concentration, flow velocity, water viscosity, filtered volume, and structural dimensions. These models achieved high prediction accuracy (R² > 0.84), enabling effective head loss prediction for these three filter types under reclaimed water conditions and providing a theoretical basis for pump station selection, filtration system performance monitoring, and backflushing schedule formulation in reclaimed water drip irrigation systems [24,25]. To further elucidate the nonlinear characteristics of the filtration process in sand filters under unconventional water conditions, García-Nieto et al. [26,27,28,29] successively applied Gradient Boosting Regression Tree (GBRT) models to quantify the filtered volume, employed Support Vector Machine (SVM) and Gaussian Process Regression (GPR) models to predict the outlet dissolved oxygen and outlet turbidity, and utilized the Differential Evolution (DE) algorithm to determine the key hyperparameters of Random Forest Regression (RFR) models, achieving an RMSE for outlet turbidity as low as 0.39 FNU, indicating high simulation accuracy. Compared with reclaimed water filtration systems, drip irrigation using sediment-laden water from the Yellow River involves limited annual irrigation events, resulting in a limited number of in situ test samples. Models such as GBRT, SVM, and GPR exhibit relatively weak stability under small-sample conditions. In contrast, the RFR model, which integrates multiple decision trees and employs random sampling and feature selection strategies, possesses robust anti-overfitting capabilities and superior generalization performance. It is more suitable for complex nonlinear modelling of small-scale datasets [30]. Furthermore, water quality characteristics and filter clogging processes under high-sediment conditions (e.g., Yellow River water) are inconsistent with those of reclaimed water. Therefore, the feasibility and adaptability of utilizing the DE/RFR model to characterize the filtration performance of multi-stage filtration systems warrant further investigation. However, the adaptability of the DE/RFR model to Yellow River sediment-laden water and its prediction accuracy for the filtration performance of multi-stage filtration systems warrant further investigation.

To this end, this study conducted in situ field experiments on the filtration performance of drip irrigation filtration systems using Yellow River sediment-laden water in the Hetao Irrigation District, Inner Mongolia. The specific objectives of this study were to: (1) elucidate the effects of inlet pressure, flow rate, sediment concentration, sediment median particle size, and operating time on the head loss and outlet sediment characteristics of the filters and the filtration system and identify the key influencing parameters and their order of importance; (2) develop predictive models for head loss and outlet sediment characteristics using Linear Regression (LR), Random Forest Regression (RFR), and a hybrid Differential Evolution-optimized Random Forest Regression (DE/RFR) algorithm and determine the most suitable model through accuracy comparison to provide technical support for the configuration and operation of sediment-laden water drip irrigation systems.

2. Materials and Methods

This study combines in situ field experiments with data-driven modelling. Based on a multi-stage filtration system composed of sand and disc filters, filtration experiments using sediment-laden water from the Yellow River were conducted, and predictive models were developed for hydraulic performance and effluent sediment characteristics. The technical workflow is shown in Figure 1.

2.1. Experimental Setup

The multi-stage filtration system performance test platform was constructed in 2025. It is located south of the second check gate of the Heji sub-main canal in the Yongji Irrigation Area of the Hetao Irrigation District, Inner Mongolia, China (40°73′ N, 107°28′ E). The platform primarily consists of a floating sewage pump (Shandong Shuimai Agricultural Development Co., Ltd., Dezhou, Shandong, China; flow rate: 60 m³/h; head: 50 m; power: 18.5 kW), a sand media filter (Shanghai Huawei Agritech Group Co., Ltd., Shanghai, China), a disc filter (Shanghai Huawei Agritech Group Co., Ltd., Shanghai, China), pressure gauges (Hongqi Instrument Co., Ltd., Yueqing, Zhejiang, China; range: 0–0.6 MPa; accuracy: ±0.4%), an electromagnetic flowmeter (Henan Zhong’an Electronic Detection Technology Co., Ltd., Zhengzhou, Henan, China; range: 9–90 m³/h; accuracy: ±2%), valves, and sediment sampling ports (Figure 2). Both the sand media filter and the disc filter are equipped with backflushing capabilities. Their filtration specifications correspond to those commonly used in drip irrigation systems supplied with Yellow River water in the Hetao Irrigation District, namely, 100 mesh (nominal filtration accuracy: 150 μm) and 120 mesh (effective filtration accuracy: 125 μm), respectively. Three pressure gauges were used to monitor the inlet pressure (H_ini), outlet pressure (H_outi), and inter-stage pressure (H_midi, i.e., at the sand media filter outlet or disc filter inlet) of the multi-stage filtration system. Additionally, three sediment sampling ports were positioned at the inlet, outlet, and inter-stage of the filtration system.

2.2. Experimental Design and Methods

To investigate the hydraulic performance response of the multi-stage filtration system under all sediment characteristics of the Heji sub-trunk canal, this study conducted experimental tests during both the summer (29 July–17 August) and winter (19 October–30 October) irrigation periods of 2025. During the summer irrigation period, the initial sediment concentration at the inlet (G_in0) ranged from 0.72 to 3.6 g/L, with a corresponding median particle size (D_50in0) of 4.70–6.56 μm; in the winter irrigation period, G_in0 ranged from 0.62 to 2.31 g/L and D_50in0 was 5.87–16.03 μm. Notably, G_in0 peaked at 3.6 g/L in late August. D_50in0 exhibited an increasing trend over time, and the average D_50in0 in the winter irrigation period was 48.3% higher than that in the summer irrigation period (Figure 3).

During the experimental process, the initial flow rate (Q₀) of the multi-stage filtration system at the beginning of operation was set at three levels: 30, 40, and 50 m³/h. Under each of the three Q₀ levels, 5–6 filtration experiments were conducted under different sediment-laden water conditions, resulting in a total of 17 filtration performance experiments (

Table 1. Initial hydraulic operating conditions and inlet sediment characteristics of the multi-stage filtration system.

Trial Number	Test Date	Initial Hydraulic Operating Conditions			Initial Inlet Sediment Characteristics
		Flow Rate (Q0) (m3/h)	Inlet Pressure (Hin0) (MPa)	Outlet Pressure (Hout0) (MPa)	Sediment Concentration (Gin0) (g/L)	Median Particle Size (D50in0) (μm)
1	14 Aug.	30	0.17	0.15	0.73	4.91
2	19 Oct.		0.18		0.66	6.59
3	23 Oct.		0.18		0.98	9.13
4	25 Oct.		0.18		1.23	10.13
5	30 Oct.		0.19		2.16	15.84
6	29 July	40	0.19	0.15	1.76	5.46
7	14 Aug.		0.19		0.98	4.71
8	20 Oct.		0.20		0.63	6.03
9	23 Oct.		0.19		1.14	10.18
10	25 Oct.		0.20		1.29	10.95
11	30 Oct.		0.20		2.18	15.91
12	14 Aug.	50	0.15	0.09	0.72	4.80
13	17 Aug.		0.15	0.09	3.60	6.56
14	20 Oct.		0.17	0.11	0.62	6.37
15	23 Oct.		0.17	0.11	0.95	8.21
16	25 Oct.		0.17	0.10	1.40	10.22
17	30 Oct.		0.17	0.10	2.21	15.25

). Specifically, when Q₀ was 30 or 40 m³/h, the initial pressure at the outlet (H_out0) was set to 0.15 MPa at the beginning of the operation. However, when Q₀ was 50 m³/h, H_out0 was maintained between 0.09 and 0.11 MPa due to the head limitations of the floating sewage pump.

The experimental procedure is as follows:

• After initiating the water pump and ensuring its stable operation, the valves shown in Figure 2 were adjusted according to the experimental design (Table 1) to set the outlet pressure and flow rate of the multi-stage filtration system to H_out0 and Q₀, respectively.
• Once the readings of the pressure gauges and electromagnetic flowmeter stabilized, the initial pressures at the inlet (H_in0), outlet (H_out0), and inter-stage (H_mid0) were recorded at the onset of the system operation. These values were then used to determine the initial head losses of the sand filter (∆H_s0), disc filter (∆H_d0), and the multi-stage filtration system (∆H₀), respectively.
• For every incremental increase of approximately 0.01 MPa in the head loss of the multi-stage filtration system (∆H_i), the pressures at the inlet (H_ini), inter-stage (H_midi), and outlet (H_outi) were recorded. Based on these readings, the head loss of the sand filter (∆H_si), the disc filter (∆H_di), and the multi-stage filtration system (∆H_i) were determined. Simultaneously, the flow rate (Q_i) and the cumulative operating time (T_i) were documented.
• Concurrently, 250 mL sediment-laden water samples were collected from the inlet, outlet, and inter-stage ports using 300 mL sampling bottles at the three designated locations.
• When ∆H_si or ∆H_di reached 0.07 MPa, the corresponding T_i was recorded as T_max, defined as the filtration cycle. The water pump was then deactivated to initiate the backflushing of the sand or disc filter until ∆H_si or ∆H_di recovered to ∆H_s0 or ∆H_d0, respectively [31].

2.3. Measured Parameters

2.3.1. Hydraulic Performance of Filters

The hydraulic performance parameters of the multi-stage filtration system primarily include the pressure indices (H_ini, H_midi, and H_outi), head losses (∆H_si, ∆H_di, and ∆H_i), and flow rate (Q_i). Specifically, the head losses during operation were calculated as: ∆H_si = H_ini − H_midi, ∆H_di = H_midi – H_outi, and ∆H_i = H_ini − H_outi. Correspondingly, the initial head losses were defined as: ∆H_s0 = H_in0 − H_mid0, ∆H_d0 = H_mid0 − H_out0, and ∆H₀ = H_in0 − H_out0.

2.3.2. Sediment Characteristic Indices

The sediment characteristic indices of the multi-stage filtration system included the sediment concentrations (G_ini, G_midi, and G_outi) and their corresponding median sediment particle sizes (D_50ini, D_50midi, and D_50outi) obtained from three sampling ports. The testing methods for sediment concentration and particle size distribution were as follows:

• Sediment concentration:

The sediment-laden water sample was shaken evenly, and the volume was fixed to 200 mL using a volumetric flask; the sample was suction-filtered using a diaphragm vacuum pump (Jianhu Mingte Glass Instrument Factory, Yancheng, Jiangsu, China; negative pressure 0.086 MPa, filter membrane pore size 4 μm); after suction filtration, the filtered sediment was placed in an aluminum box and dried at 105 °C to a constant weight, recorded as m (g); thus, the sediment concentration was calculated as 5 m (g/L) [32].

2.

Sediment particle size distribution:

A dried sediment sample of 0.5–0.6 g was weighed and subjected to ultrasonic dispersion using an automatic sample dispersion system (Bettersize Instruments Ltd., Dandong, China); subsequently, the particle size distribution of the sediment sample was measured using a laser particle size analyzer (measuring range: 0.02–2000 μm, error: ≤1%; Bettersize Instruments Ltd., Dandong, China) [33]. The D₅₀ (corresponding to 50% of the cumulative volume distribution) was adopted as the representative index to characterize the sediment particle size distribution.

2.3.3. Filtration Efficiency

The filtration efficiencies of the sand filter (E_si), disc filter (E_di), and multi-stage filtration system (E_i) were calculated using Equations (1)–(3):

E_si = (G_ini − G_midi)/G_ini × 100%

(1)

E_di = (G_midi − G_outi)/G_midi × 100%

(2)

E_i = (G_ini − G_outi)/G_ini × 100%

(3)

2.4. Mathematical Modelling Techniques

Under sediment-laden water conditions, previous studies have employed linear regression (LR) models to establish quantitative mathematical models characterizing the relationship between the head loss of the disc filter and the inlet flow rate, inlet sediment concentration, and inlet median particle size [22]. In recent years, to enhance the prediction accuracy of mathematical models, Garcia-Nieto et al. [29] proposed a hybrid model combining the differential evolution (DE) algorithm with random forest regression (RFR), denoted as DE/RFR. By utilizing the DE algorithm to optimize the hyperparameters of the RFR model, the prediction accuracy for the outlet turbidity of sand filters was significantly improved. Therefore, this study aims to develop LR, RFR, and DE/RFR models for sand filters, disc filters, and multi-stage filtration systems to calculate the head loss, outlet sediment concentration, and outlet median sediment particle size for each of these three filtration configurations across various scenarios.

2.4.1. Variables in the Model

Based on 17 independent hydraulic performance tests, data were collected at 3–5 time points during each test, yielding a cumulative total of 75 datasets (

Table 2. Means and standard deviations of the input and output variables for the respective filter units and the multi-stage filtration system.

Input Variables	Name of the Variable	Mean			Standard Deviation
		Sand Filter	Disc Filter	Filtration System	Sand Filter	Disc Filter	Filtration System
Initial flow rate (m3/h)	Q0	39.0			7.8
Operating time (h)	Ti	/			/
Initial inlet pressure (MPa)	Hin0 or Hmid0	0.18	0.16	0.18	0.01	0.02	0.02
Initial inlet sediment concentration (g/L)	Gin0 or Gmid0	1.33	1.29	1.33	0.76	0.75	0.76
Initial inlet median particle size (μm)	D50in0 or D50mid0	8.61	8.69	8.61	3.71	3.78	3.71
Output variables	Name of the variable	Mean			Standard deviation
		Sand Filter	Disc Filter	Filtration System	Sand Filter	Disc Filter	Filtration System
Head loss (MPa)	∆Hsi or ∆Hdi or ∆Hi	0.02	0.05	0.07	0.01	0.02	0.02
Outlet sediment concentration (g/L)	Gmidi or Gouti	1.29	1.27	1.27	0.75	0.72	0.72
Outlet median particle size (μm)	D50midi or D50outi	8.69	8.49	8.49	3.78	3.70	3.70

). This study selected indicators affecting the hydraulic performance of the filter as model input variables [34,35], including the flow rate (Q₀), inlet pressure (H_in0 or H_mid0), sediment concentration (G_in0 or G_mid0), and median sediment particle size (D_50in0 or D_50mid0) under initial operating conditions, along with the operating time (T_i). ∆H_si, ∆H_di, and ∆H_i, along with outlet sediment characteristics (G_midi or G_outi, D_50midi or D_50outi), were selected as output indicators. Among these, ∆H_si, ∆H_di, and ∆H_i characterize the filter head loss [36], while G_outi and D_50outi are associated with the risk of physical clogging in drip irrigation systems [2].

2.4.2. Linear Regression (LR)

Pearson correlation analysis was performed to examine the relationships among all inlet and outlet variables in the sand, disc, and multi-stage filtration systems. Based on this analysis, LR models were employed to establish quantitative mathematical relationships between the head loss indices (∆H_si, ∆H_di, and ∆H_i) and the operating time (T_i) as well as the initial parameters of the filters and the filtration system, including Q₀, H_in0 (or H_mid0), G_in0 (or G_mid0), and D_50in0 (or D_50mid0). Additionally, linear regression analysis was performed to determine the relationships between inlet and outlet sediment characteristics. The linear regression (LR) models employed in this study comprised multiple linear regression (MLR) and simple linear regression (SLR).

2.4.3. Random Forest Regression (RFR)

RFR is a non-linear statistical model grounded in the Bagging ensemble learning framework, with the regression decision tree serving as its fundamental unit. Its core mechanism employs Bootstrap sampling and random feature selection strategies to construct an ensemble of multiple decision trees, collectively forming a “forest”. Each tree was trained independently, and the final prediction was derived by averaging the outputs of multiple decision trees, which significantly mitigates the risk of overfitting while enhancing generalization capability and accuracy [37]. RFR possesses the capability to assess variable importance, enabling the quantitative identification of key features and providing a reliable basis for input variable selection [38]. The key hyperparameters of the RFR model primarily include the number of trees (n_estimators), maximum tree depth (max_depth), minimum samples required to be at a leaf node (min_samples_leaf), minimum samples required to split an internal node (min_samples_split), and maximum features considered for the best split (max_features). Their definitions and roles in the model are described in detail in Reference [39].

2.4.4. Differential Evolution (DE) Optimizer

Given that the prediction accuracy of RFR is related to the values of hyperparameters, this study adopted DE to optimize the RFR hyperparameters. DE is a meta-heuristic method that solves optimization problems by iteratively improving the quality of candidate solutions. It mainly includes four steps: initialization, mutation, crossover, and selection [40]. After initialization, the search was started, and the “mutation–crossover–selection” steps were executed until the preset termination conditions were met (reaching the maximum number of iterations, convergence accuracy of the target solution, or no significant improvement in population fitness). It has been successfully applied to the prediction of head loss, outlet dissolved oxygen, and outlet turbidity in sand filters [27,28,29].

• Initialization

According to Equation (4), an initial population was randomly generated within the search space, where NP and I denote the population size and the maximum number of iterations, respectively. Each individual in the population corresponds to an n -dimensional vector of RFR hyperparameters. In this study, the individual vector is denoted as $X_p^i$ = [n_estimators, max_depth, min_samples_leaf, min_samples_split, max_features], where $X_p^i$ represents the p-th individual in the i-th generation (p = 1, …, NP; i = 0, 1, …, I). The variable range for the initial population is [$X_j^L$, $X_j^U$], where j denotes the index of the variable within the individual.

$$X_{p,j}^0=X_j^L+ran{d}_{p,j} (0, 1)\cdot (X_j^U-X_j^L) \mathrm{for} p=1, 2, \dots , NP \mathrm{and} j=1, \dots , n$$

(4)

where $X_{p,j}^0$ is the j-th variable of the p-th individual in the initial population (the first generation); rand_p,j (0, 1) denotes a uniformly distributed random number within the range (0, 1).

2.

Mutation

The initial vectors $X_p^i$ were updated according to the mutation strategy in Equation (5) to generate the new vectors $V_p^g$.

$$V_p^i=X_{r_1}^i+F(X_{r_2}^i-X_{r_3}^i) \mathrm{for} p=1, 2, \dots , NP$$

(5)

where $X_{r_1}^i$, $X_{r_2}^i$, and $X_{r_3}^i$ are three distinct individuals randomly selected from the population; F is the scaling factor, which lies in the interval [0, 2].

3.

Recombination

The trial vectors $U_p^i$ are generated through the crossover operation between the initial vectors $X_p^i$ and the new vectors $V_p^i$.

$$U_p^i=\left\{\left.\begin{array}{l}{V}_{p,j}^i, \mathrm{if} ran{d}_{p,j} (0, 1) \le CR, \mathrm{or} j=I_{rand}\\ X_{p,j}^i, \mathrm{otherwise}\end{array}\right\}\right. \mathrm{for} p=1, 2, \dots , NP \mathrm{and} j=1 , \dots , n$$

(6)

where I_rand denotes a random dimension index; CR is the crossover rate, which lies in the interval [0, 1].

4.

Selection

In order to choose the vectors of the following generation, the trial vectors were compared to the initial vectors to retain the individuals with the best value given by the fitness function.

$$X_p^{i+1}=\left\{\left.\begin{array}{l}{U}_p^i, \mathrm{if} f(U_p^i)<f(X_p^i)\\ X_p^i, \mathrm{otherwise}\end{array}\right\}\right.$$

(7)

where f denotes the fitness function.

2.4.5. Differential Evolution-Optimized Random Forest Regression (DE/RFR)

During the construction phase of the DE/RFR model, the dataset consisting of 75 samples was partitioned into an 80% training set (60 samples) and a 20% test set (15 samples) using a random sampling method. The training set was employed for model construction and parameter learning, while the test set was utilized to evaluate the generalization ability of the model on unseen data. To enhance the generalization capability and prevent overfitting, a 5-fold cross-validation mechanism was implemented within the training set [41]. Specifically, the training set was randomly divided into five equal subsets; four subsets were rotated for training, while the remaining one served as the validation subset, with the process repeated five times. Based on the characteristics of the RFR hyperparameters and their suitability for the limited dataset of 75 samples used in this study, the initial search ranges (

Table 3. Initial ranges of hyperparameter values for the DE/RFR model.

RFR Parameters	Lower Limit	Upper Limit
n_estimators	100	500
max_depth	5	30
min_samples_split	2	10
min_samples_leaf	1	5
max_features	0.2	1

) were predefined with reference to empirical settings reported in previous studies [29,42]. These ranges provided reasonable search boundaries for the DE algorithm. The DE algorithm was employed to perform global optimization of RFR hyperparameters by iteratively generating candidate combinations. The average error of the 5-fold cross-validation served as the fitness function for the DE algorithm, guiding the population to evolve toward error minimization until the algorithm converged and established the optimal hyperparameters [43]. Finally, the RFR model was constructed based on these optimal hyperparameters, and its predictive performance was verified on the test set using various accuracy metrics. The overall modelling workflow of the DE/RFR model is illustrated in Figure 4.

2.4.6. Model Evaluation

The prediction accuracy of the LR, RFR, and DE/RFR models was evaluated using the coefficient of determination (R², Equation (8)), root mean square error (RMSE, Equation (9)), and mean absolute error (MAE, Equation (10)). Specifically, the closer the R² is to 1.0, and the closer the RMSE and MAE are to 0, the higher the prediction accuracy of the model is indicated.

$$R^2=1-{\displaystyle \frac{S{S}_{\mathrm{err}}}{S{S}_{\mathrm{tot}}}}=1-{\displaystyle \frac{{{\displaystyle {\sum }_{i=1}^n\left(t_i-y_i\right)}}^2}{{{\displaystyle {\sum }_{i=1}^n\left(t_i-\overline{t}\right)}}^2}}$$

(8)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(t_i-y_i\right)}^2}}$$

(9)

$$\mathrm{MAE}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|t_i-y_i\right|}$$

(10)

where SS_err is the residual sum of squares; SS_tot is the total sum of squares; t_i and y_i represent the measured and predicted values of the samples, respectively; $\stackrel{\mathrm{-}}{t}$ is the mean of the measured values; and n is the sample size.

2.4.7. Feature Importance Analysis

To quantify the contribution of each input feature to the DE/RFR model’s predictions, feature importance analysis was conducted based on the permutation importance method [44]. Specifically, while maintaining the fixed structure of the model trained with optimal hyperparameters, the values of a single feature in the training set were randomly permuted to recalculate the R². The resulting decrease in R² (∆R²) was employed as the indicator of variable importance, where a larger ∆R² signifies a more significant impact of that specific variable on the model’s predictive performance. In this study, the number of permutations (n_perm) was set to 50, and the mean value of these iterations was adopted as the final importance score for each feature. On this basis, each importance score was normalized, such that the relative proportion of each variable’s score represents its relative importance.

2.4.8. Data Analysis

In this study, Python (v3.9) was used to perform Pearson correlation analysis and LR model construction, the Scikit-learn library in Python was adopted to construct the RFR model, and the DE algorithm from the SciPy library was utilized to optimize hyperparameters; finally, all charts and figures were completed using Excel (v2021).

3. Results

3.1. Variation in Head Loss with Operating Time

3.1.1. Sand Filter

The variation in ∆H_si for the sand filter with increasing T_i is relatively gentle, and both ∆H_si and its fluctuations increase with the rise in Q₀ (Figure 5). When Q₀ was 30 m³/h, G_in0 ranged from 0.66 to 2.16 g/L, and D_50in0 ranged from 4.91 to 15.84 μm, the average ∆H_si was 0.014 MPa. When Q₀ was 40 m³/h, G_in0 was 0.63–2.18 g/L, and D_50in0 was 4.71–15.91 μm, the average ∆H_si was 0.024 MPa. When Q₀ was 50 m³/h, G_in0 was 0.62–3.60 g/L, and D_50in0 was 4.80–15.25 μm, the average ∆H_si was 0.031 MPa.

3.1.2. Disc Filter and Multi-Stage Filtration System

The ∆H_di of the disc filter (Figure 6) and ∆H_i of the multi-stage filtration system (Figure 7) both exhibit a linearly increasing trend with the rise in T_i, and the magnitude of the increase grows with the rise in Q₀, G_in0, and D_50in0.

When Q₀ was 30 m³/h, G_in0 was 0.66–2.16 g/L, and D_50in0 was 4.91–15.84 μm, ∆H_di ranged from 0.01 to 0.075 MPa, and ∆H_i ranged from 0.023 to 0.088 MPa; when Q₀ was 40 m³/h, G_in0 was 0.63–2.18 g/L, and D_50in0 was 4.71–15.91 μm, ∆H_di was 0.021–0.079 MPa, and ∆H_i was 0.041–0.099 MPa; and when Q₀ was 50 m³/h, G_in0 was 0.62–3.60 g/L, and D_50in0 was 4.80–15.25 μm, ∆H_di was 0.031–0.075 MPa, and ∆H_i was 0.061–0.108 MPa.

As G_in0 and D_50in0 increased, the operating time (T_i) required for ∆H_di and ∆H_i to reach 0.07 MPa gradually shortened across the three Q₀ levels. Specifically, when G_in0 sequentially increased from <1.0 g/L to 1.0–2.0 g/L and >2.0 g/L, the average T_i for ∆H_di decreased from 129 min to 51 min and 28 min, respectively, while the average T_i for ∆H_i decreased from 80 min to 33 min and 13 min. Similarly, when D_50in0 sequentially increased from 4.0–7.0 μm to 8.0–11.0 μm and >15.0 μm, the average T_i for ∆H_di decreased from 126 min to 62 min and 29 min, respectively, while the average T_i for ∆H_i decreased from 72 min to 39 min and 15 min.

3.2. Variation Patterns of Filtration Efficiency and Sediment Characteristics

At Q₀ of 30–50 m³/h, the average filtration efficiencies (E_si, E_di, and E_i) of the sand filter, disc filter, and multi-stage filtration system were 2.5%, 4.2%, and 6.4%, respectively. The filters and the filtration system exhibited a limited interception effect on sediment.

Taking Q₀ = 30 m³/h as an example, when G_ini ranges from 0.66 to 2.28 g/L and D_50ini ranges from 4.84 to 16.03 μm, G_midi and G_outi range from 0.65 to 2.22 g/L and 0.62 to 2.17 g/L, respectively, while D_50midi and D_50outi range from 4.84 to 15.92 μm and 4.78 to 15.58 μm, respectively. Both outlet sediment concentration and median particle size of the filters and the filtration system were linearly correlated with their inlet counterparts, with slopes approaching 1 (Figure 8). The sediment characteristics at each node of the filtration system did not show significant changes.

3.3. Significance Analysis of Factors Influencing the Filtration Performance

Based on the Pearson correlation analysis results (Figure 9), the head loss variations in the filters and filtration system are influenced by multiple factors. ∆H_si exhibited significant positive correlations with Q₀ (r = 0.93) and G_in0 (r = 0.25), and significant negative correlations with T_i (r = −0.36) and H_in0 (r = −0.25) (p < 0.05), while no significant correlation with D_50in0 was observed. ∆H_di was significantly and positively correlated only with T_i (r = 0.58, p < 0.01). Meanwhile, ∆H_i showed significant positive correlations with Q₀ (r = 0.47) and T_i (r = 0.38) (p < 0.01), whereas its correlations with the remaining variables were not significant.

In terms of sediment characteristics, the outlet sediment characteristics (G_midi or G_outi, D_50midi or D_50outi) of the sand filter, disc filter, and multi-stage filtration system were all significantly and positively correlated with their respective inlet sediment characteristics (G_in0 or G_mid0, D_50in0 or D_50mid0) (r = 0.993–0.995, p < 0.01) and were significantly and negatively correlated with T_i (r = −0.42 to −0.36, p < 0.05). Additionally, both D_50midi and D_50outi exhibited significant positive correlations with H_in0 (r = 0.27–0.28).

The filtration efficiency of the sand filter, disc filter, and the multi-stage filtration system is relatively low; however, the variation patterns of head loss differ. ∆H_si is significantly positively correlated with Q₀, indicating that its head loss is primarily governed by hydraulic conditions, with a relatively minor influence from sediment retention. In contrast, both ∆H_di and ∆H_i show significant positive correlations with T_i, reflecting the cumulative effects induced by continuous sediment deposition within the disc flow channels. For the multi-stage filtration system, the evolution of ∆H_i is dominated by the disc filter and is jointly influenced by both Q₀ and T_i.

3.4. Development of the LR Model

3.4.1. Head Loss of the Filters and the Filtration System

The head loss of filters results from the coupled effects of flow resistance and sediment retention. The flow rate and pressure conditions determine the instantaneous energy dissipation as the fluid passes through the porous medium, whereas sediment characteristics and operating time drive the dynamic evolution of head loss by altering the pore structure and permeability.

Building upon the aforementioned head loss formation mechanism, to accurately calculate the head loss of the filters and filtration system and quantify the combined effects of multiple factors, MLR was employed to construct regression equations for ∆H_si, ∆H_di, and ∆H_i incorporating all independent variables. The R² values were 0.916, 0.568, and 0.674, respectively (Equations (11)–(13)), and all models passed the overall significance test (p < 0.05).

∆H_si = 0.048H_in0 − 0.001G_in0 + 0.001D_50in0 + 0.0002T_i + 0.001Q₀ − 0.03  R² = 0.916

(11)

∆H_di = 0.105H_mid0 + 0.007G_mid0 + 0.001D_50mid0 + 0.019T_i + 0.001Q₀ − 0.045  R² = 0.568

(12)

∆H_i = 0.076H_in0 + 0.005G_in0 + 0.002D_50in0 + 0.02T_i + 0.002Q₀ − 0.062  R² = 0.674

(13)

3.4.2. Outlet Sediment Characteristics of the Filters and the Filtration System

Based on Figure 8 and the Pearson correlation analysis results, the inlet sediment characteristics are the primary factor governing the variations in outlet sediment characteristics (r > 0.99). The SLR equations between the outlet and inlet sediment characteristics for the sand filter (Equations (14) and (15)), disc filter (Equations (16) and (17)), and multi-stage filtration system (Equations (18) and (19)) exhibited high goodness-of-fit, with R² values ranging from 0.9862 to 0.9903.

G_midi = 0.9492 G_in0 + 0.0611  R² = 0.9875

(14)

D_50midi = 1.0147 D_50in0 − 0.0745  R² = 0.9902

(15)

G_outi = 0.9493 G_mid0 + 0.0455  R² = 0.9875

(16)

D_50outi = 0.9744 D_50mid0 + 0.0213  R² = 0.9903

(17)

G_outi = 0.9384 G_in0 + 0.0283  R² = 0.9862

(18)

D_50outi = 0.9933 D_50in0 − 0.0663  R² = 0.9878

(19)

3.5. Development of the DE/RFR Model

3.5.1. Hyperparameter Optimization of the RFR Model Based on the DE Algorithm

To improve the prediction accuracy of the hydraulic performance at the outlet of the filters and the filtration system, this study developed both RFR and DE/RFR models. The hyperparameters of the RFR model were determined within predefined ranges based on the literature experience and were kept fixed. For the DE algorithm, considering the relatively small sample size of this study, a four-level full factorial experiment was conducted for parameter tuning. Specifically, the NP was set to 5, 10, 15, and 20; the F was set to 0.4, 0.5, 0.7, and 0.9; and the CR was set to 0.5, 0.6, 0.7, and 0.9, resulting in a total of 64 parameter combinations. Based on five-fold cross-validation on the training set, the minimum RMSE for the head loss prediction model on the validation set was obtained when NP = 10, F = 0.5, and CR = 0.7. Therefore, this parameter set was selected as the optimal configuration for the DE algorithm. During the iterative process, the fitness value tended to stabilize after 10 iterations and remained constant after 20 iterations, indicating that the model had converged. Therefore, the number of iterations was set to 20. The mean error of 5-fold cross-validation was utilized as the fitness function, with a tolerance threshold of 0.01 for the fitness value between adjacent iterations. The results indicated that the optimized n_estimators for ∆H_di were 465, significantly higher than those for ∆H_si (120) and ∆H_i (149). Similarly, the max_depth for ∆H_di was 19, higher than that for ∆H_si (11) and ∆H_i (14). Regarding the low-dimensional case with five input variables, the max_features obtained via DE optimization were all close to 1, exhibiting a gradual decrease from ∆H_si (0.97) to ∆H_i (0.83) (

Table 4. Hyperparameters of the RFR model and optimal hyperparameters of the DE/RFR model.

Optimal Parameters	ΔHsi		ΔHdi		ΔHi
	RFR	DE/RFR	RFR	DE/RFR	RFR	DE/RFR
n_estimators	300	120	200	465	200	149
max_depth	15	11	12	19	15	14
min_samples_split	2	2	2	2	5	2
min_samples_leaf	1	1	2	1	2	1
max_features	0.8	0.97	0.9	0.93	0.7	0.83

).

3.5.2. Feature Importance Analysis Based on the DE/RFR Model

Given that Pearson correlation analysis and MLR are primarily based on linear assumptions, they lack the capacity to capture complex nonlinear relationships and interaction effects among variables. Consequently, the permutation importance method within the DE/RFR model was employed to more accurately quantify the relative contributions of independent variables to ∆H_si, ∆H_di, and ∆H_i (Figure 10). The importance of variables affecting Q₀ > H_in0 > D_50in0 > G_in0 > Tᵢ; Tᵢ > G_mid0 > Q₀ > D_50mid0 > H_mid0; and Tᵢ > Q₀ > G_in0 >H_in0 > D_50in0, respectively. Specifically, in the feature importance analysis for ∆H_si, Q₀ emerged as the most dominant factor with a relative importance of 85.7%. For ∆H_di, the cumulative contribution of Tᵢ, G_mid0, and Q₀ reached approximately 93.6%. Regarding the prediction of ∆H_i, Tᵢ and Q₀ were identified as the two key variables, with Tᵢ providing the largest contribution (57.2%) and Q₀ accounting for a relative importance of 29.8%.

3.6. Prediction Accuracy Comparison and Model Selection

The DE/RFR model demonstrated the highest prediction accuracy for ∆H_si, ∆H_di, and ∆H_i, yielding R² values of 0.9340, 0.7304, and 0.7078; RMSE values of 0.0017, 0.0096, and 0.0104 MPa; and MAE values of 0.0013, 0.0075, and 0.0085 MPa, respectively (

Table 5. Comparison of the prediction accuracy of MLR and DE/RFR models for ∆H_si, ∆H_di, and ∆H_i.

Output Variables	Technique	R2	RMSE (Mpa)	MAE (Mpa)
∆Hsi	MLR	0.9160	0.0023	0.0018
	RFR	0.9276	0.0018	0.0014
	DE/RFR	0.9340	0.0017	0.0013
∆Hdi	MLR	0.5680	0.0120	0.0096
	RFR	0.6716	0.0106	0.0089
	DE/RFR	0.7304	0.0096	0.0075
∆Hi	MLR	0.6740	0.0118	0.0094
	RFR	0.6796	0.0109	0.0088
	DE/RFR	0.7078	0.0104	0.0085

). Compared with the MLR and RFR models, the average R² of the DE/RFR model increased by 11.86% and 4.53%, respectively. Furthermore, the average RMSE and MAE of the DE/RFR model decreased by 19.31% and 19.74% relative to the MLR model, and by 6.52% and 8.76% relative to the RFR model.

Based on the DE/RFR model, the predicted values of ΔH_si were in close agreement with the measured data. Regarding ΔH_di and ΔH_i, the DE/RFR model effectively captured the fluctuation trends of the experimental data; however, at peak values ranging from 0.07 to 0.075 MPa for ΔH_di and 0.084 to 0.102 MPa for ΔH_i, the prediction errors were 17.6–20.1% and −19.9% to 11.9%, respectively (Figure 11).

4. Discussion

4.1. Variation Patterns of Head Loss Under Sediment-Laden Water Conditions

When G_in0, D_50in0, Q₀, and filter media size ranged from 0.6–2.5 g/L, 25–63 μm, 1.24–25 m³/h, to 0.85–3.0 mm, respectively, the ΔH_si of sand filters increased with T_i, reaching 0.07 MPa within 20–60 min of operation [45,46]. In the present study, due to the small D_50in0, the ratio of sediment size to filter media size was merely 1/851–1/125. This value is significantly lower than the range of 1/48–1/13 reported in the aforementioned studies, resulting in ineffective interception of Yellow River sediment by the filter media. Consequently, even with G_in0 and Q₀ reaching as high as 3.6 g/L and 50 m³/h, respectively, ΔH_si remained relatively stable as T_i increased, maintaining values of 0.025–0.029 MPa after 24 min of operation. When Q₀ was 30–50 m³/h, ΔH_si exhibited fluctuations (Figure 5). This phenomenon was attributed to the dynamic attachment, deposition, and detachment of sediment particles at the micro-scale within the filter bed. The analysis indicates that the fluctuation amplitude of ΔH_si is influenced by the flow rate. Under low flow conditions, the hydrodynamic shear is relatively weak, resulting in a relatively stable particle deposition process and smaller fluctuations in ΔH_si. Under high flow conditions, particles migrate deeper into the filter layer and scouring of previously deposited particles occurs, thereby intensifying the fluctuations in ΔH_si. Similarly, the Pearson correlation analysis for the sand filter in this study indicated that ΔH_si was significantly positively correlated with both Q₀ and G_in0 (p < 0.05). This finding is consistent with the conclusions of Ren et al. [47] and Zhang et al. [48]: as G_in0 increased from 0.3 g/L to 0.8 g/L and Q₀ from 12 m³/h to 24 m³/h, ΔH_si rose from 0.045 MPa to 0.067 MPa after 20 min of operation.

In contrast to the sand filter, experimental data from this study indicated that ΔH_di increased with the increase in Tᵢ, and the T_max shortened as both G_in0 and D_50in0 increased. For example, when G_in0 increased from <1.0 g/L to >2.0 g/L, T_max decreased from 129 min to 28 min; similarly, when D_50in0 increased from 4.0–7.0 μm to >15.0 μm, T_max decreased from 126 min to 29 min (Figure 6). Under sediment-laden water conditions, the accumulation of particles within the disc filter channels is the primary cause of increased head loss. Higher sediment concentration increases the particle flux per unit time, while larger particles are more prone to local deposition at the inlet and within the filter channels, thereby accelerating the clogging process. Similarly, Yang et al. [20] observed that, under conditions where Q₀ was 30 m³/h and D₅₀ was 40–80 μm, when G_in0 increased from 0.2 g/L to 0.4 g/L, ΔH_di increased with Tᵢ, and T_max decreased from 15–18 min to 4–5 min. Similarly, Li et al. [22] confirmed via regression analysis that ∆H_di was positively correlated with G_in0 and D_50in0 for both linear and annular flow channels. Notably, the Pearson correlation analysis in this study indicated that ΔH_di was significantly influenced only by Tᵢ (p < 0.05). This phenomenon was attributed to the linear cumulative effect of Tᵢ during the monitoring process, which accounted for a contribution rate as high as 77.8%. Consequently, the strong temporal characteristics masked the significant impact of the initial conditions (G_in0 and D_50in0) on the increase in ΔH_di. Furthermore, the integration of Pearson correlation analysis and feature importance analysis yielded consistent conclusions, indicating that ΔH_si, ΔH_di, and ΔHᵢ were primarily influenced by Q₀, by Tᵢ and G_mid0, and by Tᵢ and Q₀, respectively. Given the performance disparities among the aforementioned filters, the ΔHᵢ in a multi-stage filtration system combining sand and disc filters is predominantly attributed to the disc filter. Consequently, the overall operational performance of the system is determined by the disc filter [31,49].

4.2. Analysis of Outlet Sediment Characteristics and Filtration Performance Under Sediment-Laden Water Conditions

Analysis of 225 sediment-laden water samples revealed that for the sand filter, disc filter, and multi-stage filtration system, both the sediment concentration and median particle diameter at the outlet exhibited linear correlations with those at the inlet (Figure 8). The average E_si, E_di, and E_i for these three filtration configurations were merely 2.5%, 4.2%, and 6.4%, respectively. This is attributed to the fact that, from July to October 2025, the incoming sediment particle sizes in the Heji diversion canal were consistently smaller than the filtration accuracy of the disc filter. This indicated that the filters and the filtration system with this specific filtration accuracy provided ineffective interception of the Yellow River sediment in the 2025 experimental area. Similarly, Yang et al. [21] found that a linear disc filter with a filtration accuracy of 125 μm exhibited a filtration efficiency of merely 3.4–7.9% for sediment-laden water with a D₅₀ of approximately 40–50 μm. Yuan et al. [50] revealed that clogging in disc filters is primarily concentrated in the upper filter element and channel inlets, where, in suboptimally designed filters, over 1/3 of the element and 81% of the flow channels fail to intercept sediment, resulting in a physical clogging rate of merely 25% even when ΔH_di reaches 0.06 MPa. It is precisely for this reason that the disc filter in the present study exhibited low filtration efficiency, yet a substantial increase in ΔH_di with respect to Tᵢ.

Relevant studies indicate that, when the sediment D₅₀ ranges from 49 to 66 μm, a filter with a filtration accuracy of 75–100 μm is required; whereas, for a D₅₀ of 82–100 μm, a filtration accuracy of 125 μm is more suitable [19,23]. However, increasing the filtration accuracy often leads to problems such as rapid clogging and frequent backflushing. Hou et al. [2] reported that the D₅₀ of sediment deposited in the flow channels of emitters (rated flow: 1.6–2.8 L/h) operated for one year in the Hetao Irrigation District was 22.2–30.5 μm, implying that most sediment with a D₅₀ smaller than 22.2 μm could pass through the emitters. In the present study, although the E_i of the multi-stage filtration system was relatively low, the D₅₀ of the outlet sediment was merely 4.63–15.56 μm, which allowed these particles to pass through the emitters, posing a low risk of clogging.

4.3. Prediction Model for Filtration Performance of Multistage Filtration Systems Using Yellow River Water

Based on experimental data, MLR, RFR, and DE/RFR models were developed to predict ∆H_si, ∆H_di, and ∆H_i under Yellow River water conditions. The prediction accuracy followed the order: DE/RFR > RFR > MLR. Specifically, the RMSE of the DE/RFR model was reduced by an average of 19.3% compared to the MLR model and by 6.5% compared to the RFR model. Constrained by the assumption of linear relationships between dependent and independent variables, the MLR model often struggles to effectively capture and characterize the intrinsic features of data under highly complex non-linear conditions [51]. In contrast, the Random Forest Regression (RFR) model, based on ensemble learning, can accurately capture complex high-dimensional non-linear relationships between inputs and outputs by constructing and aggregating multiple decision trees [52,53]. However, the accuracy of RFR is closely related to its hyperparameter settings (e.g., number of trees and depth) [54]. While Grid Search (GS) is a conventional method for determining hyperparameters, integrating meta-heuristic algorithms—such as Differential Evolution (DE) and Genetic Algorithms (GAs)—with mathematical models has recently become a vital approach to further enhancing model performance [55]. Moradi et al. [56] analyzed operational data from a water treatment plant and demonstrated that prediction models using influent water quality, chemical parameters, and flow rate as inputs could accurately predict the unit filter run volume of sand filters. Among them, the Grid Search-optimized Random Forest (GS-RFR) model exhibited the best performance, with its RMSE reduced by 55.5–57.9% compared to the MLR model. In this study, the DE algorithm was employed to achieve adaptive global optimization of RFR hyperparameters, thereby improving the prediction accuracy of the RFR model. Similarly, Garcia-Nieto et al. [29] developed linear regression models (Ridge, Lasso, and Elastic-net) and a DE/RFR model. In predicting the effluent turbidity of sand filters, the RMSE of the DE/RFR model was 46.3–46.5% lower than that of the linear regression models.

4.4. Application of the DE/RFR Model to Drip Irrigation Systems in the Yellow River Basin

Based on the ΔH_di calculation model (DE/RFR), this study further developed a Filtration Cycle (T_max) Prediction Model (FCPM), transforming the forward prediction of ΔH_di into an inverse solution for T_max. Regarding model construction, building upon the established DE/RFR model, this study coupled the Differential Evolution (DE) algorithm with the Limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm with Box constraints (L-BFGS-B). Specifically, the DE algorithm was first utilized to perform global optimization under the given input parameters (Q₀, H _in0, D_50in0, G_in0) to locate the optimal solution region, followed by local gradient refinement using the L-BFGS-B algorithm. By minimizing the sum of squared residuals between the predicted and target values of ΔH_di, the precise calculation of T_max under given operating conditions was achieved. In 2021, the sediment concentration in the Main Canal of the Hetao Irrigation District ranged from 0.70 to 3.90 g/L between April and November, with a D₅₀ of 18–35 μm [57]. Using these water–sediment characteristics as initial input parameters for the FCPM, under the conditions of H_in0 = 0.19 MPa and Q₀ = 40 m³/h, the predicted T_max ranged from 25 to 115 min (Figure 12), exhibiting a seasonal trend of initially increasing and then decreasing. During the spring irrigation period (April–May), due to the high sediment concentration (3.5–3.9 g/L), and the autumn irrigation period (September–November), due to the large D₅₀ (approx. 35 μm), the T_max ranged from 25 to 59 min; in the summer irrigation period (June–August), due to the lower sediment concentration (0.7–2.2 g/L) and smaller D₅₀ (approx. 18 μm), the T_max increased to 42–118 min. Backflushing of filtration systems typically causes reductions in outlet pressure and flow rate, thereby compromising the uniformity of irrigation and fertilization in drip irrigation systems. Based on the DE/RFR and FCPMs developed in this study, the variations in head loss with T_i and the T_max of the filters and the filtration system can be predicted under diverse sediment characteristics and operating conditions. Therefore, these predictions provide critical technical support for optimizing the configuration (e.g., filter specifications and combination schemes) and operational strategies (e.g., pump pressure regulation, fertilization timing, and backwashing frequency) of multi-stage filtration systems for sediment-laden water drip irrigation.

5. Conclusions

This study investigated the effects of Q₀, H₀, D_50in0, G_in0, and T_i on the head loss (ΔH_si, ΔH_di, and ΔH_i) and outlet sediment characteristics (G_midi, D_50midi, or G_outi, D_50outi) of the filters and the filtration system through field tests in the Hetao Irrigation District, Inner Mongolia, China, using Yellow River sediment-laden water. The primary conclusions are as follows:

• Under sediment concentrations of 0.62–3.6 g/L, median particle sizes of 4.70–16.03 μm, and flow rates of 30–50 m³/h, the ΔH_si remained stable over the T_i. In contrast, the ΔH_di reached the backflushing threshold of 0.07 MPa after 16–235 min of operation, increasing with operating time. The magnitude of this increase grew with higher flow rates, sediment concentrations, and median particle sizes. Furthermore, by integrating Pearson correlation analysis and DE/RFR feature importance analysis, the factors influencing the head loss of the filters and the filtration system were identified. Specifically, ΔH_si, ΔH_di, and ΔH_i were primarily influenced by flow rate, sediment concentration and operating time, and flow rate and operating time, respectively.
• The inlet and outlet sediment characteristics of the filters and the filtration system were significantly and linearly correlated. The average filtration efficiencies of the sand filter, disc filter, and multi-stage filtration system were 2.5%, 4.2%, and 6.4%, respectively. The outlet median particle size (4.63–15.56 μm) remained within the safety threshold (22.2–30.5 μm).
• The Random Forest Regression model optimized by the Differential Evolution algorithm (DE/RFR) developed in this study exhibited high accuracy in predicting ΔH_si, ΔH_di, and ΔH_i. The model achieved R² values ranging from 0.71 to 0.93 and RMSE values from 0.0017 to 0.0104 MPa. The FCPM, developed on the basis of the DE/RFR model, can calculate T_max for the filters and the filtration system under different operating conditions. This model can provide technical support for the configuration of filter specifications and combination modes in sediment-laden drip irrigation systems, as well as for the formulation of operational strategies such as pump pressure regulation, fertilization timing, and backwashing frequency.
• This study is constrained by variations in the inflow conditions of the Yellow River and the frequency of water distribution, resulting in a relatively limited sample size of in situ experiments. Future work is suggested to conduct multi-year repeated experiments combined with quantitative sediment addition tests, in order to expand the dataset for the DE/RFR model, thereby further improving its prediction accuracy and generalization performance for head loss.