Method for predicting penetration curve and solute transport parameters of karst pipeline

Abstract

The invention belongs to the technical field of solute transport prediction by a deep learning method, and particularly relates to a method for predicting a penetration curve and solute transport parameters of a karst pipeline, which applies a Bayesian optimized multilayer perceptron (MLP) model to predict complete penetration curves of two karst pipeline structures under nine different water flow conditions. And then simulating the predicted penetration curve by adopting a temporary storage model to obtain solute transport parameters. And for part of working conditions, a random forest and a support vector regression model are synchronously adopted to predict a complete penetration curve, MLP is compared with RF and SVR, and the superior performance of the MLP is verified. The prediction performance of the MLP on the penetration curve and the solute transport parameter is compared with the prediction value of the penetration curve and the solute transport parameter directly using other flows, and the superior performance of the MLP is verified.

Description

Technical Field

[0001] The present invention belongs to the technical field of solute transport prediction using deep learning methods, and specifically relates to a method for predicting karst pipeline penetration curves and solute transport parameters. Background Art

[0002] The transport of solutes that are temporarily stored in karst conduits is significantly affected by multiple factors, especially flow rate. Temporary storage refers to the temporary retention of solutes in a storage area before returning to the main conduit. Storage areas in karst conduits are mainly caused by local changes in the conduit geometry (solution pools, dissolution pores or top holes) or irregularities in the conduit surface (ripples, deposited sediments, cave collapses, etc.). Flow rate affects convection processes through flow velocity and longitudinal shear diffusion through spatial variations in flow velocity; however, the relationship between temporary storage and flow rate is relatively complex. During heavy rainstorms, the flow rate in karst conduits varies greatly. Therefore, the solute transport process in karst conduits exhibits considerable variability and complexity.

[0003] In practice, karst conduit tracer tests may not be conducted under expected flow conditions, resulting in unexpected solute transport parameters derived from interpreting the test breakthrough curve. Ideally, tracer tests should be conducted within the expected flow range. However, budget and time constraints often make this impractical. Therefore, predictions of solute transport parameters at specific flow rates are necessary. This relies on understanding the relationship between solute transport parameters and flow rate. However, studies have shown that the relationship between solute transport parameters and flow rate is not robust and consistent, making prediction of these parameters challenging. Therefore, predicting breakthrough curves at different flow rates becomes crucial. Previous studies have analyzed the relationship between characteristic parameters of breakthrough curves and flow rate using tracer tests, but these were only able to predict characteristic parameters or portions of the breakthrough curve. To date, no studies have predicted and interpreted complete breakthrough curves under different flow conditions to derive solute transport parameters at different flow rates, which is crucial for predicting contamination in karst groundwater. Summary of the Invention

[0004] The present invention discloses a method for predicting the penetration curve and solute transport parameters of karst pipelines, which can predict the penetration curve and solute transport parameters of unknown flow rate based on the penetration curve of known flow rate, and the prediction performance is better than traditional machine learning methods.

[0005] To achieve the above object, the technical solution of the present invention is:

[0006] A method for predicting karst pipeline penetration curves and solute transport parameters comprises the following steps:

[0007] Step 1: Use the penetration curve data of two pipelines under nine different flow conditions to build a multi-layer perceptron (MLP) model;

[0008] Step 2: Optimize the hyperparameters of the multilayer perceptron (MLP) model.

[0009] Step 3: After training the multi-layer perceptron (MLP) model with optimized hyperparameters, it is used to predict penetration curves for different flow rates, and the prediction performance is evaluated and compared with the baseline machine learning model.

[0010] Step 4: Use the temporary storage model to simulate the predicted breakthrough curve to obtain the model parameters, which are the predicted solute transport parameters;

[0011] Step 5: Compare the prediction errors of the breakthrough curve and solute transport parameters with the errors of directly using the breakthrough curve and solute transport parameters of other flow rates.

[0012] Preferably, in the step 1:

[0013] (1) The multi-layer perceptron MLP model has a dense layer, and the output calculation formula of the dense layer is as follows:

[0014] h W,b (X)=f(XW+b) Formula (1);

[0015] In formula (1), X is the input vector, h W,b is the output value, W is the weight matrix, which contains all connection weights except the bias neuron connection weight, b is the bias vector, which contains all connection weights between the bias neuron and the artificial neuron, and f is the activation function of the multilayer perceptron neuron, with f as the rectified linear unit function, that is, ReLU(z)=max(0,z);

[0016] (2) The architecture of the multilayer perceptron (MLP) model has two hidden layers, uses a rectified linear unit activation function for nonlinear feature learning, and finally an output layer with one neuron for predicting concentration; the number of neurons in each hidden layer, i.e., the hidden layer size, is the same to ensure consistent processing power across the network; the multilayer perceptron is trained using the RMSprop optimizer, with rho set to 0.9, a learning rate set to 0.001, and a mean square error (MSE) as the loss function, i.e., formula (2); the multilayer perceptron model is set to train for a maximum of 1000 training cycles, and an early stopping mechanism is added, with a patience value of 50 steps to prevent overfitting;

[0017]

[0018] In formula (2), o represents the observed value, p represents the predicted value, and n represents the number of concentration data in the predicted breakthrough curve;

[0019] (3) The penetration curve data comes from the literature. By generalizing the field karst pipeline into an indoor pipeline system, using polyvinyl chloride (PVC) hoses to represent the karst pipeline, and using cubic water tanks to represent the karst pool structure developed in the pipeline, indoor tracer experiments were carried out under three pipeline structures and nine water flow conditions based on the indoor pipeline system to obtain the above penetration curve. The experimental pipeline system includes a constant head water supply tank, a PVC hose connected to the water supply tank, and a cubic symmetrical or asymmetrical water tank installed on the PVC hose. The three pipeline structures include: a single pipeline without a water tank, a symmetrical water tank-pipe with a symmetrical water tank installed, and an asymmetric water tank-pipe with an asymmetric water tank installed. Among them, the cubic symmetrical water tank has relative inlets / outlets in the middle of the front and back walls of the water tank, while the inlets / outlets on the front and back walls of the cubic asymmetric water tank are located at the lower left corner of the front wall and the lower right corner of the back wall respectively. The side length of the two water tanks is 0.1 m. For the asymmetric water tank, the deviation l of the inlet and outlet of the water tank perpendicular to the main flow direction of the pipeline is 0.071 m, while for the symmetric water tank, l is 0.

[0020] (4) In each tracer experiment, 2.5 ml of 100 g / L sodium chloride solution was injected as the tracer. The tracer concentration was obtained by measuring the conductivity. The tracer concentration was measured at the outlet of the PVC hose and at a position 40 m away from the injection point. The flow rate was measured by a turbine flow meter with an accuracy of ±1.0%. To ensure the stability of the flow measurement, the flow meter was installed 6 m away from the outlet of the water supply tank. The tracer injection point was set 2 m away from the flow meter outlet to ensure the stability of the water flow in the pipe at this location. The length of the PVC hose from the tracer injection point to the outlet was 101.1 m:

[0021] (5) Pipe structures are classified according to two key properties: the volume of the tank V and the deviation l between the tank inlet and outlet. For a single pipe, l and V are both 0. For symmetrical tank-pipes (b1-b9), the deviation l is 0 and the tank volume V is 0.001m 3 For the asymmetric tank-pipe (c1-c9), the deviation l is 0.071m and the tank volume V is 0.001m 3 ;

[0022] (6) Three prediction scenarios S1, S2, and S3 are used. The multi-layer perceptron (MLP) model uses different data sets and input variables in different scenarios. The outlet tracer concentration is the model output variable. Scenario S1 uses the penetration curve data from a single pipeline structure and focuses on the outlet concentration C. i The time after injection t i and flow rate Q iScenario S2 uses the penetration curve data of two pipeline structures and takes l as an additional input variable to characterize the difference between the two pipeline structures. Scenario S3 integrates the data of all three pipeline structures and takes both V and l as input variables to characterize the differences between the three pipeline structures:

[0023] (7) Divide the data into a calibration set and a prediction set to fine-tune the parameters of the multilayer perceptron (MLP) model and predict the penetration curve; in the calibration phase, the calibration set is further divided so that 80% of the data is used as a training set and 10% of the data is used as a validation set to terminate the training process when necessary (early stopping mechanism), and the remaining 10% of the data is used as a test set during the hyperparameter optimization process;

[0024] Before training, all input data are normalized according to their minimum and maximum values in the training set so that their values are between 0 and 1.

[0025] Preferably, the step 2 includes a Bayesian optimization method, specifically:

[0026] Step 1. Consider an objective function f(x) on a bounded domain X, where x represents a set of hyperparameters. The hyperparameters range from 1 to 64 neurons in each hidden layer and the batch size ranges from 16 to 256. The objective function is chosen to be the Nash-Sutcliffe efficiency (NSE) and the square of the Pearson correlation coefficient (R). 2 The sum of o and p, where o and p have the same meaning as in formula (2), and the objective function for hyperparameter search is calculated separately on the optimization set:

[0027]

[0028]

[0029] Step 2. Define the observation values of N points as where y i =f(x i )+ε i , ε i The variance is Gaussian noise, in Bayesian optimization, is 0, and N is initially set to 25;

[0030] Step 3. Fit a probabilistic surrogate model to the target observations: The probabilistic surrogate model is a Gaussian process that outputs a predictive distribution for the potential value of the target function at each input point based on the observations in step 2. Subsequently, the target value, i.e., the maximum value of f, is obtained. The predictive distribution is a Gaussian distribution with a mean μ(x *) and variance σ 2 (x * ), that is, p(f(x * )|D,x * )=N(μ(x * ),σ 2 (x * )), where the mean and variance of the posterior distribution of the random variable are:

[0031]

[0032]

[0033] in, d=2here;

[0034] K ij =k(x i ,x j ); k(x,x′)

[0035] is the prior kernel function;

[0036] Step 4. The next point to be selected is determined by the acquisition function. The prediction distribution in step 3 summarizes the uncertainty about the target function and is used to calculate the acquisition function α(·) on the domain X. The next point x is determined by selecting the independent variable that maximizes the acquisition function α(·). N+1 ; The acquisition function is selected as the expected improvement EI; once a new observation is obtained, the operations from step 2 to step 4 are repeated iteratively; the expected improvement EI acquisition function is given by the following expression:

[0037] α(x * )=(μ(x * )-υ)Φ(Z)+σ(x * )φ(Z) Equation (7);

[0038] Z=(μ(x * )-υ) / σ(x * ) Formula (8);

[0039] in, is the best observed value obtained so far, Φ(·) and φ(·) represent the cumulative distribution function CDF and probability density function PDF of the standard Gaussian distribution, respectively.

[0040] Preferably, the step three includes: setting three indicators for evaluating the performance of the multilayer perceptron MLP model in predicting the penetration curve to be NSE, R 2 And the root mean square error RMSE:

[0041]

[0042] Support vector regression and random forest were chosen as benchmark machine learning models for comparison.

[0043] Preferably, the step 4 includes:

[0044] The predicted breakthrough curve was simulated using a temporary storage model to calibrate the solute transport parameters. The temporary storage model generalizes solute transport as occurring within the main pipeline. Simultaneously, a secondary exchange process occurs in the storage area caused by the dead-end channel and the water tank. This exchange process allows the solute to slowly flow out of the storage area and gradually re-enter the main pipeline, resulting in the tailing of the breakthrough curve. The model equation is as follows:

[0045]

[0046]

[0047] Where t is time [T], x is distance [L], C and C s are the concentrations of solute in the main channel and storage area [M / L 3 ];A and A s are the cross-sectional areas of the main channel and storage area [L 2 ]; Q is the flow rate [L 3 / T]; D is the diffusion coefficient [L 2 / T], α is the exchange coefficient [T -1 ].

[0048] The predicted model parameters were compared with the actual values, and the prediction error was calculated according to the following formula:

[0049]

[0050] Among them, P pre is the predicted value of each model parameter, P act is the actual value.

[0051] The beneficial effects of the method for predicting karst pipeline penetration curves and solute transport parameters of the present invention are as follows:

[0052] The present invention is theoretically feasible, convenient in practice, and short in time. It can predict the breakthrough curve and solute transport parameters of an unknown flow rate based on the breakthrough curve of a known flow rate, and its prediction performance is better than that of traditional machine learning methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 This is a flow chart of the method for predicting the penetration curve and solute transport parameters of karst pipelines under different flow conditions according to the present invention;

[0054] Figure 2 Experimental penetration curves of three pipe structures at different flow rates: (a) penetration curve of a single pipe; (b) penetration curve of a symmetrical water tank-pipe; (c) penetration curve of an asymmetrical water tank-pipe.

[0055] Figure 3 Schematic diagram of the experimental setup.

[0056] Figure 4 During the calibration phase, the calibration set is further partitioned into graphs.

[0057] Figure 5 Comparison of optimized hyperparameters of the MLP models established for b1-b9(a) and c1-c9(b) in three scenarios.

[0058] Figure 6 The experimental concentration data, predicted concentration data, and concentration data simulated by the temporary storage model (TSM) for the symmetric water tank-pipeline under all three scenarios; the black solid line represents the experimental concentration data; the dot symbols represent the predicted concentration data under the three scenarios, and the solid lines of the same color represent the concentration data obtained by TSM simulation.

[0059] Figure 7 The experimental concentration data, predicted concentration data, and concentration data simulated by the temporary storage model (TSM) for the asymmetric water tank-pipeline under all three scenarios; the black solid line represents the experimental concentration data; the dot symbols represent the predicted concentration data under the three scenarios, and the solid lines of the same color represent the concentration data obtained by TSM simulation.

[0060] Figure 8 The prediction accuracy (NSE, R 2 and RMSE) with flow rate.

[0061] Figure 9 Penetration curves of cases b5 and c5 predicted using MLP, RF, and SVR in scenario S2.

[0062] Figure 10 Temporarily stores model parameters for prediction of pipelines b1-b9.

[0063] Figure 11 Temporarily stores model parameters for prediction of pipelines c1-c9.

[0064] Figure 12 Comparison of prediction errors of cases b1-b9 in scenarios S2, S3, and S4.

[0065] Figure 13Comparison of prediction errors of cases c1-c9 in scenarios S2, S3, and S4.

[0066] In the figure: 1. Water supply tank; 2. PVC pipe; 3. Valve; 4. Flow meter; 5. Tracer injection point; 6. Water tanks in series; 7. Optional water tank types; 8. Exhaust pipe. DETAILED DESCRIPTION

[0067] The following description is merely a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

[0068] The following embodiments may be understood as individually expressing a part of a local structure or method of the present invention, or may be understood as a combination of the embodiments to explain the connotation of a larger structure or method of the present invention.

[0069] Example 1

[0070] A method for predicting karst conduit penetration curves and solute transport parameters, such as Figure 1 As shown, the following steps are included:

[0071] Step 1: Use the penetration curve data of two pipelines under nine different flow conditions to build a multi-layer perceptron (MLP) model;

[0072] Step 2: Optimize the hyperparameters of the multilayer perceptron (MLP) model.

[0073] Step 3: After training the multi-layer perceptron (MLP) model with optimized hyperparameters, it is used to predict penetration curves for different flow rates, and the prediction performance is evaluated and compared with the baseline machine learning model.

[0074] Step 4: Use the temporary storage model to simulate the predicted breakthrough curve to obtain the model parameters, which are the predicted solute transport parameters;

[0075] Step 5: Compare the prediction errors of the breakthrough curve and solute transport parameters with the errors of directly using the breakthrough curve and solute transport parameters of other flow rates.

[0076] Example 2

[0077] Based on Example 1, this embodiment discloses that the step 1 includes:

[0078] (1) The multilayer perceptron (MLP) model has dense layers, where all neurons in a layer are fully connected to every neuron in the previous layer. The output of each dense layer is calculated as follows:

[0079] h W,b(X)=f(XW+b) Formula (1);

[0080] In formula (1), X is the input vector, h W,b is the output value, W is the weight matrix, which contains all connection weights except the bias neuron connection weight, b is the bias vector, which contains all connection weights between the bias neuron and the artificial neuron, and f is the activation function of the multilayer perceptron neuron, which is the rectified linear unit function in this invention, that is, ReLU(z)=max(0,z);

[0081] (2) The architecture of the multilayer perceptron (MLP) model has two hidden layers, uses a rectified linear unit activation function for nonlinear feature learning, and finally an output layer with one neuron for predicting concentration; the number of neurons in each hidden layer, i.e., the hidden layer size, is the same to ensure consistent processing power across the network; the multilayer perceptron (MLP) model is trained using the RMSprop optimizer, with rho set to 0.9, a learning rate set to 0.001, and the mean square error (MSE) as the loss function, i.e., formula (2); the multilayer perceptron (MLP) model is set to train for a maximum of 1000 training cycles, and an early stopping mechanism is added with a patience value of 50 steps to prevent overfitting; this means that if the loss value on the validation set does not decrease within 50 consecutive training cycles, training will stop early:

[0082]

[0083] In formula (2), o represents the observed value, p represents the predicted value, and n represents the number of concentration data in the predicted breakthrough curve;

[0084] (3) The penetration curve data used in the present invention are as follows Figure 2 As shown in the figure (the penetration curve data comes from the literature: Zhao X, Chang Y*, Wu J, Xue X. Effects of flow rate variation on solute transport in akarst conduit with a pool. Environmental Earth Sciences, 2019, 78(7)), the field karst pipeline is generalized into an indoor pipeline system, polyvinyl chloride (PVC) hose is used to represent the karst pipeline, and a cubic water tank is used to represent the karst pool structure developed in the pipeline. Based on the indoor pipeline system, indoor tracer experiments are carried out under three pipeline structures and nine water flow conditions to obtain the above penetration curve; the experimental pipeline system consists of a fixed head water supply tank, a long PVC hose and a cubic symmetrical or asymmetrical water tank ( Figure 3), 3 types of pipeline structures including single pipeline, symmetrical water tank-pipe and asymmetrical water tank-pipe, cubic symmetrical water tank ( Figure 3 The medium B tank) has opposite inlets / outlets in the middle of the front and back walls of the tank (with the inlet and outlet connected in series to the PVC hose), while the cubic asymmetrical tank ( Figure 3 (Tank A) The inlets and outlets on the front and rear walls are located at the lower left corner of the front wall and the lower right corner of the rear wall, respectively. Both tanks have a side length of 0.1 m. For the asymmetric tank, the deviation l between the inlet and outlet perpendicular to the main flow direction of the pipe is 0.071 m, while for the symmetric tank, l is 0.

[0085] (4) In each tracer experiment, 2.5 ml of 100 g / L sodium chloride solution was injected as the tracer. The tracer concentration was obtained by measuring the conductivity. The tracer concentration was measured at the pipe outlet and 40 m away from the injection point. The flow rate was measured by a turbine flowmeter (LWGY series) with an accuracy of ±1.0%. To ensure the stability of the flow measurement, the flowmeter was installed 6 m away from the water supply tank outlet. The tracer injection point was set 2 m away from the flowmeter outlet to ensure the stability of the water flow in the pipe at this location. The length of the pipe from the tracer injection point to the outlet was 101.1 m. The symbols for the three pipe structures under different flow conditions are shown in Table 1:

[0086] Table 1 Symbols for pipeline structures at different flow rates

[0087]

[0088] (6) Table 2 classifies the pipe structures according to two key properties: the volume of the tank V and the deviation l between the tank inlet and outlet. For a single pipe, l and V are both 0. For symmetrical tank-pipes (b1-b9), the deviation l is 0 and the tank volume V is 0.001m 3 For the asymmetric tank-pipe (c1-c9), the deviation l is 0.071m and the tank volume V is 0.001m 3 ;

[0089] Table 2 Characterization of differences in different pipeline structures

[0090]

[0091] (6) Three prediction scenarios (S1, S2, and S3) were used. The multi-layer perceptron (MLP) model used different data sets and input variables in different scenarios, as shown in Table 3. The outlet concentration was the model output variable. Scenario S1 used the penetration curve data from a single pipeline structure and focused on the outlet concentration (C i ) and the time after injection (t i ) and flow rate (Q i); scenario S2 uses the penetration curve data of two pipeline structures and uses l as an additional input variable to characterize the difference between the two pipeline structures; scenario S3 integrates the data of all three pipeline structures and uses both V and l as input variables to characterize the differences between the three pipeline structures:

[0092] Table 3 Input and output variables under three prediction scenarios

[0093]

[0094] (7) The data is divided into a calibration set and a prediction set, the purpose of which is to fine-tune the parameters of the multilayer perceptron (MLP) model and predict the penetration curve, respectively. As shown in Table 1, b1-b9 and c1-c9, there are a total of 18 independent prediction sets. Each scenario contains 18 prediction sets, which means that a total of 54 multilayer perceptron (MLP) models need to be built and trained.

[0095] During the calibration phase, the calibration set is further divided into ( Figure 4 :The data used is divided into four parts, which are used for training, early stopping mechanism, hyperparameter optimization and prediction. When b5 is the prediction set, for scenario S1, the calibration set is b1-b4 and b6-b9; for scenario S2, the calibration set is b1-b4, b6-b9 and c1-c9; for scenario S3, the calibration set is a1-a9, b1-b4, b6-b9 and c1-c9), so that 80% of the data is used as the training set, 10% of the data is used as the validation set to terminate the training process (early stopping mechanism) when necessary, and the remaining 10% of the data is used as the test set in the hyperparameter optimization process;

[0096] Before training, all input data are normalized according to their minimum and maximum values in the training set so that their values are between 0 and 1.

[0097] Example 3

[0098] The second step comprises:

[0099] Before training a Multilayer Perceptron (MLP) model, various hyperparameters must be set. The batch size directly impacts the stability of gradient estimation and computational throughput, while the hidden layer size determines the model's representational capabilities, training efficiency, and generalization capabilities. As key MLP hyperparameters, both require systematic optimization to achieve a balance between learning dynamics and model performance.

[0100] (1) Bayesian Optimization (BO)

[0101] Bayesian optimization (BO) is a global optimization algorithm for black-box functions, typically used when the objective function evaluation is very expensive. This paper uses the Bayesian optimization algorithm for hyperparameter tuning, aiming to improve model performance and reduce the risks associated with poor hyperparameter selection. The Bayesian optimization algorithm for hyperparameter tuning operates through four systematic steps, with the probabilistic surrogate model (step 3) and the acquisition function (step 4) being the two key components:

[0102] Step 1. Consider an objective function f(x) over some bounded domain X. Here, x represents a set of hyperparameters (hidden layer size and batch size). The hyperparameters range as follows: the number of neurons in each hidden layer (hidden layer size) varies from 1 to 64, and the batch size ranges from 16 to 256. The objective function is chosen to be the Nash-Sutcliffe efficiency (NSE) (Formula (3)) and the square of the Pearson correlation coefficient (R 2 )(Formula (4)), where o and p have the same meanings as in Formula (2). The objective function used for hyperparameter search is calculated separately on the optimization set.

[0103]

[0104]

[0105] Step 2. Define the observation values of N points as where y i =f(x i )+ε i , ε i The variance is Gaussian noise, in Bayesian optimization, is 0, and N is initially set to 25;

[0106] Step 3. Fit a probabilistic proxy model to the target observations: In the present invention, the probabilistic proxy model is a Gaussian process (GP). Based on the observations in step 2 (prior knowledge in Bayesian optimization), the Gaussian process outputs a predictive distribution for the potential value of the objective function at each input point. Subsequently, the target value (the maximum value of f) is obtained. The predictive distribution is a Gaussian distribution with mean μ(x*) and variance σ 2 (x*), that is, p(f(x * )|D,x * )=N(μ(x * ),σ 2 (x * )), where the mean and variance of the posterior distribution of the random variable are:

[0107]

[0108] in,

[0109] K ij =k(x i ,x j ); k(x,x′)

[0110] is a priori kernel function, such as radial basis function (RBF); I is the identity matrix;

[0111] Step 4. The next point to be selected is determined by the acquisition function. The prediction distribution in step 3 summarizes the uncertainty about the target function and is used to calculate the acquisition function α(·) on the domain X. The next point x is determined by selecting the independent variable that maximizes the acquisition function α(·). N+1 ; In the present invention, the acquisition function is selected as the expected improvement (EI); once this new observation value is obtained, the operations from step 2 to step 4 are iteratively repeated; the expected improvement (EI) acquisition function is given by the following expression:

[0112] α(x * )=(μ(x * )-υ)Φ(Z)+σ(x * )φ(Z) Equation (7);

[0113] Z=(μ(x * )-υ) / σ(x * ) Formula (8);

[0114] in, is the best observed value obtained so far, Φ(·) and φ(·) represent the cumulative distribution function (CDF) and probability density function (PDF) of the standard Gaussian distribution, respectively.

[0115] In summary, at each iteration of the optimization process, Bayesian optimization (BO) builds a surrogate model to select the next appropriate point. The Bayesian optimization process consists of at least 50 optimization steps, initially with 25 steps of random exploration followed by another 25 Bayesian optimization steps. The optimization process is set to terminate after 20 consecutive steps without improvement or when the total number of steps reaches an upper limit of 150.

[0116] (2) Genetic Algorithm (GA)

[0117] In addition to the Bayesian optimization algorithm, we also implemented an evolutionary search strategy, specifically a genetic algorithm (GA), for comparative analysis in two specific cases (b5 and c5 in Scenario S2). GAs operate through a biologically inspired mechanism: a population is initialized in an N-dimensional solution space and the solutions are iteratively optimized through selection, crossover, and mutation operators. These evolutionary processes prioritize the survival and recombination of individuals with high fitness, ultimately converging to an optimal solution that maximizes an objective function. Notably, the objective function used in the GA is the same as that used in Bayesian optimization.

[0118] Regarding the implementation of the genetic algorithm, the following parameter settings were used: population size of 30, a total of 15 generations, crossover probability (CXPB) of 0.5, mutation probability (MUTPB) of 0.3, mutation probability of each gene (indpb) of 0.2, and tournament selection size (tournsize) of 5.

[0119] Furthermore, based on the two optimization algorithms, the two hyperparameters of the multilayer perceptron (MLP) were globally optimized, and the optimal value of each hyperparameter is shown in Table 4. Table 5 shows the prediction performance of the multilayer perceptron on cases b5 and c5. Among the two optimization algorithms, Bayesian optimization (BO) showed higher prediction performance on both the optimization set and the prediction set compared with the genetic algorithm (GA). In addition, the time consumed by Bayesian optimization is much less than that of the genetic algorithm (see Table 4). Taking into account the prediction performance of the model on the prediction set and the time consumption during the optimization process, the present invention selects the Bayesian optimization algorithm as the method for searching the MLP hyperparameters.

[0120] Table 4 Optimal values of hyperparameters in the MLP model determined by the two optimization algorithms for cases b5 and c5 in scenario S2

[0121]

[0122] Table 5 Prediction performance of MLP based on two optimization algorithms for cases b5 and c5 in scenario S2

[0123]

[0124] Figure 5Box plots are shown for the hidden layer size, batch size, optimal number of iterations, and maximum number of iterations obtained by Bayesian optimization (BO) for the multilayer perceptron (MLP) model in three scenarios. For cases b1-b9, most of the maximum and optimal number of iterations in scenarios S2 and S3 are smaller than those in scenario S1. This is consistent with the significant improvement in model performance from scenario S1 to scenario S3. Hidden layer sizes mostly fall within the range of 43 to 64 and remain consistent across the three scenarios. In most cases, the batch size in scenario S2 is smaller than that in scenarios S1 and S3. For cases c1-c9, the ranges of batch size, optimal number of iterations, and maximum number of iterations are roughly similar across the three scenarios. In scenarios S2 and S3, the hidden layer size is slightly larger than that in scenario S1.

[0125] Example 4

[0126] The step 3 includes: setting three indicators for evaluating the performance of the multilayer perceptron MLP model in predicting the penetration curve: NSE (Formula (3)), R 2 (Formula (4)) and root mean square error (RMSE):

[0127]

[0128] Support Vector Regression (SVR) and Random Forest (RF) were selected as benchmark machine learning models for comparison;

[0129] The predicted concentration data and experimental concentration data of all cases under the three scenarios are as follows: Figures 6 and 7 shown. Figure 8 It shows that in all three scenarios, NSE, R 2 And the change of RMSE with the increase of flow rate.

[0130] (1) The prediction accuracy in scenario S1 is poor, especially for cases b1-b2, b9, c1-c2, and c9. 2 The values are all lower than 0.7( Figure 8 The predicted peak concentration is significantly lower than the experimental value ( Figures 6 and 7 ).like Figure 8 As shown, for almost all cases, the NSE and R 2 The values are larger than those in scenario S1, indicating that the prediction accuracy has been significantly improved. This improvement is particularly evident in the two lowest traffic flows (b1-b2 and c1-c2). The prediction performance of scenarios S2 and S3 is good, and the NSE and R of b2-b8 and c2-c8 are 2 The value is close to or greater than 0.9. In addition, in scenario S2, the NSE and R 2The values are close to or greater than 0.95. In scenario S3, the values of b5-b8 and c3-c5 are also close to or greater than 0.95. Compared with scenario S1, the predicted peak concentrations of the penetration curves in scenarios S2 and S3 are closer to the measured values ( Figures 6 and 7 ).

[0131] (2) Compared with scenario S2, the prediction accuracy of almost all class b cases in scenario S3 has been slightly improved, except for b6. For example, the NSE value of b5 has increased from 0.96 in scenario S2 to 0.98 in scenario S3. This is because the number of new data points (a1-a9) in scenario S3 is smaller than the number of new data points (c1-c9) in scenario S2. In contrast, the prediction accuracy of class c cases in scenario S3 has been slightly reduced or improved compared with scenario S2. This may be due to the difference between the single pipe and the asymmetric tank-pipe in scenario S3, and the input features may not be able to accurately capture this difference, resulting in a certain error.

[0132] (3) Figure 9 Table 6 shows the prediction results of each model. Among the three models, the multilayer perceptron (MLP) has the best prediction performance. Compared with the benchmark machine learning models (random forest RF, support vector regression SVR), for cases b5 and c5 in scenario 2, MLP has lower RMSE and R 2 Or NSE is higher. Although RF and SVR show high prediction accuracy on the optimization set, their performance on the prediction set is relatively poor.

[0133] Table 6 Prediction performance of MLP, RF, and SVR for penetration curves of cases b5 and c5 in scenario S2

[0134]

[0135] Example 5

[0136] The fourth step includes:

[0137] The predicted breakthrough curve was simulated using a transient storage model (TSM) to calibrate solute transport parameters. The transient storage model generalizes solute transport as occurring within the main pipeline, with secondary exchange processes occurring in the storage area created by the dead-end channel and the water tank (solution pool). This exchange process allows the solute to slowly flow out of the storage area and gradually re-enter the main pipeline, resulting in the tailing of the breakthrough curve. In this invention, only conservative tracers were used, and reactive transport processes such as attenuation and delay were negligible. The model equation is as follows:

[0138]

[0139]

[0140] Where t is time [T], x is distance [L], C and C s are the concentrations of solute in the main channel and storage area [M / L 3 ];A and A s are the cross-sectional areas of the main channel and storage area [L 2 ]; Q is the flow rate [L 3 / T]; D is the diffusion coefficient [L 2 / T], α is the exchange coefficient [T -1 ].

[0141] Furthermore, the comparison between the predicted model parameters and the actual values in the three scenarios can be seen Figure 10 and Figure 11 The transient storage model in OTIS failed to successfully simulate the predicted breakthrough curves for scenarios c1, c2, and c9 in S1. Therefore, the model parameters for these cases were not analyzed.

[0142] Subsequently, the predicted model parameters were compared with the actual values, and the prediction error was calculated according to the following formula:

[0143]

[0144] Among them, P pre is the predicted value of each model parameter, P act The prediction errors of the model parameters are listed in Tables 7 and 8.

[0145] Figure 10 The predicted and actual values of the model parameters for cases b1-b9 in scenarios S1, S2, and S3 are shown, while Table 7 lists the prediction errors of the model parameters. Figure 10 As shown in Table 7, the predicted values of the model parameters in scenarios S2 and S3 are closer to the actual values than those in scenario S1, indicating a significant improvement in prediction performance. Table 7 further shows that the model parameter prediction errors in scenarios S3 or S2 are the lowest in all cases.

[0146] Table 7. Prediction errors of the parameters of the temporary storage model in the symmetric tank-pipe; bold text indicates the minimum error for each case in the three scenarios, and brackets indicate that the predicted value is lower than the actual value.

[0147]

[0148]

[0149] according to Figure 10In most cases, the predicted values of the diffusion coefficient D were significantly higher than the actual values. In scenario S2, the errors ranged from 33.5% to 325.3%, and in scenario S3, the errors ranged from 4.7% to 264.3% (Table 7). These results indicate that MLP overestimates the D values. In addition, in all cases, the predicted values of the main channel cross-sectional area A were very close to the actual values ( Figure 10 ), which means that the MLP is able to accurately predict the value of A. The error range of scenario S1 is between 0.5% and 7.5%, the error range of scenario S2 is between 0.1% and 5.0%, and the error range of scenario S3 is between 0.2% and 3%. In addition, the cross-sectional area A of the storage area in scenarios S3 and S2 is s The predicted value is close to the actual value ( Figure 10 In scenario S3, the errors range from 0.5% to 20.7%, while in most cases of scenario S2, the prediction errors are below 20% (Table 7), which indicates that MLP predicts A in scenarios S3 and S2. s In addition, for the exchange coefficient α, except for cases b1 and b9, the predicted values in scenarios S2 and S3 are very close to the actual values, with prediction errors below 30% (Table 7), which shows that the prediction error of α is small under intermediate flow conditions.

[0150] Figure 11 The predicted and actual values of the model parameters for cases c1-c9 in scenarios S1, S2, and S3 are shown, while Table 8 lists the prediction errors of the model parameters. Figure 11 As shown in Table 8, the predicted values of the model parameters in scenarios S2 and S3 are closer to the actual values than those in scenario S1, indicating a significant improvement in prediction performance. Table 8 further shows that for the model parameters, scenario S2 or scenario S3 has the smallest prediction error.

[0151] Table 8. Prediction errors of the model parameters for temporary storage in an asymmetric tank-pipe; bold text indicates the minimum error for each case in the three scenarios, and brackets indicate that the predicted value is lower than the actual value.

[0152]

[0153] like Figure 11 As shown in Table 8, in most cases, the predicted values of the diffusion coefficient D in scenarios S2 and S3 are much higher than the actual values, with only a few cases where the predicted values are only slightly higher. In scenario S2, the errors range from 25.2% to 112.6%, and in scenario S3, the errors range from 8.4% to 276.7%, indicating that the MLP overestimates the D value. In all three prediction scenarios, the predicted values of the main channel cross-sectional area A are very close to the actual values ( Figure 11), the error range of scenario S1 is 0.2% to 3.5%, the error range of scenario S2 is 0.0% to 6.6%, and the error range of scenario S3 is 0.3% to 2.0% (Table 8), indicating that MLP can accurately predict the A value under different flow rates. Overall, in scenarios S2 and S3, the cross-sectional area A of the storage area is s The predicted value is close to the actual value ( Figure 11 In scenario S2, except for c2 and c9, the prediction errors of other cases are all below 20%; in scenario S3, the prediction errors of c1 and c2 are 32.2% and 39.1% respectively, while the prediction errors of the remaining cases are all below 20% (Table 8); this shows that MLP is effective in predicting A in scenarios S3 and S2. s The prediction errors for the values are mostly small. For most cases, the prediction error for the exchange coefficient α is below 50% in scenarios S2 and S3, with prediction errors for c5–c8 below 20% in scenario S2 and c3, c4, c6, and c7 also below 20% in scenario S3 (Table 8).

[0154] Example 6

[0155] The step five includes:

[0156] This approach, called prediction scenario S4, uses the measured penetration curve and model parameters at a specific flow rate as the predicted penetration curve and parameters for other flow rates in the same pipeline. For example, the measured penetration curve and model parameters for cases b1 are used as the predicted penetration curve and parameters for cases b2-b9. In scenario S4, eight errors are calculated for each case (Equation 12). Figures 12 to 13 The prediction errors of scenarios S2, S3, and S4 are presented and compared, which helps to intuitively evaluate the model performance in each scenario.

[0157] Furthermore, in all cases, the prediction error of the main channel cross-sectional area A in scenarios S2 and S3 is lower than that in scenario S4. In scenarios S2, S3, and S4, the predicted value of A is almost the same as the actual value. Regarding the exchange coefficient α, in almost all cases, the prediction error of scenarios S2 and S3 is within the prediction error of scenario S4. In almost all cases, the storage area cross-sectional area A in scenarios S2 and S3 is s The prediction error is also within the error range of scenario S4. In addition, the predicted penetration curves in scenarios S2 and S3 are basically consistent with the measured penetration curves, and the prediction error is small; while directly using the penetration curves of other flow rates as the predicted penetration curves deviates significantly from the measured penetration curves ( Figure 2 ), the prediction error of the breakthrough curve is large. These results further highlight the ability of the multi-layer perceptron (MLP) model algorithm in accurately predicting the breakthrough curve and solute transport parameters of karst pipelines under different flow conditions.

Claims

1. A method for predicting karst pipeline penetration curves and solute transport parameters, characterized by: The steps include: Step 1: Use the penetration curve data of two pipelines under nine different flow conditions to build a multi-layer perceptron (MLP) model; Step 2: Optimize the hyperparameters of the multilayer perceptron (MLP) model. Step 3: After training the multi-layer perceptron (MLP) model with optimized hyperparameters, it is used to predict penetration curves for different flow rates, and the prediction performance is evaluated and compared with the baseline machine learning model. Step 4: Use the temporary storage model to simulate the predicted breakthrough curve to obtain the model parameters, which are the predicted solute transport parameters; Step 5: Compare the prediction errors of the breakthrough curve and solute transport parameters with the errors of directly using the breakthrough curve and solute transport parameters of other flow rates.

2. The method for predicting karst pipeline penetration curve and solute transport parameters according to claim 1, characterized in that: In the step 1: (1) The multi-layer perceptron MLP model has a dense layer, and the output calculation formula of the dense layer is as follows: h W,b (X) = f(XW + b) Equation (1); In formula (1), X is the input vector, h W,b is the output value, W is the weight matrix, which contains all connection weights except the bias neuron connection weight, b is the bias vector, which contains all connection weights between the bias neuron and the artificial neuron, and f is the activation function of the multilayer perceptron neuron, with f as the rectified linear unit function, that is, ReLU(z)=max(0,z); (2) The architecture of the multilayer perceptron (MLP) model has two hidden layers, uses a rectified linear unit activation function for nonlinear feature learning, and finally an output layer with one neuron for predicting concentration; the number of neurons in each hidden layer, i.e., the hidden layer size, is the same to ensure consistent processing power across the network; the multilayer perceptron is trained using the RMSprop optimizer, with rho set to 0.9, a learning rate set to 0.001, and a mean square error (MSE) as the loss function, i.e., formula (2); the multilayer perceptron model is set to train for a maximum of 1000 training cycles, and an early stopping mechanism is added, with a patience value of 50 steps to prevent overfitting; In formula (2), o represents the observed value, p represents the predicted value, and n represents the number of concentration data in the predicted breakthrough curve; (3) The penetration curve data comes from the literature. By generalizing the field karst pipeline into an indoor pipeline system, using polyvinyl chloride (PVC) hoses to represent the karst pipeline, and using cubic water tanks to represent the karst pool structure developed in the pipeline, indoor tracer experiments were carried out under three pipeline structures and nine water flow conditions based on the indoor pipeline system to obtain the above penetration curve. The experimental pipeline system includes a constant head water supply tank, a PVC hose connected to the water supply tank, and a cubic symmetrical or asymmetrical water tank installed on the PVC hose. The three pipeline structures include: a single pipeline without a water tank, a symmetrical water tank-pipe with a symmetrical water tank installed, and an asymmetric water tank-pipe with an asymmetric water tank installed. Among them, the cubic symmetrical water tank has relative inlets / outlets in the middle of the front and back walls of the water tank, while the inlets / outlets on the front and back walls of the cubic asymmetric water tank are located at the lower left corner of the front wall and the lower right corner of the back wall respectively. The side length of the two water tanks is 0.1 m. For the asymmetric water tank, the deviation l of the inlet and outlet of the water tank perpendicular to the main flow direction of the pipeline is 0.071 m, while for the symmetric water tank, l is 0. (4) In each tracer experiment, 2.5 ml of 100 g / L sodium chloride solution was injected as the tracer. The tracer concentration was obtained by measuring the conductivity. The tracer concentration was measured at the outlet of the PVC hose and at a position 40 m away from the injection point. The flow rate was measured by a turbine flow meter with an accuracy of ±1.0%. To ensure the stability of the flow measurement, the flow meter was installed 6 m away from the outlet of the water supply tank. The tracer injection point was set 2 m away from the flow meter outlet to ensure the stability of the water flow in the pipe at this location. The length of the PVC hose from the tracer injection point to the outlet was 101.1 m: (5) Pipe structures are classified according to two key properties: the volume of the tank V and the deviation l between the tank inlet and outlet. For a single pipe, l and V are both 0. For symmetrical tank-pipes (b1-b9), the deviation l is 0 and the tank volume V is 0.001m 3 For the asymmetric tank-pipe (c1-c9), the deviation l is 0.071m and the tank volume V is 0.001m 3 ; (6) Three prediction scenarios S1, S2, and S3 are used. The multi-layer perceptron (MLP) model uses different data sets and input variables in different scenarios. The outlet tracer concentration is the model output variable. Scenario S1 uses the penetration curve data from a single pipeline structure and focuses on the outlet concentration C. i The time after injection t i and flow rate Q i Scenario S2 uses the penetration curve data of two pipeline structures and takes l as an additional input variable to characterize the difference between the two pipeline structures. Scenario S3 integrates the data of all three pipeline structures and takes both V and l as input variables to characterize the differences between the three pipeline structures: (7) Divide the data into a calibration set and a prediction set to fine-tune the parameters of the multilayer perceptron (MLP) model and predict the penetration curve; in the calibration phase, the calibration set is further divided so that 80% of the data is used as a training set and 10% of the data is used as a validation set so that the training process can be terminated when necessary (early stopping mechanism), and the remaining 10% of the data is used as a test set during the hyperparameter optimization process; Before training, all input data are normalized according to their minimum and maximum values in the training set so that their values are between 0 and 1.

3. The method for predicting karst pipeline penetration curve and solute transport parameters according to claim 2, characterized in that: The second step includes the Bayesian optimization method, specifically: Step 1. Consider an objective function f(x) on a bounded domain X, where x represents a set of hyperparameters. The hyperparameters range as follows: the number of neurons in each hidden layer varies from 1 to 64, and the batch size ranges from 16 to 256. The objective function is chosen to be the Nash-Sutcliffe efficiency (NSE) and the square of the Pearson correlation coefficient (R). 2 The sum of o and p, where o and p have the same meaning as in formula (2), and the objective function for hyperparameter search is calculated separately on the optimization set: Step 2. Define the observation values of N points as where y i =f(x i )+ε i , ε i The variance is Gaussian noise, in Bayesian optimization, is 0, and N is initially set to 25; Step 3. Fit a probabilistic surrogate model to the target observations: The probabilistic surrogate model is a Gaussian process that outputs a predictive distribution for the potential value of the target function at each input point based on the observations in step 2. Subsequently, the target value, i.e., the maximum value of f, is obtained. The predictive distribution is a Gaussian distribution with a mean μ(x * ) and variance σ 2 (x * ), that is, p(f(x * )|D,x * )=N(μ(x * ),σ 2 (x * )), where the mean and variance of the posterior distribution of the random variable are: in, K ij =k(x i ,x j ); k(x,x′) is the prior kernel function; Step 4. The next point to be selected is determined by the acquisition function. The prediction distribution in step 3 summarizes the uncertainty about the target function and is used to calculate the acquisition function α(·) on the domain X. The next point x is determined by selecting the independent variable that maximizes the acquisition function α(·). N+1 ; The acquisition function is selected as the expected improvement EI; once a new observation is obtained, the operations from step 2 to step 4 are repeated iteratively; the expected improvement EI acquisition function is given by the following expression: a(x * )=(μ(x * )-υ)Φ(Z)+σ(x * )φ(Z) Equation (7); Z=(μ(x * )-υ) / σ(x * ) expression(8); in, is the best observed value obtained so far, Φ(·) and φ(·) represent the cumulative distribution function CDF and probability density function PDF of the standard Gaussian distribution, respectively.

4. A method for predicting karst pipeline penetration curves and solute transport parameters according to claim 3, characterized in that: The step three includes: setting three indicators for evaluating the performance of the multi-layer perceptron MLP model in predicting the penetration curve: NSE, R 2 And the root mean square error RMSE: Support vector regression and random forest were chosen as benchmark machine learning models for comparison.

5. The method for predicting karst pipeline penetration curve and solute transport parameters according to claim 4, characterized in that: The fourth step includes: The predicted breakthrough curve was simulated using a temporary storage model to calibrate the solute transport parameters. The temporary storage model generalizes solute transport as occurring within the main pipeline. Simultaneously, a secondary exchange process occurs in the storage area caused by the dead-end channel and the water tank. This exchange process allows the solute to slowly flow out of the storage area and gradually re-enter the main pipeline, resulting in the tailing of the breakthrough curve. The model equation is as follows: Where t is time [T], x is distance [L], C and C s are the concentrations of solutes in the main channel and storage area [M / L 3 ];A and A s are the cross-sectional areas of the main channel and storage area [L 2 ]; Q is the flow rate [L 3 / T]; D is the diffusion coefficient [L 2 / T], α is the exchange coefficient [T -1 ]. The predicted model parameters were compared with the actual values, and the prediction error was calculated according to the following formula: Among them, P pre is the predicted value of each model parameter, P act is the actual value.

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