A method for predicting the permeability coefficient of viscous coarse-grained soil based on a physically constrained neural network

Abstract

The application discloses a viscous coarse-grained soil permeability coefficient prediction method based on a physically constrained neural network, and comprises the following steps: carrying out indoor viscous coarse-grained soil seepage test, establishing a viscous coarse-grained soil permeability coefficient formula considering porosity and particle size distribution characteristics, and then constructing a hybrid model containing a physical driving term and a neural network data driving term; on the basis of the seepage test, porosity, particle size distribution characteristics and corresponding permeability coefficient data of the viscous coarse-grained soil sample are supplemented by numerical simulation technology and literature investigation to form a complete data set, and are divided into a training set and a test set; model training is carried out by combining Bayesian optimization to dynamically search for optimal hyperparameters, so that a loss function curve and the permeability coefficient of the viscous coarse-grained soil under different porosities and particle size distribution characteristics are obtained; and test shows that the application realizes high-precision prediction of the viscous coarse-grained soil permeability coefficient, and solves the problems of insufficient physical constraint and low prediction precision of traditional methods.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of environmental geotechnical engineering and artificial intelligence, specifically involving a method for predicting the permeability coefficient of coarse-grained cohesive soil based on a physically constrained neural network. Background Technology

[0002] Cohesive coarse-grained soil, as a core material for dam rockfill and roadbed filling, directly affects the seepage stability of engineering projects due to its permeability coefficient. Traditional research relies on laboratory permeability tests, obtaining data through steady-state flow experiments. However, cohesive coarse-grained soil suffers from limitations such as difficult sample preparation, lengthy testing cycles, high costs, and difficulty in covering complex particle size distributions and pore structures. Existing permeability coefficient prediction methods mostly rely on empirical formulas to establish mathematical models that fit experimental data. While computationally efficient, these methods do not consider the influence of fine particles in cohesive coarse-grained soil on permeability properties, resulting in significant prediction errors. Furthermore, the formula parameters require manual calibration and lack adaptive optimization capabilities. Numerical simulation methods based on simulation technology can characterize particle-pore interactions, but simulations are time-consuming, and modeling errors can easily lead to prediction biases. With the development of data-driven technologies, machine learning methods (such as BP neural networks and SVM) have been used for permeability coefficient prediction. However, these methods are trained using simple mean square error (MSE) and lack constraints on the physical mechanisms of permeability in cohesive coarse-grained soil. This results in poor generalization ability of the models in scenarios with abrupt changes in particle size distribution and complex pore structures, and the inability to explain the physical rationality of the prediction results. Therefore, there is an urgent need to propose a method for predicting the permeability coefficient of coarse-grained cohesive soil based on a physically constrained neural network. Summary of the Invention

[0003] To address the problems of existing technologies, this invention proposes a method for predicting the permeability coefficient of coarse-grained cohesive soils based on physically constrained neural networks. Considering the influence of fine particles on the permeability properties of coarse-grained cohesive soils, a permeability coefficient formula suitable for coarse-grained cohesive soils is established based on porosity and particle size distribution characteristics. Numerical simulation technology is introduced to expand the dataset and reduce experimental costs. The design of multiple loss terms simultaneously constrains model training from dimensions such as accuracy, physical consistency, generalization, and uncertainty, effectively improving the model's robustness and interpretability. Embedding it into a neural network effectively solves the problem of inconsistent accuracy, computational efficiency, and generalization ability in existing methods for predicting the permeability coefficient of coarse-grained cohesive soils. This method can quickly predict the permeability coefficient of coarse-grained cohesive soils under different porosities and particle size distribution characteristics, providing a data-efficient and physically interpretable technical solution for predicting the permeability coefficient of coarse-grained cohesive soils, significantly reducing experimental costs and improving engineering prediction accuracy.

[0004] Technical solution

[0005] A method for predicting the permeability coefficient of coarse-grained cohesive soil based on a physically constrained neural network includes the following steps:

[0006] Step 1: Conduct indoor seepage tests on cohesive coarse-grained soil and establish a formula for the permeability coefficient of cohesive coarse-grained soil that takes into account porosity and particle size distribution characteristics.

[0007] Step 2: Construct a hybrid model containing both physics-driven terms and neural network data-driven terms;

[0008] Step 3: Based on the seepage test in Step 1, supplement the collection of porosity, particle size distribution characteristics and corresponding permeability coefficient data of cohesive coarse-grained soil samples through numerical simulation technology and literature review to form a complete dataset, which is then divided into training set and test set.

[0009] Step four: Combine Bayesian optimization to dynamically search for the optimal hyperparameters and train the hybrid model from step two;

[0010] Step 5: Output the loss function curves for the training and test sets based on the training results to obtain the predicted permeability coefficients under different porosities and gradation characteristics. k pred .

[0011] Step 6: Conduct indoor tests based on the actual engineering situation, compare the predicted values ​​with the measured values, verify the model's generalization ability, and ensure its engineering application capability.

[0012] Specifically, in step one,

[0013] The indoor cohesive coarse-grained soil seepage test includes the following steps: setting up the test apparatus, preparing the test materials, and designing the test and calculating the permeability coefficient.

[0014] The test apparatus includes a sample loading system, a pressurized water supply system, a dynamic load control system, and a data acquisition system.

[0015] The sample loading system is custom-made from 5mm thick transparent plexiglass and includes: an inlet, a buffer zone, a permeable base plate, a soil chamber, and a drain.

[0016] The pressurized water supply system includes an air compressor, air pressure pipes, and a water-air exchange tank. To ensure water pressure stability, a dual pressure regulating valve is used to control the gas pressure inside the water-air exchange tank.

[0017] The dynamic load control system uses a small eccentric vibration motor to simulate vibration load.

[0018] The data acquisition system consists of a pore water pressure gauge, a soil-water separation device, and a flow measurement device.

[0019] The test material was coarse-grained clay, and the soil sample was dried, crushed, and sieved.

[0020] The experimental design and permeability coefficient calculation involved designing seepage tests on cohesive coarse-grained soils with different porosities and particle size distributions. The permeability coefficient was calculated based on water flow rate, collection time, cross-sectional area of ​​the test soil chamber, length of each soil layer, and the head difference between the soil layers. The permeability coefficient calculation formula is as follows:

[0021] No. a Permeability coefficient of stratified soil in the test k a for;

[0022]

[0023] in, Q Water flow rate; L a For the first a The length of each test soil layer; A The cross-sectional area of ​​the test soil silo; h a For the first a The water head difference between the soil layers in the test; t For time; n This represents the total number of soil layers.

[0024] The particle size distribution characteristics include: average particle size of coarse clay, average particle size of fine clay, and uniformity coefficient.

[0025] The permeability coefficient formula for coarse-grained cohesive soil, considering porosity and particle size distribution characteristics, is established based on the results of indoor seepage tests on coarse-grained cohesive soil. Specifically, after obtaining the permeability coefficient under different porosities and particle size distribution characteristics using equation (2), the empirical formula method is used to fit and obtain the permeability coefficient formula for coarse-grained cohesive soil. Considering that coarse-grained cohesive soil has a certain fine particle content, reflected by porosity and uniformity coefficient, the expression is shown in equation (3):

[0026]

[0027] In the formula, A The coefficients are linear. n e Effective porosity; d 50 The average particle size of coarse-grained clay. C u The coefficient of non-uniformity is denoted as . i , α , β , m It is a dimensionless parameter.

[0028] The effective porosity n eTo account for the ineffective porosity caused by the bound water film of fine particles in coarse-grained clays, the double-layer parameter method shows that it is mainly affected by the average particle size of the fine particles:

[0029]

[0030] In the formula, n Input a variable for porosity; C s The average particle size of the fine particles; or , c It is a dimensionless parameter.

[0031] Specifically, in step two,

[0032] The hybrid model includes physical driving terms and data driving terms, and the formula for predicting the permeability coefficient is shown in equation (5).

[0033]

[0034] Among them, physics-driven items k h The permeability coefficient of coarse-grained soil considering porosity and particle size distribution characteristics is calculated using the formula (Equation (3)); the data-driven term NN( n , d 50 , C s , C u , i () is a neural network; i These are the parameters of the neural network.

[0035] By implementing the aforementioned residual correction mechanism, collaborative optimization is achieved, addressing the issues of low prediction accuracy and poor applicability to complex gradation characteristics in traditional pure physical models, as well as the lack of interpretability and weak generalization ability in pure data-driven models.

[0036] Specifically, in step three, the numerical simulation technology includes the following processes: model establishment and supplementary working condition design.

[0037] The model is built based on PFC 5.0. 2D An indoor simulation model for seepage testing of coarse-grained cohesive soil was established. The microscopic parameters of the material were calibrated through biaxial tests. The calibrated parameters were then used for seepage test simulation. The permeability coefficient obtained from the seepage test was compared with the results obtained from the indoor test to ensure the reliability of the parameters.

[0038] The method for establishing the simulation model for the seepage test of the indoor cohesive coarse-grained soil is as follows:

[0039] In PFC 5.0 2DThe software constructs the model boundary using wall elements based on the size of the soil chamber sample. It generates soil particles representing each particle size group based on the porosity and gradation of cohesive coarse-grained soil and fills them in layers. It generates a seepage grid using Gmsh2D, traverses the seepage grid cells, calculates the effective porosity in the model, and then solves the permeability coefficient according to the formula for the permeability coefficient of cohesive coarse-grained soil that considers the porosity and particle gradation characteristics. Finally, it applies seepage force to obtain the indoor seepage test simulation model of cohesive coarse-grained soil.

[0040] The dimensions of the model soil chamber are consistent with those of the indoor test, which are 1.25m in length and 0.5m in width.

[0041] The supplementary working condition design involves designing a series of coarse-grained soil permeability test conditions with different porosities and particle size distribution characteristics based on the obtained microscopic parameters, conducting numerical simulations, and obtaining the corresponding permeability coefficients.

[0042] The literature review involved consulting previous literature and compiling permeability test data for coarse-grained clays, including porosity and particle size distribution characteristics.

[0043] The complete dataset was obtained through three methods: indoor experiments, numerical simulation, and literature review. It includes data on porosity, average particle size of coarse-grained clay, average particle size of fine-grained clay, uniformity coefficient, and corresponding permeability coefficient, which are then combined to form the complete dataset.

[0044] Specifically, in step four, the processing includes: forward propagation, calculation of the composite loss function, gradient calculation and joint optimization of parameters, and dynamic adjustment of hyperparameters through Bayesian optimization.

[0045] Specifically as follows:

[0046] Forward propagation: Define the input layer, hidden layer, and output layer; initialize the parameters of the permeability coefficient formula for coarse-grained cohesive soil. A , i , α , β , m、h , c Initialize neural network parameters i Input training data and calculate k h Formula terms and neural network terms.

[0047] Composite loss function calculation: Calculate the mean squared error term, physical constraint penalty term, parameter regularization term, uncertainty quantification term, and dynamic equilibrium term, and sum them to obtain the total loss, as shown in equation (6):

[0048]

[0049] In the formula, l 1 represents the weighting coefficient of the physical constraint penalty term; l 2 represents the weighting coefficient of the parameter regularization term; l 3 represents the weighting coefficient of the uncertainty quantification term; l 4 represents the weighting coefficient of the dynamic balance term; This is the mean square error term; This is a physical constraint penalty term; For parameter regularization terms; This is a quantification term for uncertainty. This is a dynamic equilibrium term.

[0050] Specifically, the mean squared error term minimizes the mean squared error between the predicted and actual values, as expressed in equation (7):

[0051]

[0052] In the formula, For the first i The true permeability coefficient of coarse-grained soil for each sample; For the first i The predicted permeability coefficient of each sample is expressed as shown in equation (8); N This represents the total number of training samples used in the calculation.

[0053]

[0054] In the formula, For the first i Each sample physical driving item k h The calculated value is shown in equation (9); For the first i The neural network calculates the value for each sample.

[0055]

[0056] in, The expression is shown in equation (10):

[0057]

[0058] The physical constraint penalty term is used to ensure that the parameters conform to physical meaning, and its expression is shown in equation (11):

[0059]

[0060] In the formula, q max , q min for A The upper and lower limits of the preset reasonable value range.

[0061] The parameter regularization constraint constrains the parameter amplitude to prevent overfitting, and the expression is shown in equation (12):

[0062]

[0063] The uncertainty quantification term assumes that the neural network output follows a Gaussian distribution. The mean and variance are optimized by maximum likelihood estimation, and the standard deviation is ensured to be non-negative. The expression is shown in equation (13).

[0064]

[0065] In the formula, m The mean permeability coefficient predicted by the model is equal to k h The sum of the calculated value and the neural network calculated value; s This represents the standard deviation of the predicted results.

[0066] The dynamic equilibrium term is calculated. k h The correlation between the calculated value and the neural network calculated value is used to prevent the physical terms and neural network terms from canceling each other out. The expression is shown in equation (14):

[0067]

[0068] Gradient calculation and joint parameter optimization: including k h Parameter gradient and neural network parameter gradient.

[0069] The k h The parameter gradient is calculated using the chain rule to determine the gradient of the composite loss with respect to the physical parameters, and the parameters are updated using gradient descent. A , i , α , β , m、h , c The gradient is calculated using formulas (15) to (21).

[0070]

[0071] The gradient of the neural network parameters is calculated using automatic differentiation (AutoDiff) to evaluate the loss on the neural network parameters. i The gradient (see Equation (22)) is obtained, and the parameters are updated using gradient descent.

[0072]

[0073] Bayesian optimization for dynamic hyperparameter adjustment: The gradient descent method dynamically adjusts the learning rate through Bayesian optimization to ensure model convergence. Specifically: A Bayesian optimization strategy is employed, assuming the validation set loss follows a Gaussian process. A surrogate model is constructed, and the validation set is used to evaluate the current hyperparameter performance. The expected improvement criterion is used to select the hyperparameter combination that maximizes the expected improvement, thereby efficiently coordinating the loss weights. l 1. l 2. l 3. l 4) and learning rate.

[0074] The loss weights take values ​​in the range [0, 1]. The learning rate takes values ​​in the range [10, 1]. -5 10 -3 ].

[0075] Beneficial effects

[0076] (1) This invention uses numerical simulation technology to establish a seepage test model for cohesive coarse-grained soil, which can efficiently obtain the permeability coefficient of cohesive coarse-grained soil under different porosity and gradation levels, thus overcoming the disadvantages of high cost and long time consumption in obtaining samples from indoor tests.

[0077] (2) A formula for the permeability coefficient of cohesive coarse-grained soil that takes into account both porosity and particle size distribution characteristics was established. This formula can reflect the influence of fine particle content and improve the reliability and accuracy of the permeability coefficient prediction of cohesive coarse-grained soil. Furthermore, by constructing a hybrid physical-data driven model, the seepage prediction formula is combined with the neural network residual correction, which not only preserves the interpretability of the physical model but also uses the data driven model to capture complex nonlinear relationships.

[0078] (3) Based on the traditional mean square error, a physical constraint penalty term (forcing the parameters of the seepage prediction formula to conform to the physical meaning), a parameter regularization term (suppressing overfitting), an uncertainty quantification term (characterizing the prediction confidence interval) and a dynamic balance term (coordinating the physical term and the data term) are introduced to enhance the physical consistency and generalization ability of the model and avoid non-physical output.

[0079] (4) Improve model training efficiency by dynamically searching for the optimal hyperparameters through Bayesian optimization, and ensure the optimal balance between physical constraints and data fitting. Attached Figure Description

[0080] Figure 1 A flowchart of a method for predicting the permeability coefficient of coarse-grained cohesive soil based on a Bayesian optimization-physical constraint neural network is provided for an embodiment of the present invention.

[0081] Figure 2 A schematic diagram of an indoor seepage test of coarse-grained cohesive soil provided in an embodiment of the present invention;

[0082] Figure 3The influence of porosity and particle size distribution characteristics on permeability coefficient is provided in the embodiments of the present invention; wherein, (a) is the relationship between permeability coefficient and effective porosity, (b) is the relationship between permeability coefficient and non-uniformity coefficient, and (c) is the relationship between permeability coefficient and average particle size of coarse-grained soil.

[0083] Figure 4 A schematic diagram of a numerical model for an indoor cohesive coarse-grained soil seepage test provided in an embodiment of the present invention;

[0084] Figure 5 This is a design diagram of the permeability coefficient prediction algorithm framework for coarse-grained cohesive soil based on a physically constrained neural network, provided in an embodiment of the present invention.

[0085] Figure 6 Loss function curves for the training and test sets provided in this embodiment of the invention;

[0086] Figure 7 A comparison chart of predicted and actual permeability coefficients under different porosities and gradation characteristics provided in this embodiment of the invention;

[0087] Figure 8 A comparison chart of prediction results for verifying the generalization ability of the model provided in this embodiment of the invention. Detailed Implementation

[0088] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0089] This embodiment provides a method for predicting the permeability coefficient of coarse-grained cohesive soil based on a physically constrained neural network, such as... Figure 1 As shown, it includes the following steps:

[0090] Step 1: Conduct indoor seepage tests on cohesive coarse-grained soil and establish a formula for the permeability coefficient of cohesive coarse-grained soil that takes into account porosity and particle size distribution characteristics.

[0091] The indoor cohesive coarse-grained soil seepage test includes the following steps: setting up the test apparatus, preparing the test materials, and designing the test and calculating the permeability coefficient.

[0092] An indoor seepage test setup for coarse-grained cohesive soil was constructed, including: a sample loading system, a pressurized water supply system, a dynamic load control system, and a data acquisition system. For example... Figure 2 As shown.

[0093] The sample loading system is custom-made from 5mm thick transparent plexiglass, and its components include: a water inlet, a buffer zone, a permeable base plate, a soil chamber, and a drainage outlet.

[0094] The pressurized water supply system consists of an air compressor, air pressure pipes, and a water-air exchange tank. To ensure the stability of the water pressure, a dual pressure regulating valve is used to control the gas pressure inside the water-air exchange tank.

[0095] The dynamic load control system uses a small eccentric vibration motor to simulate vibration load.

[0096] The data acquisition system consists of a pore water pressure gauge, a soil-water separation device, and a flow measurement device.

[0097] After the soil sample loading system completes the soil sample loading, the pressurized water supply system injects stable pressurized water into the buffer zone through the inlet. The water seeps into the soil sample through the permeable bottom plate. The dynamic load control system can simulate dynamic load action. After the seepage water flows out of the drain outlet, it passes through the soil-water separation device and the flow measurement device in sequence. At the same time, the pore water pressure gauge monitors the pore water pressure in real time, realizing the collection and recording of seepage data of cohesive coarse-grained soil.

[0098] In this embodiment, the test material was moderately to strongly weathered siliceous mudstone, and the soil sample was dried, crushed, and sieved.

[0099] It should be noted that, in this embodiment, the experimental design and permeability coefficient calculation are as follows: 15 groups of seepage tests were designed for coarse-grained cohesive soils with different porosities and particle size distribution characteristics. The permeability coefficient was calculated based on water flow rate, water collection time, cross-sectional area of ​​the test soil silo, length of the test soil layers, and the head difference between the test soil layers. The permeability coefficient calculation formula is as follows:

[0100]

[0101] In the formula, Q Water flow rate; L a For the first a The length of each test soil layer; A The cross-sectional area of ​​the test soil silo; h a For the first a The water head difference between the soil layers in the test; t For time; n This represents the total number of soil layers. k a For the first a Permeability coefficient of the soil in each test layer; k The test soil permeability coefficient.

[0102] In this embodiment, the total number of soil layers n It consists of 5 layers, with the length of each layer being...L a The cross-sectional area of ​​the test soil silo is 25cm. A It is 0.196m 2 ,time t It takes 20 minutes.

[0103] It should be noted that in this embodiment, the particle size distribution characteristics are the average particle size of coarse clay, the average particle size of fine clay, and the coefficient of non-uniformity.

[0104] The permeability coefficient is closely related to porosity and particle size distribution characteristics. For coarse-grained clays, the influence of various parameters on the permeability coefficient should be considered even more. Specifically, in this embodiment, based on equation (2), the influence of porosity and particle size distribution characteristics on the permeability coefficient is obtained experimentally as follows: Figure 3 As shown:

[0105] from Figure 3 (a) It can be seen that the permeability coefficient and the effective porosity are positively correlated. Form can effectively represent i , m It is a dimensionless parameter.

[0106] from Figure 3 (b) It can be seen that the permeability coefficient and the non-uniformity coefficient have an exponential relationship. The larger the non-uniformity coefficient, the more significant the difference in particle size. Small particles fill the pores of large particles, reducing pore connectivity and effective channels, thereby weakening the permeability.

[0107] from Figure 3 (c) It can be seen that the larger the average particle size of cohesive coarse-grained soil, the wider the pore channels, the smaller the fluid flow resistance, and the higher the permeability coefficient, which is also a positive correlation.

[0108] According to the research, the empirical formula method can be used to combine the fitting relationship between various control parameters and permeability coefficient, and then establish a permeability coefficient formula for coarse-grained clay that takes into account porosity and particle size distribution characteristics.

[0109] In this embodiment, considering that coarse-grained clay has a certain fine-grain content, and reflecting this through porosity and uniformity coefficient, the formula expression for the permeability coefficient of coarse-grained clay considering porosity and particle size distribution characteristics is shown in equation (3):

[0110]

[0111] In the formula, A The coefficients are linear. n e Effective porosity; d 50 The average particle size of coarse-grained clay. C uThe coefficient of non-uniformity is denoted as . i , α , β , m It is a dimensionless parameter.

[0112] It should be noted that in this embodiment, the effective porosity is... n e To account for the ineffective porosity caused by the bound water film of fine particles in coarse-grained clays, the double-layer parameter method shows that it is mainly affected by the average particle size of the fine particles:

[0113]

[0114] In the formula, n Input a variable for porosity; C s The average particle size of the fine particles; or , c It is a dimensionless parameter.

[0115] Step 2: Construct a hybrid model containing physics-driven terms and neural network data-driven terms. This model includes physics-driven terms and data-driven terms, and its expression is shown in equation (5):

[0116]

[0117] In the formula, the physical driving term k h The formula for the permeability coefficient of coarse-grained clay considering porosity and particle size distribution characteristics is given by equation (3); the data-driven term NN( n , d 50 , C s , C u , i () is a neural network; i These are the parameters of the neural network. A collaborative optimization mechanism is achieved through residual correction, addressing the problems of low prediction accuracy and poor applicability to complex gradation features in traditional pure physical models, as well as the lack of interpretability and weak generalization ability in pure data-driven models.

[0118] Step 3: Based on the seepage test, supplement the data on porosity, particle size distribution characteristics and corresponding permeability coefficient of cohesive coarse-grained soil samples by numerical simulation technology and literature review to form a complete dataset, which is then divided into training set and test set.

[0119] It should be noted that, in this embodiment, the numerical simulation technology includes model building and supplementary working condition design. The model size and functions implemented in the numerical simulation are consistent with those in the indoor test in step one.

[0120] The model is built based on PFC 5.0.2D An indoor simulation model for seepage testing of coarse-grained cohesive soil was established. The microscopic parameters of the material were calibrated through biaxial tests. The calibrated parameters were then used for seepage test simulation. The permeability coefficient obtained from the seepage test was compared with the results obtained from the indoor test to ensure the reliability of the parameters.

[0121] The method for establishing a simulation model for seepage tests on coarse-grained cohesive soils in the laboratory is as follows:

[0122] In PFC 5.0 2D The software constructs the model boundary using wall elements based on the dimensions of the soil chamber sample. Based on the porosity and gradation of cohesive coarse-grained soil, representative soil particles for each particle size group are generated and layered. A seepage mesh is generated using Gmsh2D, and the effective porosity in the model is calculated by traversing the seepage mesh elements. Then, the permeability coefficient is solved using the formula for the permeability coefficient of cohesive coarse-grained soil considering porosity and particle gradation characteristics. Seepage force is applied to obtain a simulation model for the indoor seepage test of cohesive coarse-grained soil. Specifically, the obtained numerical model for the indoor seepage test of cohesive coarse-grained soil is as follows: Figure 4 As shown.

[0123] It should be noted that in this embodiment, the dimensions of the model soil chamber are consistent with those of the indoor test, which are 1.25m in length and 0.5m in width.

[0124] The microstructure parameters of the material are shown in Table 1.

[0125] Table 1. Microscopic parameters of cohesive coarse-grained soil materials

[0126]

[0127] The supplementary working condition design involves designing a series of coarse-grained soil permeability test conditions with different porosities and particle size distribution characteristics based on the obtained microscopic parameters, conducting numerical simulations, and obtaining the corresponding permeability coefficients.

[0128] The literature review involved consulting previous literature and compiling permeability test data for coarse-grained clays, including porosity and particle size distribution characteristics.

[0129] Ultimately, the complete dataset was obtained through three methods: indoor experiments, numerical simulation, and literature review. It includes data on porosity, average particle size of coarse-grained clay, average particle size of fine-grained clay, uniformity coefficient, and corresponding permeability coefficient, which were then combined to form the complete dataset.

[0130] In this embodiment, the complete dataset includes 15, 30, and 85 samples from indoor experiments, numerical simulation techniques, and literature surveys, respectively.

[0131] The dataset was divided into a training set and a test set in a 9:1 ratio, with 117 samples in the training set and 13 samples in the test set, and the data was normalized.

[0132] Step 4: Train the model by dynamically searching for the optimal hyperparameters using Bayesian optimization, including forward propagation, calculation of the composite loss function, gradient calculation and joint parameter optimization, and dynamic adjustment of hyperparameters using Bayesian optimization.

[0133] In this embodiment, the framework design for model training is as follows: Figure 5 As shown.

[0134] In this embodiment, forward propagation is used as the input training data, and the input layer, hidden layer, and output layer are set to initialize the parameters of the permeability coefficient formula for coarse-grained cohesive soil. A , i , α , β , m、h , c Initialize neural network parameters i ,calculate k h Formula terms and neural network terms.

[0135] It should be noted that in this embodiment, the input training data includes porosity, average particle size of coarse-grained clay, average particle size of fine-grained clay, non-uniformity coefficient, and corresponding permeability coefficient data.

[0136] It should be noted that in this embodiment, the input layer is set as a 4-dimensional vector. n , d 50 , C u , C s The hidden layer is set to 2 layers, with 64 nodes in each layer, and the output layer is set to a 1-dimensional vector. k pred ].

[0137] In this embodiment, the composite loss function is calculated by calculating the mean squared error term, physical constraint penalty term, parameter regularization term, uncertainty quantification term, and dynamic equilibrium term according to the composite loss function formula, and then summing them to obtain the total loss, as shown in equation (6):

[0138]

[0139] In the formula, l 1 represents the weighting coefficient of the physical constraint penalty term; l 2 represents the weighting coefficient of the parameter regularization term; l 3 represents the weighting coefficient of the uncertainty quantification term; l 4 represents the weighting coefficient of the dynamic balance term; This is the mean square error term; This is a physical constraint penalty term; For parameter regularization terms; This is a quantification term for uncertainty. This is a dynamic equilibrium term.

[0140] It should be noted that in this embodiment, the mean square error term can minimize the mean square error between the predicted value and the true value, as expressed in equation (7):

[0141]

[0142] In the formula, For the first i The true permeability coefficient of coarse-grained soil for each sample; For the first i The predicted permeability coefficient of each sample is expressed as shown in equation (8); N In this embodiment, the total number of training samples used in the calculation is [number]. N The value is 117.

[0143]

[0144] In the formula, For the first i Each sample physical driving item k h The calculated value is shown in equation (9); For the first i The neural network calculates the value for each sample.

[0145]

[0146] in, The expression is shown in equation (10):

[0147]

[0148] It should be noted that in this embodiment, the physical constraint penalty term ensures that the parameters conform to physical meaning, and the expression is shown in equation (11):

[0149]

[0150] In the formula, q max , q min for A The upper and lower limits of the preset reasonable value range.

[0151] It should be noted that in this embodiment, parameter regularization constrains the parameter amplitude to prevent overfitting, as shown in equation (12):

[0152]

[0153] It should be noted that in this embodiment, the uncertainty quantification term assumes that the neural network output follows a Gaussian distribution, optimizes the mean and variance through maximum likelihood estimation, and ensures that the standard deviation is non-negative. The expression is shown in equation (13):

[0154]

[0155] In the formula, m The mean permeability coefficient predicted by the model is equal to k h The sum of the calculated value and the neural network calculated value; s This represents the standard deviation of the predicted results.

[0156] It should be noted that, in this embodiment, the dynamic balance term is calculated... k h The correlation between the calculated value and the neural network calculated value is used to prevent the physical terms and neural network terms from canceling each other out. The expression is shown in equation (14):

[0157]

[0158] It should be noted that in this embodiment, gradient calculation and parameter joint optimization include... k h Parameter gradient and neural network parameter gradient.

[0159] It should be noted that, in this embodiment, k h The parameter gradient is calculated using the chain rule to determine the gradient of the composite loss with respect to the physical parameters, and the parameters are updated using gradient descent. A , i , α , β , m、h , c The gradient is calculated using formulas (15) to (21).

[0160]

[0161] It should be noted that, in this embodiment, the gradient of the neural network parameters is calculated using automatic differentiation (AutoDiff) to determine the loss on the neural network parameters. i The gradient (see Equation (22)) is obtained, and the parameters are updated using gradient descent.

[0162]

[0163] It should be noted that in this embodiment, the gradient descent method dynamically adjusts the learning rate through Bayesian optimization to ensure model convergence.

[0164] It should be noted that in this embodiment, the dynamic adjustment of Bayesian optimization hyperparameters is as follows: A Bayesian optimization strategy is adopted, assuming the validation set loss follows a Gaussian process. A surrogate model is constructed, the validation set is used to evaluate the current hyperparameter performance, and the combination of hyperparameters that maximizes the expected improvement is selected through the expected improvement criterion, thereby efficiently coordinating the loss weights. l 1. l 2. l 3. l 4) and learning rate.

[0165] It should be noted that, in this embodiment, the neural network structure parameters include the number of hidden layers and the number of nodes.

[0166] It should be noted that, in this embodiment, to ensure that the model output conforms to physical meaning and improve the interpretability of the model, non-negative number constraints are used to... i , α , β , m、h , c ≥0.

[0167] After Bayesian optimization in this embodiment: Loss Weight l 1. l 2. l 3. l The values ​​for 4 are 0.075, 0.234, 0.164, and 0.327; the learning rate is 6.64 × 10⁴. -4 .

[0168] Step 5: Output the loss function curves for the training and test sets based on the training results, and obtain the predicted permeability coefficient values ​​under different porosities and gradation characteristics. k pred .

[0169] It should be noted that in this embodiment, the loss function curves for the training set and the test set are output based on the training results, as shown below. Figure 6 As shown.

[0170] It should be noted that in this embodiment, the predicted permeability coefficients are obtained under different porosities and gradation characteristics. k pre and compared with the true value, for example Figure 7 As shown.

[0171] Step 6: Conduct indoor tests based on the actual engineering situation to verify the generalization ability of the model. Specifically, conduct more than 5 groups of tests to determine the permeability coefficient of coarse-grained soil under different particle size distributions and porosities, and compare the predicted values ​​with the measured values. The average relative error of the predicted permeability coefficient in the indoor tests should be ≤9%.

[0172] Specifically, 10 groups of coarse-grained soil specimens with different gradations and porosities were prepared, the permeability coefficient was measured, and model predictions were made and compared.

[0173] Traditional empirical formula prediction methods compared include the Terzaghi formula, Hazen formula, and Kozeny-Carman formula. The average error of the method in this invention is compared with that of traditional empirical formula prediction methods, and the results are as follows: Figure 8 As shown.

[0174] The average prediction errors of traditional empirical formula prediction methods are Terzaghi formula (10.5%), Hazen formula (13.07%), and Kozeny-Carman formula (11.15%), respectively. In contrast, the average error of the method in this embodiment is only 4.47%. It can be seen that the method in this embodiment has higher accuracy than other traditional empirical formula prediction methods and can have better generalization ability while ensuring prediction accuracy.

[0175] The innovative aspects of this invention are as follows:

[0176] (1) This invention uses numerical simulation technology to establish a seepage test model for cohesive coarse-grained soil, which can efficiently obtain the permeability coefficient of cohesive coarse-grained soil under different porosity and gradation levels, thus overcoming the disadvantages of high cost and long time consumption in obtaining samples from indoor tests.

[0177] (2) A formula for the permeability coefficient of cohesive coarse-grained soil that comprehensively considers porosity and particle size distribution characteristics was established, and the influence of fine particle content was reflected, which improved the reliability and accuracy of the permeability coefficient prediction of cohesive coarse-grained soil. By constructing a hybrid physical-data driven model, the formula for the prediction of the permeability coefficient of cohesive coarse-grained soil that considers porosity and particle size distribution characteristics was combined with the neural network residual correction, which not only preserved the interpretability of the physical model, but also used the data driven model to capture complex nonlinear relationships.

[0178] (3) Based on the traditional mean square error, a physical constraint penalty term (forcing the parameters of the permeation coefficient prediction formula to conform to the physical meaning), a parameter regularization term (suppressing overfitting), an uncertainty quantification term (characterizing the prediction confidence interval), and a dynamic balance term (coordinating the physical term and the data term) are introduced to enhance the physical consistency and generalization ability of the model and avoid non-physical output.

[0179] (4) Improve model training efficiency by dynamically searching for the optimal hyperparameters through Bayesian optimization, and ensure the optimal balance between physical constraints and data fitting.

[0180] Compared to physically constrained neural network techniques in other fields, this invention considers the influence of fine particles in coarse-grained cohesive soils on permeability. Based on porosity and particle size distribution characteristics, it proposes a permeability coefficient formula suitable for coarse-grained cohesive soils and introduces numerical simulation technology to expand the dataset, reducing experimental costs. The design of multiple loss terms simultaneously constrains model training from dimensions such as accuracy, physical consistency, generalization ability, and uncertainty, effectively improving the model's robustness and interpretability. This invention effectively solves the problem of inconsistent accuracy, computational efficiency, and generalization ability in existing coarse-grained cohesive soil permeability coefficient prediction. It can quickly predict the permeability coefficient of coarse-grained cohesive soils under different porosities and particle size distribution characteristics, providing a data-efficient and physically interpretable technical solution for permeability coefficient prediction in coarse-grained cohesive soils, significantly reducing experimental costs and improving engineering prediction accuracy.

[0181] The embodiments of this application have been described above with reference to the accompanying drawings. However, this application is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of this application without departing from the spirit and scope of the claims, and all of these forms are within the protection scope of this application.

Claims

1. A method for predicting the permeability coefficient of coarse-grained cohesive soil based on a physically constrained neural network, characterized in that, Includes the following steps: Step 1: Conduct indoor seepage tests on cohesive coarse-grained soil and establish a formula for the permeability coefficient of cohesive coarse-grained soil that takes into account porosity and particle size distribution characteristics. Step 2: Construct a hybrid model containing both physics-driven terms and neural network data-driven terms; Step 3: Based on the seepage test in Step 1, supplement the data on porosity, particle size distribution characteristics and corresponding permeability coefficient of cohesive coarse-grained soil samples by numerical simulation technology and literature review to form a complete dataset, which is then divided into training set and test set. Step four: Combine Bayesian optimization to dynamically search for the optimal hyperparameters and train the hybrid model from step two; Step 5: Output the loss function curves for the training and test sets based on the training results to obtain the predicted permeability coefficients under different porosities and gradation characteristics. k pred ; Step 6: Conduct indoor tests based on the actual engineering situation, compare the predicted values ​​with the measured values, verify the model's generalization ability, and ensure its engineering application capability; In step one, The seepage test of the indoor cohesive coarse-grained soil includes the following steps: setting up the test apparatus, preparing the test materials, and designing the test and calculating the permeability coefficient; The test material was coarse-grained clay, and the soil sample was dried, crushed, and sieved. The experimental design and permeability coefficient calculation involved designing seepage tests on cohesive coarse-grained soils with different porosities and particle size distributions. The permeability coefficient was calculated based on water flow rate, collection time, cross-sectional area of ​​the test soil chamber, length of each soil layer, and the head difference between the soil layers. The permeability coefficient calculation formula is as follows: No. a Permeability coefficient of stratified soil in the test k a for; in, Q Water flow rate; L a For the first a The length of each test soil layer; A The cross-sectional area of ​​the test soil silo; h a For the first a The water head difference between the soil layers in the test; t For time; n This represents the total number of soil layers. The particle size distribution characteristics include: average particle size of coarse-grained clay, average particle size of fine-grained clay, and uniformity coefficient. The permeability coefficient formula for coarse-grained clay considering porosity and particle size distribution characteristics is established based on the results of indoor coarse-grained clay permeability tests. That is, after obtaining the permeability coefficient under different porosity and particle size distribution characteristics using equation (2), the empirical formula method is used to fit and obtain the permeability coefficient formula for coarse-grained clay. Considering that coarse-grained clay has a certain fine particle content, which is reflected by porosity and non-uniformity coefficient, the expression is shown in equation (3): In the formula, A The coefficients are linear. n e Effective porosity; d 50 The average particle size of coarse-grained clay. C u The coefficient of uniformity; i , α , β , m It is a dimensionless parameter; The effective porosity n e To account for the ineffective porosity caused by the bound water film of fine particles in coarse-grained clays, the double-layer parameter method shows that it is mainly affected by the average particle size of the fine particles: In the formula, n Input a variable for porosity; C s The average particle size of the fine particles; η , γ It is a dimensionless parameter; In step two, The hybrid model includes physical driving terms and data driving terms, and the formula for predicting the penetration coefficient is shown in equation (5): Among them, physics-driven items k h The permeability coefficient of coarse-grained cohesive soil is calculated based on the formula considering porosity and particle size distribution characteristics; data-driven term NN( n , d 50 , C s , C u , θ () is a neural network; θ These are the parameters of the neural network.

2. The method for predicting the permeability coefficient of coarse-grained cohesive soil based on a physically constrained neural network according to claim 1, characterized in that, The test apparatus includes a sample loading system, a pressurized water supply system, a dynamic load control system, and a data acquisition system. The sample loading system is made of 5mm thick transparent plexiglass and includes: an inlet, a buffer zone, a permeable base plate, a soil chamber, and a drain outlet; The pressurized water supply system includes an air compressor, a pressure pipe, and a water-air exchange tank; to ensure the stability of the water pressure, a dual pressure regulating valve is used to control the gas pressure in the water-air exchange tank. The dynamic load control system simulates vibration loads using a small eccentric vibration motor. The data acquisition system includes: a pore water pressure gauge, a soil-water separation device, and a flow measurement device; After the soil sample loading system completes the soil sample loading, the pressurized water supply system injects stable pressurized water into the buffer zone through the inlet. The water seeps into the soil sample through the permeable bottom plate. The dynamic load control system can simulate dynamic load action. After the seepage water flows out of the drain outlet, it passes through the soil-water separation device and the flow measurement device in sequence. At the same time, the pore water pressure gauge monitors the pore water pressure in real time, realizing the collection and recording of seepage data of cohesive coarse-grained soil.

3. The method for predicting the permeability coefficient of coarse-grained cohesive soil based on a physically constrained neural network according to claim 1, characterized in that, In step three, The numerical simulation technology includes the following processes: model building and supplementary working condition design; The model is built based on PFC 5.

0. 2D An indoor simulation model for seepage test of cohesive coarse-grained soil was established. The microscopic parameters of the material were calibrated through biaxial tests. The calibrated parameters were used for seepage test simulation. The permeability coefficient obtained from the seepage test was compared with the results obtained from the indoor test to ensure the reliability of the parameters. The method for establishing the simulation model for the seepage test of the indoor cohesive coarse-grained soil is as follows: In PFC 5.0 2D The software constructs the model boundary using wall elements based on the size of the soil chamber sample. Based on the porosity and gradation of cohesive coarse-grained soil, it generates soil particles representing each particle size group and fills them in layers. It generates a seepage grid using Gmsh2D, traverses the seepage grid cells, calculates the effective porosity in the model, and then solves the permeability coefficient according to the formula for the permeability coefficient of cohesive coarse-grained soil that considers the porosity and particle gradation characteristics. It then applies seepage force to obtain the indoor seepage test simulation model of cohesive coarse-grained soil. The dimensions of the model soil chamber are consistent with those of the indoor test, being 1.25m in length and 0.5m in width. The supplementary working condition design involves designing a series of cohesive coarse-grained soil permeability test conditions with different porosities and particle size distribution characteristics based on the obtained microscopic parameters, conducting numerical simulations, and obtaining the corresponding permeability coefficients. The literature review involved consulting previous literature and compiling permeability test data for coarse-grained clays, including porosity and particle size distribution characteristics. The complete dataset was obtained through three methods: indoor experiments, numerical simulation, and literature review. It includes data on porosity, average particle size of coarse-grained clay, average particle size of fine-grained clay, uniformity coefficient, and corresponding permeability coefficient.

4. The method for predicting the permeability coefficient of coarse-grained cohesive soil based on a physically constrained neural network according to claim 1, characterized in that, Step four includes the following processes: forward propagation, calculation of the composite loss function, gradient calculation and joint optimization of parameters, and dynamic adjustment of hyperparameters through Bayesian optimization. Specifically as follows: Forward propagation: Define the input layer, hidden layer, and output layer; initialize the parameters of the permeability coefficient formula for coarse-grained cohesive soil. A , i , α , β , m、η , γ Initialize neural network parameters θ Input training data and calculate k h Formula terms, neural network terms; Composite loss function calculation: Calculate the mean squared error term, physical constraint penalty term, parameter regularization term, uncertainty quantification term, and dynamic equilibrium term, and sum them to obtain the total loss, as shown in equation (6): In the formula, λ 1 represents the weighting coefficient of the physical constraint penalty term; λ 2 represents the weighting coefficient of the parameter regularization term; λ 3 represents the weighting coefficient of the uncertainty quantification term; λ 4 represents the weighting coefficient of the dynamic balance term; This is the mean square error term; This is a physical constraint penalty term; For parameter regularization terms; This is a quantification term for uncertainty. It is a dynamic equilibrium term; Among them, the mean square error term minimizes the mean square error between the predicted value and the true value, and its expression is shown in equation (7): In the formula, For the first i The true permeability coefficient of coarse-grained soil for each sample; For the first i The predicted permeability coefficient of each sample is expressed as shown in equation (8); N The total number of training samples used in the calculation; In the formula, For the first i Each sample physical driving item k h The calculated value is shown in equation (9); For the first i The neural network calculates the value for each sample; in, The expression is shown in equation (10): The physical constraint penalty term is used to ensure that the parameters conform to physical meaning, and its expression is shown in equation (11): In the formula, q max , q min for A Preset upper and lower limits of a reasonable value range; The parameter regularization constraint constrains the parameter amplitude to prevent overfitting, and the expression is shown in equation (12): The uncertainty quantification term assumes that the neural network output follows a Gaussian distribution. The mean and variance are optimized by maximum likelihood estimation, and the standard deviation is ensured to be non-negative. The expression is shown in equation (13). In the formula, μ The mean permeability coefficient predicted by the model is equal to k h The sum of the calculated value and the neural network calculated value; σ The standard deviation of the predicted results; The dynamic equilibrium term is calculated. k h The correlation between the calculated value and the neural network calculated value is used to prevent the physical terms and neural network terms from canceling each other out. The expression is shown in equation (14): Gradient calculation and joint parameter optimization: including k h Parameter gradient and neural network parameter gradient; The k h The parameter gradient is calculated using the chain rule to determine the gradient of the composite loss with respect to the physical parameters, and the parameters are updated using gradient descent. A , i , α , β , m、η , γ The gradient is calculated using formulas (15) to (21): The gradient of the neural network parameters is calculated using automatic differentiation to determine the loss of the neural network parameters. θ The gradient is calculated, and the parameters are updated using gradient descent. Bayesian optimization for dynamic hyperparameter adjustment: The gradient descent method dynamically adjusts the learning rate through Bayesian optimization to ensure model convergence; specifically: a Bayesian optimization strategy is adopted, assuming that the validation set loss follows a Gaussian process, a surrogate model is constructed, the validation set is used to evaluate the current hyperparameter performance, and the hyperparameter combination that maximizes the expected improvement is selected through the expected improvement criterion, thereby efficiently coordinating the loss weights (…). λ 1. λ 2. λ 3. λ 4) and learning rate; the loss weight ranges from [0,1]; the learning rate ranges from [10] to [10]. -5 10 -3 ].

5. The method for predicting the permeability coefficient of coarse-grained cohesive soil based on a physically constrained neural network according to claim 4, characterized in that, After Bayesian optimization, the loss weights λ 1. λ 2. λ 3. λ The values ​​for 4 are 0.075, 0.234, 0.164, and 0.327; the learning rate is 6.64 × 10⁴. -4 .

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