Bayesian and physics-informed neural network based prediction method for permeability coefficient of sand

Abstract

This invention discloses a method for predicting the permeability coefficient of sand based on Bayesian and physical information neural networks, comprising the following steps: constructing a physical-data hybrid-driven augmented dataset, the dataset including measured anchor point data and physical augmentation data generated based on seepage physical mechanisms; constructing a dual-channel feature decoupling and fusion neural network, extracting gradation morphology features and soil compaction features through particle size and structure channels respectively, and fusing them at a deep layer; introducing a multi-physical constraint embedding mechanism and a physical constraint activation layer, constructing a total loss function, and using the HMC algorithm to perform posterior inference on network parameters to obtain a parameter sample set; and predicting the permeability coefficient based on the parameter sample set. This invention effectively solves the overfitting problem caused by small samples in geotechnical engineering through a dual-channel architecture and data augmentation strategy, and eliminates the generation of non-physical prediction values ​​through both network architecture and physical loss embedding, achieving a synergy between high-precision prediction and uncertainty quantification, greatly improving engineering adaptability.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of geotechnical engineering seepage analysis and artificial intelligence, specifically involving a method for predicting the permeability coefficient of sand based on Bayesian and physical information neural networks. Background Technology

[0002] The permeability coefficient is a crucial hydrogeological parameter, serving as a quantitative indicator of soil permeability and a fundamental parameter essential for seepage calculations. Accurately obtaining the permeability coefficient directly impacts the reliability of dam seepage control design, foundation pit dewatering schemes, and groundwater management. Accurately determining the soil permeability coefficient is a critical task during construction design. Common methods for obtaining permeability coefficients include theoretical analysis, experimental methods, and inversion methods. Traditional methods, such as indoor constant head or variable head tests, require the preparation of standard samples, exhibit significant size effects for wide-graded sands, and involve long test cycles and high labor costs. Theoretical analysis methods rely on empirical formulas (such as the Kozeny-Carman equation) based on the assumption of a uniform sphere, leading to prediction errors for sands with high non-uniformity and failing to couple the nonlinear effects of void ratio and particle size distribution. While field pumping tests can reflect actual working conditions, they require the deployment of monitoring well groups, resulting in high project costs and limiting large-scale application. These methods all face the common challenges of long testing cycles, high costs, and poor adaptability, necessitating data-driven technological innovation.

[0003] In recent years, deep learning models (such as ELM and random forests) have improved prediction accuracy through nonlinear mapping, but they still have fundamental shortcomings: First, purely data-driven models lack seepage physical constraints and may output negative or extreme values ​​that violate Darcy's law; second, they cannot quantify the uncertainty of predictions, making it difficult to support safety decisions in high-risk engineering projects; and third, they do not demonstrate a coupling mechanism that integrates sand particle size distribution and void ratio, limiting their generalization ability. To address these issues, there is an urgent need for a method to predict the permeability coefficient of sand that can achieve "high-precision prediction + uncertainty quantification + physical consistency assurance." Summary of the Invention

[0004] To address the shortcomings of existing technologies, the present invention aims to provide a method for predicting the permeability coefficient of sand based on Bayesian and physical information neural networks. Through the synergy of dual-channel feature architecture, physical equation embedding, and Bayesian inference, it achieves a triple breakthrough of "high-precision prediction + uncertainty quantification + physical consistency guarantee".

[0005] This invention provides a method for predicting the permeability coefficient of sandy soil based on Bayesian and physical information neural networks, comprising the following steps:

[0006] S1. Construct a physics-data hybrid-driven augmented dataset; the dataset includes laboratory-measured sand gradation data and physical augmentation data generated based on seepage physical mechanisms; preprocess the dataset;

[0007] S2. Construct a dual-channel feature decoupling and fusion neural network; the network includes independent and parallel particle size feature extraction channels and structural feature extraction channels, which respectively learn the sand gradation features and structural features, and fuse them in a deep layer to establish a nonlinear mapping relationship between input features and permeability coefficient;

[0008] S3. Introduce a multi-physical constraint embedding mechanism, construct a total loss function, and train a dual-channel feature decoupling and fusion neural network based on the enhanced dataset;

[0009] S4. Optimize the neural network using the Hamiltonian Monte Carlo algorithm to obtain the posterior sample set of network parameters;

[0010] S5. Based on the parameter posterior sample set, the actual permeability coefficient of sandy soil is predicted by using a dual-channel feature decoupling and fusion neural network.

[0011] In step S1, the gradation characteristics include effective particle size. , ... ;

[0012] The method for generating the physical augmentation data is as follows: random sampling is performed within a reasonable parameter space based on physical equations such as Kozeny-Carman, theoretical permeability coefficient is calculated and random noise is superimposed, and then mixed with measured data to form an augmentation dataset.

[0013] The preprocessing involves cleaning the sand data, eliminating outliers and missing values, and performing Z-score standardization to eliminate the influence of dimensions.

[0014] The specific architecture of the dual-channel feature decoupling and fusion neural network described in step S2 includes: a particle size feature extraction channel: used to receive particle size data of various grades of sand ( , ... The particle size feature extraction channel extracts the nonlinear morphological features of the gradation curve through multiple fully connected layers; the particle size feature extraction channel contains two hidden layers with 16 and 8 nodes respectively; the structural feature extraction channel receives structural parameters such as the void ratio (e) of sand (1 dimension in total) and extracts the compaction state features of the soil through an independent fully connected layer; the structural feature extraction channel contains two hidden layers with 8 and 4 nodes respectively; the feature fusion module concatenates the 8-dimensional feature vector output by the particle size feature extraction channel with the 4-dimensional feature vector output by the structural feature extraction channel to form a 12-dimensional comprehensive feature vector, which is input to a fusion hidden layer containing 8 nodes; the physical constraint output layer has a physical constraint activation function (Softplus) connected to the end of the feature fusion module, which forces the network output value to always be greater than zero to satisfy the physical definition of non-negative permeability coefficient.

[0015] Except for the output layer, all other layers of the network use the hyperbolic tangent function tanh as the activation function.

[0016] In step S3, the total loss function is expressed using the following formula: ;in, For data likelihood loss; This is the a priori regularization loss; Loss due to physical constraints; Weights for physical constraint loss;

[0017] The multi-physics constraint embedding mechanism specifically involves introducing several physical models. Based on each physical model, the absolute difference between the theoretical penetration coefficient and the penetration coefficient predicted by the dual-channel feature decoupling and fusion neural network is calculated as the residual, expressed using the following formula: ,in, The residuals calculated for the k-th physical model are... The theoretical permeability coefficient calculated for the k-th physical model. The penetration coefficient is predicted by the dual-channel feature decoupling and fusion neural network; then, the residuals are weighted and summed according to preset weights, and the result is used as the physical constraint loss. The physical model includes the Kozeny-Carman equation, the Terzaghi formula, empirical formula one for the permeability coefficient of coarse-grained soil, and empirical formula two for the permeability coefficient of coarse-grained soil; physical constraint loss. Express it using the following formula: ;in, The residuals are calculated based on the Kozeny-Carman equation; The residuals are calculated based on the Terzaghi formula; The residual is calculated based on the empirical formula for the permeability coefficient of coarse-grained soil. The residual is calculated based on the second empirical formula for the permeability coefficient of coarse-grained soil. This is the first weight value; This is the second weight value; This is the third weight value; It is the fourth weight value;

[0018] The data likelihood loss Express it using the following formula: ;in, This represents the expected value of the data likelihood. These are the parameters of the neural network, including weights and biases; Let X be the likelihood function; X is the input feature. The variational posterior distribution of the model parameters;

[0019] The prior regularization loss Express it using the following formula: ;in, It is the prior distribution; The Kullback-Leibler divergence; The variational posterior distribution of the model parameters;

[0020] Based on the augmented dataset obtained in step S1, the training of the dual-channel feature decoupling and fusion neural network is completed by minimizing the total loss function.

[0021] The Kozeny-Carman equation is expressed using the following formula: Where n is porosity, and ; ε is the theoretical permeability coefficient calculated based on the Kozeny-Carman equation; e is the void ratio.

[0022] The Terzaghi formula is expressed using the following equation: ;in, The theoretical permeability coefficient is calculated based on the Terzaghi formula; e is the void ratio.

[0023] The empirical formula for the permeability coefficient of coarse-grained soil is expressed by the following formula: ;in, The theoretical permeability coefficient is calculated based on the empirical formula for the permeability coefficient of coarse-grained soil. Let be the curvature coefficient, and ; The coefficient of non-uniformity is, and ;

[0024] The empirical formula for the permeability coefficient of coarse-grained soil, as described in Formula 2, is expressed by the following formula: ;in, The theoretical permeability coefficient is calculated based on empirical formula two for the permeability coefficient of coarse-grained soil.

[0025] In step S4, the Hamiltonian Monte Carlo (HMC) algorithm is used to perform posterior sampling on the parameters of the neural network to generate a posterior sample set of the parameters.

[0026] Step S4 specifically involves: first, constructing the probability distribution of the HMC system simulation parameter space, for the neural network parameters... The joint probability distribution is defined as follows: ; r is the introduced momentum variable; denoted as , where W represents the neural network parameters; W represents the weights in the neural network parameters; and b represents the biases in the neural network parameters. It is the likelihood function; It is a joint probability distribution;

[0027] Then, based on the HMC algorithm, the following parameters are preset: number of sampling steps, step size, number of warm-up steps, and total number of samples;

[0028] Based on the preset parameters, the HMC algorithm is used to generate a posterior sample set of the parameters, which constitutes the posterior distribution of the model parameters.

[0029] Step S5 is as follows:

[0030] Collect the various distribution parameters and porosity e of the sand to be predicted to obtain 11-dimensional input features. The parameters are input into the corresponding channels of the trained dual-channel network; based on the Bayesian neural network framework, the parameter samples in the posterior sample set of the parameters are used as network parameters to generate several sets of permeability coefficient prediction values.

[0031] Statistical analysis was performed on the obtained permeability coefficient predictions to obtain the predictions and confidence intervals.

[0032] This invention discloses a method for predicting the permeability coefficient of sand based on Bayesian and physical information neural networks, solving the problem of inaccurate permeability coefficient prediction in practical engineering. This invention effectively extracts the decoupling features of the sand skeleton and pores through a dual-channel architecture, eliminates negative prediction phenomena through the Softplus physical constraint layer, and combines physically enhanced data with Bayesian inference to provide a high-precision and reliable prediction tool with uncertainty quantification capabilities for practical engineering applications. Attached Figure Description

[0033] Figure 1 This is a schematic flowchart of the method of the present invention;

[0034] Figure 2 This is a schematic diagram of the entire data flow in an embodiment of the present invention;

[0035] Figure 3 This is a comparison chart of the final predicted value and the actual value in an embodiment of the present invention. Detailed Implementation

[0036] This invention provides a method for predicting the permeability coefficient of sandy soil based on Bayesian and physical information neural networks, the flowchart of which is shown below. Figure 1 As shown, it includes the following steps:

[0037] S1. Construct a physics-data hybrid-driven augmented dataset; the dataset includes laboratory-measured sand gradation data and physical augmentation data generated based on seepage physical mechanisms; preprocess the dataset;

[0038] In step S1, the gradation characteristics include effective particle size. , ... ;

[0039] The method for generating the physical augmentation data is as follows: random sampling is performed within a reasonable parameter space based on physical equations such as Kozeny-Carman, theoretical permeability coefficient is calculated and random noise is superimposed, and the data is mixed with measured data to form an augmentation dataset;

[0040] The preprocessing involves cleaning the sand data to remove outliers and missing values, and performing Z-score standardization to eliminate the influence of dimensions. During model training, the data is standardized to convert it into a distribution with zero mean and unit variance. This step eliminates the influence of dimensions between different features, ensuring that all features are within the same scale range. This helps accelerate the model's convergence process and improves training efficiency and prediction accuracy.

[0041] S2. Construct a dual-channel feature decoupling and fusion neural network; the network includes independent and parallel particle size feature extraction channels and structural feature extraction channels, which respectively learn the sand gradation features and structural features, and fuse them in a deep layer to establish a nonlinear mapping relationship between input features and permeability coefficient;

[0042] The specific architecture of the dual-channel feature decoupling and fusion neural network described in step S2 includes: a particle size feature extraction channel: used to receive particle size data of various grades of sand. , ... The system comprises 10 dimensions, extracting nonlinear morphological features of the gradation curve through multiple fully connected layers. The particle size feature extraction channel includes two hidden layers with 16 and 8 nodes respectively. The structural feature extraction channel receives structural parameters (1 dimension) such as the void ratio (e) of the sand, extracting soil compaction features through an independent fully connected layer. This channel also includes two hidden layers with 8 and 4 nodes respectively. The feature fusion module concatenates the 8-dimensional feature vector output from the particle size feature extraction channel with the 4-dimensional feature vector output from the structural feature extraction channel to form a 12-dimensional comprehensive feature vector, which is then input into a fusion hidden layer containing 8 nodes. The physical constraint output layer connects to a physical constraint activation function (Softplus) at the end of the feature fusion module, forcing the network output value to always be greater than zero to satisfy the physical definition of a non-negative permeability coefficient. Except for the output layer, all other layers use the hyperbolic tangent function tanh as the activation function.

[0043] In the hidden layers of the dual-channel network, the hyperbolic tangent function (tanh) is selected. Tanh has a non-zero gradient, and its derivative ranges from (0,1), making it more suitable for Bayesian network training processes that require fine-tuning of weights compared to other activation functions (such as ReLU). Simultaneously, tanh's saturation region restricts the feature output to the range of [-1,1], effectively suppressing numerical instability in intermediate layers caused by extreme granular data distributions. In the final output layer of the network, this invention innovatively introduces the Softplus function (i.e., Softplus). Since the permeability coefficient of sand is strictly greater than zero in its physical definition, conventional activation functions may output negative values ​​(i.e., non-physical interpretation). Softplus has a value range greater than 0 and possesses monotonically increasing and smoothly differentiable properties. By applying the Softplus constraint at the output, the non-negativity of the prediction result is enforced from the bottom layer of the network architecture, completely eliminating the "negative permeability coefficient" error that may occur in purely data-driven models, and achieving a deep integration of physical mechanisms and deep learning.

[0044] S3. Introduce a multi-physical constraint embedding mechanism, construct a total loss function, and train a dual-channel feature decoupling and fusion neural network based on a sand dataset;

[0045] In step S3, the total loss function is expressed using the following formula: ;in, For data likelihood loss; This is the a priori regularization loss; Loss due to physical constraints; Weights for physical constraint loss;

[0046] The multi-physics constraint embedding mechanism specifically involves introducing several physical models. Based on each physical model, the absolute difference between the theoretical penetration coefficient and the penetration coefficient predicted by the dual-channel feature decoupling and fusion neural network is calculated as the residual, expressed using the following formula: ,in, The residuals calculated for the k-th physical model are... The theoretical permeability coefficient calculated for the k-th physical model. The initial fully connected neural network predicts the penetration coefficient; then, the residuals are weighted and summed according to preset weights, and the result is used as the physical constraint loss. The physical model includes the Kozeny-Carman equation, the Terzaghi formula, empirical formula one for the permeability coefficient of coarse-grained soil, and empirical formula two for the permeability coefficient of coarse-grained soil; physical constraint loss. Express it using the following formula: ;in, The residuals are calculated based on the Kozeny-Carman equation; The residuals are calculated based on the Terzaghi formula; The residual is calculated based on the empirical formula for the permeability coefficient of coarse-grained soil. The residual is calculated based on the second empirical formula for the permeability coefficient of coarse-grained soil. This is the first weight value; This is the second weight value; This is the third weight value; It is the fourth weight value;

[0047] The data likelihood loss Express it using the following formula: ;in, This represents the expected value of the data likelihood. These are the parameters of the neural network, including weights and biases; Let X be the likelihood function; X is the input feature. The variational posterior distribution of the model parameters;

[0048] The prior regularization loss Express it using the following formula: ;in, It is the prior distribution; The Kullback-Leibler divergence; The variational posterior distribution of the model parameters;

[0049] Based on the sand dataset obtained in step S1, the training of the dual-channel feature decoupling and fusion neural network is completed by minimizing the total loss function.

[0050] The Kozeny-Carman equation is expressed using the following formula: Where n is porosity, and ; The theoretical permeability coefficient is calculated based on the Kozeny-Carman equation.

[0051] The Terzaghi formula is expressed using the following equation: ;in, The theoretical permeability coefficient is calculated based on the Terzaghi formula; e is the void ratio.

[0052] The empirical formula for the permeability coefficient of coarse-grained soil is expressed by the following formula: ;in, The theoretical permeability coefficient is calculated based on the empirical formula for the permeability coefficient of coarse-grained soil. Let be the curvature coefficient, and ; The coefficient of non-uniformity is, and ;

[0053] The empirical formula for the permeability coefficient of coarse-grained soil, as described in Formula 2, is expressed by the following formula: ;in, The theoretical permeability coefficient is calculated based on empirical formula two for the permeability coefficient of coarse-grained soil.

[0054] S4. The dual-channel feature decoupling and fusion neural network is optimized using the Hamiltonian Monte Carlo algorithm to obtain the network posterior parameter sample set;

[0055] In step S4, the Hamiltonian Monte Carlo (HMC) algorithm is used to perform posterior sampling on the parameters of the fully connected neural network to generate a posterior sample set of the parameters.

[0056] Step S4 specifically involves: first, constructing the probability distribution of the HMC system simulation parameter space, for the neural network parameters... The joint probability distribution is defined as follows: ; r is the introduced momentum variable; denoted as , where W represents the neural network parameters; W represents the weights in the neural network parameters; and b represents the biases in the neural network parameters. It is the likelihood function; It is a joint probability distribution;

[0057] Then, based on the HMC algorithm, the following parameters are preset: number of sampling steps, step size, number of warm-up steps, and total number of samples;

[0058] Based on the preset parameters, the HMC algorithm is used to generate a posterior sample set of the parameters, which constitutes the posterior distribution of the model parameters.

[0059] S5. Based on the parameter posterior sample set, the actual permeability coefficient of sandy soil is predicted by using a dual-channel feature decoupling and fusion neural network.

[0060] Step S5 is as follows:

[0061] Collect the various distribution parameters and porosity e of the sand to be predicted to obtain 11-dimensional input features. The parameters are input into the corresponding channels of the trained dual-channel network; based on the Bayesian neural network framework, the parameter samples in the posterior sample set of the parameters are used as network parameters to generate several sets of permeability coefficient prediction values.

[0062] The obtained permeability coefficient predictions are statistically analyzed to obtain the final prediction result.

[0063] The method of the present invention will be further described below with reference to the embodiments:

[0064] The schematic diagram of the entire data flow in this embodiment is as follows: Figure 2 As shown.

[0065] To construct a physically complete training space, this embodiment first selected 93 representative sets of laboratory measured data as "anchor samples," covering various gradation characteristics from fine sand to coarse sand (as shown in Table 1). Based on this, the sample space was expanded and enhanced using the seepage physical mechanism, constructing a physical-data hybrid-driven augmented dataset containing approximately 400 sets of samples.

[0066] Table 1. Example of experimental anchor point sample data

[0067]

[0068] Based on the above anchor point samples, random sampling was performed within a reasonable parameter range using physical equations such as the Kozeny-Carman model, and Gaussian noise was superimposed to generate a total of 400 enhanced samples. Some of the enhanced data are shown in Table 2. The value includes both theoretical value and noise.

[0069] Table 2 Examples of Physical Enhancement Sample Data

[0070]

[0071] This data is divided into the training set and the validation set of the model in an 8:2 ratio.

[0072] The raw data is cleaned to eliminate outliers and missing values. This mainly involves global physical rule filtering, which sets reasonable physical thresholds based on the physical properties of the sand to filter out data points that do not conform to physical laws. Requirements include a particle size value greater than 0 and a permeability coefficient greater than [missing value]. cm / s.

[0073] Standardizing the data transforms it into a distribution with zero mean and unit variance. This step eliminates the influence of different dimensions on the data, helping to accelerate the model's convergence process.

[0074] This embodiment employs a dual-channel feature decoupling and fusion neural network. Specifically, channel one is the particle size extraction channel, and the input is... , ... The particle size characteristics are analyzed, and the nonlinear morphological features of the gradation curve are extracted through a multi-layer fully connected structure. Channel 2 is the structure extraction channel, which selects the void ratio e as input to extract the soil compaction state features. Feature fusion: The high-dimensional features extracted from the two channels are spliced ​​together in a deep layer and input into the fusion layer for regression prediction. A Softplus activation function is introduced into the final output layer of the network to force the output result to always be greater than 0 from the architecture, ensuring that the prediction result conforms to the physical definition of the permeability coefficient and completely eliminating the negative value problem that may be generated by traditional models.

[0075] During training, the physical constraint loss The first weight value in the expression The second weight value is 0.6. The third weight value is 0.2. The fourth weight value is 0.1. It is 0.1;

[0076] In this embodiment, the key parameters of the HMC algorithm are set as follows:

[0077] Sampling steps: L = 100 steps / sample, to ensure sufficient exploration of the parameter space.

[0078] Step size: step_size=0.0001, controls the integration accuracy and stability.

[0079] Warm-up steps: burn=200, discard initial unstable samples.

[0080] Total number of samples: num_samples=400, generating enough samples to approximate the posterior distribution.

[0081] With the above settings, the HMC algorithm generates a parameter set, which constitutes the posterior distribution of the model parameters.

[0082] The prediction results of the examples show that the model of the present invention achieves extremely high prediction accuracy on the validation set.

[0083] The mean squared error (MSE) on the validation set is as low as 0.0003, and the mean absolute error (MAE) is 0.0099. For example... Figure 3 As shown, the predicted points are closely distributed around the ideal fitting line, indicating that the model captures the seepage law of sandy soil very well.

[0084] In the extremely low permeability range ( Thanks to the introduction of physically augmented data and the constraints of the Softplus physical constraint layer, as well as the embedding of physical formulas in the loss function, the model of this invention exhibits excellent robustness. Unlike traditional methods that are prone to negative values ​​or drastic fluctuations, the predicted values ​​of this model remain positive and the trend is accurate, effectively solving the problem of low-value prediction divergence under small sample conditions.

[0085] The uncertainty of the prediction is quantified by calculating the prediction standard deviation and the 95% confidence interval. Despite the introduction of noise into the data, the 95% confidence interval generated by this invention can still effectively cover most of the actual observations, and the lower bound of the interval is limited by physical constraints to avoid non-physical negative values. This provides a reliable safety boundary reference for the design of foundation pit dewatering.

[0086] The physical formulas and parameters used in the prediction model of this invention can be determined according to the actual situation. Although this embodiment uses the Kozeny-Carman equation and the Terzaghi empirical formula as examples, in practical applications, the physical constraint system needs to be adjusted according to factors such as sand type (e.g., gradation uniformity, fine particle content) and engineering scenario (e.g., foundation pit depth, hydraulic gradient).

[0087] Based on the validation set, the Kozeny-Carman equation and Terzaghi empirical formula commonly used in geotechnical engineering were used as comparative example 1. The permeability coefficient was calculated by directly substituting the gradation parameters. The results compared with the method of this invention are shown in Table 3.

[0088] Table 3 Comparison of partial results between the present invention and Comparative Example 1

[0089] Analysis shows that the Kozeny-Carman equation and the Terzaghi empirical formula, due to their assumption of a uniform sphere, struggle to capture the nonlinear coupling effects between particles under complex gradation conditions, resulting in significant errors on non-uniform sand samples. In contrast, the method of this invention extracts gradation features through a neural network, achieving significantly higher prediction accuracy than traditional empirical formulas, demonstrating strong adaptability to gradation conditions.

[0090] To verify the effectiveness of the proposed dual-channel architecture and physical constraint mechanism, a traditional single-channel fully connected neural network (ANN) is used as a comparative example 2. This comparative example adopts a common 11-16-16-8-1 layer structure (11 indicates...). , ... The method used in this invention (which has 11 input features) does not employ the feature decoupling dual-channel design of this invention, nor does it set a physical constraint activation function (Softplus) in its output layer. Furthermore, it does not introduce a physical residual loss function term during training, nor does it use Bayesian inference sampling. Based on the same physical augmentation dataset, the prediction results of Comparative Example 2 and the method of this invention are compared on the validation set. The comprehensive evaluation index comparison results are shown in Table 4, and the prediction results of some representative samples are shown in Table 5.

[0091] Table 4 Performance Comparison of the Invention and Comparative Example 2

[0092]

[0093] Table 5 Comparison of results between the present invention and Comparative Example 2

[0094]

[0095] Analysis of the data in Tables 4 and 5 reveals that although Comparative Example 2 (ANN) performed reasonably well in terms of overall error index (MSE=0.0004) thanks to training with the augmented dataset, it exhibited a fatal negative prediction result (-0.001971) in the extremely low permeability range (such as sample 359). From a physical perspective, the permeability coefficient represents the soil's ability to conduct fluids; a negative value clearly violates fundamental laws of seepage mechanics, such as Darcy's law, and constitutes an erroneous "non-physical interpretation." This profoundly reflects that purely data-driven models, lacking constraints on the solution space boundaries, are prone to falling into mathematical fitting traps and losing physical accuracy in sparse data regions.

[0096] To more comprehensively evaluate the advancement of the method of this invention, this embodiment further introduces Comparative Example 3 (GA-BP neural network) for comparison. Comparative Example 3 adopts the same conventional fully connected neural network layer architecture (11-16-16-8-1) as Comparative Example 2. Its training process does not introduce a physical residual loss function term and does not use Bayesian inference sampling. It only optimizes the initial weights through the global optimization mechanism of the simulated genetic algorithm (GA), which represents the performance ceiling of current pure data-driven methods at the parameter optimization level. According to the prediction results of the validation set, the comprehensive evaluation index comparison between the method of this invention and Comparative Example 3 is shown in Tables 6 and 7.

[0097] Table 6 Performance Comparison of the Invention and Comparative Example 3

[0098]

[0099] Table 7 Comparison of results between the present invention and Comparative Example 3

[0100]

[0101] Analysis of Tables 6 and 7 shows that Comparative Example 3 (GA-BP), after global optimization using a genetic algorithm, did indeed achieve a statistical error index (MSE=0.0003) comparable to that of this invention, demonstrating its powerful numerical fitting capabilities. However, this purely mathematical fitting cannot mask the lack of a physical mechanism. In extremely low permeability ranges (such as sample 359), the GA-BP model still outputs a negative prediction result (-0.002569). This phenomenon profoundly reveals the fatal flaw of existing technologies: even with perfectly optimized algorithm parameters, without physical constraints, the model cannot understand the physical law that "permeability coefficient is always greater than zero," thus producing non-physical interpretations under boundary conditions.

[0102] In contrast, the method of this invention not only maintains extremely high prediction accuracy (MSE=0.0003), but more importantly, thanks to the strong synergy between the physical constraint layer (Softplus) introduced at the end of the dual-channel architecture and the PINN loss function, the model accurately outputs a positive prediction (0.004790) on sample 359. This proves that while ensuring high accuracy, this invention constructs a robust physical defense, completely eliminating "counterintuitive predictions." Furthermore, comparative samples 1, 2, and 3 can only output a single predicted value (point estimate), failing to reflect the risks brought about by parameter variability. This invention, however, provides a measure of the uncertainty of the predicted value through Bayesian inference, and its 95% confidence interval generated by HMC sampling achieves a prediction confidence coverage of 0.90 on the validation set. In high-risk scenarios such as deep foundation pit dewatering design, this invention can both avoid the negative decision disaster that occurs in GA-BP and provide a risk quantification reference, possessing irreplaceable engineering practical value.

Claims

1. A method for predicting the permeability coefficient of sandy soil based on Bayesian and physical information neural networks, characterized in that, Includes the following steps: S1. Construct a physics-data hybrid-driven augmented dataset; the dataset includes laboratory-measured sand gradation data and physical augmentation data generated based on seepage physical mechanisms; preprocess the dataset; S2. Construct a dual-channel feature decoupling and fusion neural network; the network includes independent and parallel particle size feature extraction channels and structural feature extraction channels, which respectively learn the sand gradation features and structural features, and fuse them in a deep layer to establish a nonlinear mapping relationship between input features and permeability coefficient; S3. Introduce a multi-physical constraint embedding mechanism, construct a total loss function, and perform preliminary training on the dual-channel feature decoupling and fusion neural network based on the sand and soil dataset; S4. The Hamilton-Monte Carlo algorithm is used to perform posterior sampling of the neural network parameters to generate a posterior sample set of the parameters; S5. Based on the parameter posterior sample set, a dual-channel feature decoupling and fusion neural network is used to predict the test sample and output the predicted mean and confidence interval of the permeability coefficient. The specific architecture of the dual-channel feature decoupling and fusion neural network described in step S2 includes: a particle size feature extraction channel: used to receive particle size data of various grades of sand. , ... The system comprises 10 dimensions and extracts nonlinear morphological features of the gradation curve through multiple fully connected layers. The particle size feature extraction channel includes two hidden layers with 16 and 8 nodes respectively. The structural feature extraction channel receives structural parameters of the sand's void ratio and extracts soil compaction features through independent fully connected layers. This channel also includes two hidden layers with 8 and 4 nodes respectively. The feature fusion module concatenates the 8-dimensional feature vector output from the particle size feature extraction channel with the 4-dimensional feature vector output from the structural feature extraction channel to form a 12-dimensional comprehensive feature vector, which is then input to a fusion hidden layer containing 8 nodes. The physical constraint output layer connects to a physical constraint activation function at the end of the feature fusion module, forcing the network output value to always be greater than zero to satisfy the physical definition of a non-negative permeability coefficient. Except for the physical constraint output layer, all other layers use the hyperbolic tangent function tanh as the activation function. The multi-physics constraint embedding mechanism specifically involves introducing several physical models, and based on each physical model, calculating the absolute difference between the theoretical permeability coefficient and the permeability coefficient predicted by the neural network as a residual, expressed using the following formula: ,in, The residuals calculated for the k-th physical model are... The theoretical permeability coefficient calculated for the k-th physical model. The permeability coefficient is predicted by the neural network; then, the residuals are weighted and summed according to preset weights, and the result is used as the physical constraint loss. The physical model includes the Kozeny-Carman equation, the Terzaghi formula, empirical formula one for the permeability coefficient of coarse-grained soil, and empirical formula two for the permeability coefficient of coarse-grained soil; physical constraint loss. Express it using the following formula: ;in, The residuals are calculated based on the Kozeny-Carman equation; The residuals are calculated based on the Terzaghi formula; The residual is calculated based on the empirical formula for the permeability coefficient of coarse-grained soil. The residual is calculated based on the empirical formula for the permeability coefficient of coarse-grained soil. This is the first weight value; This is the second weight value; This is the third weight value; It is the fourth weight value; The Kozeny-Carman equation is expressed using the following formula: Where n is porosity, and ; ε is the theoretical permeability coefficient calculated based on the Kozeny-Carman equation; e is the void ratio. The Terzaghi formula is expressed using the following equation: ;in, The theoretical permeability coefficient is calculated based on the Terzaghi formula; The empirical formula for the permeability coefficient of coarse-grained soil is expressed by the following formula: ;in, The theoretical permeability coefficient is calculated based on the empirical formula for the permeability coefficient of coarse-grained soil. Let be the curvature coefficient, and ; The coefficient of non-uniformity is, and ; The empirical formula for the permeability coefficient of coarse-grained soil, as described in Formula 2, is expressed by the following formula: ;in, The theoretical permeability coefficient is calculated based on empirical formula two for the permeability coefficient of coarse-grained soil; n is the porosity, and .

2. The method according to claim 1, characterized in that, The method for constructing the physical-data hybrid-driven augmented dataset in step S1 is as follows: using laboratory measured data as anchor points; randomly sampling within a preset particle size and porosity parameter space based on the Kozeny-Carman equation and the Terzaghi formula, calculating the theoretical permeability coefficient value, and superimposing random noise conforming to a normal distribution to simulate experimental errors, thereby generating physical augmented data; The physical augmentation data is mixed with the measured data to form an expanded training sample set; in step S1, the gradation feature includes the effective particle size. , ... ; The preprocessing involves cleaning the sand data, eliminating outliers and missing values, and performing Z-score standardization to eliminate the influence of dimensions. The elimination of outliers specifically involves setting a physical law threshold based on the physical properties of sand and soil, and filtering out non-compliant data points. The samples in the sand and soil dataset contain effective particle sizes. , ... And the 11-dimensional eigenvector of the porosity.

3. The method for predicting the permeability coefficient of sand based on Bayesian and physical information neural networks according to claim 1, characterized in that, In step S3, the total loss function is expressed using the following formula: ;in, For data likelihood loss; This is the a priori regularization loss; Loss due to physical constraints; Weights for physical constraint loss; Based on the sand dataset obtained in step S1, the training of the dual-channel feature decoupling and fusion neural network is completed by minimizing the total loss function.

4. The method for predicting the permeability coefficient of sand based on Bayesian and physical information neural networks according to claim 3, characterized in that, The data likelihood loss Express it using the following formula: ;in, This represents the expected value of the data likelihood. These are the parameters of the neural network, including weights and biases; Let X be the likelihood function; X is the input feature. Let be the variational posterior distribution of the model parameters.

5. The method for predicting the permeability coefficient of sand based on Bayesian and physical information neural networks according to claim 3, characterized in that, The prior regularization loss Express it using the following formula: ;in, It is the prior distribution; The Kullback-Leibler divergence; Let be the variational posterior distribution of the model parameters.

6. The method for predicting the permeability coefficient of sand based on Bayesian and physical information neural networks according to claim 1, characterized in that, In step S4, the Hamiltonian Monte Carlo algorithm is used to perform posterior sampling of the neural network parameters to generate a posterior sample set of the parameters, specifically as follows: First, the probability distribution of the simulation parameter space of the HMC system is constructed. For the neural network parameters... The joint probability distribution is defined as: ; r is the introduced momentum variable; denoted as , where W represents the neural network parameters; W represents the weights in the neural network parameters; and b represents the biases in the neural network parameters. It is the likelihood function; It is a joint probability distribution; Then, based on the HCM algorithm, the following parameters are preset: number of sampling steps, step size, number of warm-up steps, and total number of samples; Based on the preset parameters, the HMC algorithm is used to generate a posterior sample set of the parameters, which constitutes the posterior distribution of the model parameters.

7. The method for predicting the permeability coefficient of sand based on Bayesian and physical information neural networks according to claim 1, characterized in that, Step S5 is as follows: Collect the distribution parameters and porosity e of the sand to be predicted, and construct a feature vector that conforms to the input of a dual-channel network. Based on the dual-channel feature decoupling and fusion neural network, use the parameter samples in the posterior sample set of the parameters as the network parameters for the input features. Several sets of permeability coefficient prediction values ​​are generated; each prediction value corresponds one-to-one with a parameter sample in the posterior sample set of the parameters; the obtained sets of permeability coefficient prediction values ​​are statistically analyzed to obtain the prediction mean and confidence interval, which are used as the final prediction results.

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