A method for predicting the undrained shear response of gas-bearing clay and quantifying uncertainty

Abstract

This invention discloses a method for predicting and quantifying the uncertainty of undrained shear response of aerated clay. The method constructs a hybrid framework of "physical benchmark + machine learning correction," including: generating a physical benchmark dataset of saturated clay based on a modified Cambridge model; constructing a random forest biased learner with the response change rate as the learning objective, and training the model by fusing physical and experimental data using a domain weighting strategy; reconstructing the full-path response trajectory of aerated clay using the trained model through rolling integrals; generating multiple predicted trajectories by perturbing key parameters based on Monte Carlo simulation, and constructing prediction intervals through quantile statistics to quantify uncertainty. This invention achieves high-precision, physically reasonable continuous prediction under limited data conditions and provides a quantitative measure of uncertainty, improving the reliability of the prediction results.

Description

Technical Field

[0001] This invention relates to the field of geotechnical engineering structural modeling and intelligent analysis technology, and in particular to a method for predicting and quantifying the uncertainty of undrained shear response of aerated clay. Background Technology

[0002] Aerated clay is widely distributed in environments such as the seabed and lake bottom. The small amount of gas inside it can significantly alter the mechanical response of the soil under undrained shear, making prediction complex and uncertain. Accurate prediction of its behavior is crucial for the safety of marine engineering.

[0003] Existing prediction methods mainly fall into two categories, but both have limitations: First, there are physics-based constitutive modeling methods, such as those described in CN121685819A, which construct improved models like teardrop-shaped yield surfaces to describe the behavior of aerated clay. However, these models typically have many parameters and heavily rely on complete experimental data for accurate calibration, making them difficult to apply when real-world data is scarce. Second, there are purely data-driven machine learning methods. While possessing strong nonlinear fitting capabilities, these methods are prone to overfitting and have poor generalization ability when aerated clay experimental data is limited. More importantly, their predictions lack physical constraints, potentially leading to unreasonable "black box" results.

[0004] In summary, given the limited experimental data available, existing methods struggle to achieve a comprehensive prediction of the undrained behavior of aerated clay that combines high accuracy, physical plausibility, and quantifiable uncertainty. Summary of the Invention

[0005] The main objective of this invention is to provide a method for predicting and quantifying the uncertainty of undrained shear response of aerated clay, aiming to solve the problem that existing methods are unable to achieve full-path prediction of the undrained behavior of aerated clay with high accuracy, physical rationality, and quantifiable uncertainty.

[0006] To achieve the above objectives, this invention provides a method for predicting and quantifying the uncertainty of undrained shear response in aerated clay. The method constructs a hybrid framework integrating physical mechanisms and data-driven approaches, specifically including the following steps:

[0007] Step 1: Based on the constitutive model of saturated clay, generate a physically consistent undrained response simulation dataset;

[0008] Step 2: Construct a machine learning model as a bias learner. Use the simulated dataset and the measured test data of aerated clay in Step 1 as training samples, and take the response deviation of aerated clay relative to the saturated clay benchmark as the learning objective to train and obtain a bias prediction model.

[0009] Step 3: Using the trained bias prediction model, the full-path undrained response trajectory of the aerated clay is reconstructed step by step from the initial state through rolling integrals.

[0010] Step 4: The input parameters are perturbed based on Monte Carlo simulation to generate multiple predicted trajectories, and a prediction interval is constructed using quantile statistics to quantify the reliability of the prediction.

[0011] Optionally, in step 1, the constitutive model of the saturated clay is a modified Cambridge model, whose material parameters include the compression index λ, the resilience index κ, the critical stress ratio M, the normal consolidation intercept N, and the Poisson's ratio v; and by changing the overconsolidation ratio in the range of 1.0 to 1.3, different stress history conditions are simulated to generate a physically consistent undrained response simulation dataset.

[0012] Optionally, in step 2, the bias learner is a random forest model, whose input feature vector includes current and historical mechanical state variables, strain increments, and initial condition parameters; the learning objective is the strain-normalized response rate of the air-bearing clay relative to the saturated soil baseline, which includes the deviatoric stress change rate dq / dε. a Effective average stress change rate dp' / dε a and the rate of change of excess pore water pressure du w / dε a .

[0013] Optionally, three independent random forest regressors are used to predict the three response rate targets, namely the rate of change of partial stress dq / dε. a Effective average stress change rate dp' / dε a and the rate of change of excess pore water pressure du w / dε a .

[0014] Optionally, in step 2, the deviation prediction model step adopts a domain-weighted training strategy.

[0015] Optionally, in step 3, the specific process of rolling integration is as follows: the loading path is discretized into K axial strain increment steps; for the k-th step, the current state characteristics are input into the deviation prediction model to obtain the predicted response rate value f for the k-th step. θ (x k Then, through the formula... Calculate the response state for the current step, where y k =[q k ,p' k ,u w k ] T Let Δε represent the response state vector at the k-th increment step. ak This represents the axial strain increment at the k-th increment step.

[0016] Optionally, step 4 specifically includes:

[0017] Step 41: Determine the key input parameters and their nominal values. The key input parameters include the overconsolidation ratio OCR0 and the initial pore pressure u. w0 and nominal initial gas content ψ0;

[0018] Step 42: Randomly sample from the preset probability distribution of the key input parameters to generate n sets of perturbation input samples, where n≥300;

[0019] Step 43: For each group of perturbation samples, perform steps 2-3 to obtain a predicted trajectory, and obtain a total of n groups of trajectory samples;

[0020] Step 44: At the specified axial strain, calculate the mean, 5th percentile, and 95th percentile of the n sample values ​​for each response variable;

[0021] Step 45: Based on the percentiles and mean, construct the upper and lower bounds of the prediction interval for the response variable across the entire strain range.

[0022] Optionally, step 45 constructs the upper and lower bounds of the prediction interval in the following manner:

[0023] First, calculate the relative lower deviation at the strain endpoint. and relative deviation :

[0024] ;

[0025] Then, an transition function is used. The relative deviation at the terminal is smoothly expanded along the axial strain axis to obtain the prediction interval of the weighted path:

[0026] ;

[0027] in, For the mean response trajectory, Its terminal value.

[0028] Optionally, in step 45, the prediction intervals for different nominal initial gas contents ψ0 are obtained by performing Monte Carlo simulations under multiple typical gas contents conditions to obtain the corresponding prediction interval widths.

[0029] Furthermore, in order to achieve the above objectives, the present invention provides a system for predicting and quantifying the undrained shear response of aerated clay, the system comprising a memory, a processor, and a program stored in the memory and executable on the processor, the program being loaded and executed by the processor to implement the method for predicting and quantifying the undrained shear response of aerated clay as described in any of the preceding claims.

[0030] Beneficial effects:

[0031] (1) This invention pioneers a hybrid framework of "physical benchmark model + machine learning bias correction". By utilizing the mature modified Cambridge model to generate a large amount of physically consistent synthetic data, it effectively compensates for the lack of real experimental data on aerated clay. The unique "domain weighted training strategy" enables the model to accurately learn the bias caused by aerated characteristics from limited experimental data while adhering to basic physical laws. This method breaks through the bottlenecks of traditional physical models relying on a large amount of data for calibration and pure data-driven models relying on a large number of samples, and can still achieve high-precision prediction of the whole path response under the real-world condition of scarce data.

[0032] (2) Unlike the traditional "end-to-end" pure data-driven method for predicting stress-strain relationships, this invention uses the strain-normalized response rate as the learning target of the machine learning model and reconstructs the complete response path through the "rolling integral" operation. This "differential learning-integral reconstruction" mechanism essentially imposes constraints on the continuity, smoothness, and evolution logic of the predicted trajectory, effectively avoiding non-physical prediction results that violate the common sense of mechanics, such as oscillations and abrupt changes, which may be generated by pure black box models, and ensuring the physical credibility of the output path.

[0033] (3) This invention introduces an uncertainty quantification module based on Monte Carlo simulation and quantile statistics. By perturbing key input parameters (such as gas content and overconsolidation ratio) and statistically analyzing the quantiles of a large number of predicted trajectory samples, prediction intervals with different confidence levels can be constructed. For the first time, a quantitative reliability index is provided for machine learning prediction models of aerated clay, enabling engineers to intuitively assess the fluctuation range of predicted values, providing a key basis for risk assessment and decision-making, and significantly improving the practical value of the prediction results.

[0034] (4) The hybrid framework proposed in this invention has good modularity and universality. The physical benchmark model can be replaced with other constitutive models according to the specific soil type; the bias learner can also use different machine learning algorithms; and the uncertainty quantification module is independent of the specific prediction model and can be easily transferred to the prediction and uncertainty analysis of other geotechnical engineering parameters, providing a technical path that can be referenced for research in related fields. Attached Figure Description

[0035] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0036] Figure 1 This is a flowchart illustrating an embodiment of a method for predicting and quantifying the uncertainty of undrained shear response of aerated clay according to the present invention.

[0037] Figure 2 This is a schematic diagram of the training framework for the random forest biased learner in this invention;

[0038] Figure 3 This is a comparison chart of the root mean square error (RMSE) of different candidate learning models in this invention.

[0039] Figure 4 This is a comparison chart of the mean absolute error (MAE) of different candidate learning models in this invention;

[0040] Figure 5 This is a comparison chart of the errors and time consumption of different candidate learning models in this invention.

[0041] Figure 6 This is a comparison of the effects of different domain weighting ratios β / α on the deviatoric stress q-effective average stress p′ under the working conditions ψ=5.5% and OCR=1.12 in this invention, on the predicted response under different conditions.

[0042] Figure 7 This invention relates to the effect of different domain weighting ratios β / α on deviatoric stress q-axial strain ε under the working conditions ψ=5.5% and OCR=1.12. a A comparison of the impact of different β / α on the predicted response;

[0043] Figure 8 To illustrate the effect of different domain weighted ratios β / α on excess pore water pressure u under the operating conditions ψ=5.5% and OCR=1.12 in this invention. w –Axial strain ε a A comparison chart of the impact of the response;

[0044] Figure 9 This is a comparison diagram showing the influence of different domain weighting ratios β / α on the response path of deviatoric stress q–effective average stress p′ under the working conditions ψ=3.3% and OCR=1.11 in this invention;

[0045] Figure 10 The weighted ratio β / α of different domains affects the deviatoric stress q–axial strain ε under the working conditions ψ=3.3% and OCR=1.11 in this invention.a A comparison chart of the impact of the response;

[0046] Figure 11 The influence of different domain weighted ratios β / α on excess pore water pressure u under the operating conditions ψ=3.3% and OCR=1.11 in this invention. w –Axial strain ε a A comparison chart of the impact of the response;

[0047] Figure 12 This is a diagram showing the construction results of the effective stress path prediction interval under the working condition ψ=5.5% and OCR=1.12 in this invention;

[0048] Figure 13 The deviatoric stress q – axial strain ε is the stress under the working conditions ψ=5.5% and OCR=1.12 in this invention. a The result of constructing the prediction interval;

[0049] Figure 14 The excess pore water pressure u_w – axial strain ε under the working conditions ψ=5.5% and OCR=1.12 in this invention is... a The result of constructing the prediction interval;

[0050] Figure 15 This is a diagram showing the construction results of the effective stress path prediction interval under the working condition ψ=3.3% and OCR=1.11 in this invention;

[0051] Figure 16 The deviatoric stress q – axial strain ε under the working conditions ψ=3.3% and OCR=1.11 in this invention is... a The result of constructing the prediction interval;

[0052] Figure 17 The excess pore water pressure u in this invention is defined under the operating conditions ψ=3.3% and OCR=1.11. w –Axial strain ε a The result of constructing the prediction interval.

[0053] The realization of the objective, functional features and advantages of the present invention will be further explained in conjunction with the embodiments and with reference to the accompanying drawings. Detailed Implementation

[0054] It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0055] See Figure 1 This invention provides a flowchart illustrating a method for predicting the path behavior of undrained aerated clay and quantifying its uncertainty based on physical mechanisms and machine learning. The method includes the following steps:

[0056] Step 1 - Generating Physical Baseline: Based on the constitutive model of saturated clay, a physically consistent undrained response simulation dataset is generated. Specifically, based on a mature elastoplastic constitutive model of saturated clay (such as the modified Cambridge model), a large number of undrained shear response trajectories of saturated clay under different initial conditions (mainly referring to the overconsolidation ratio OCR varying within the range of 1.0 to 1.3) are generated through numerical simulation. Each trajectory fully includes the effective mean stress p', deviatoric stress q, and excess pore water pressure u. w With axial strain ε a Continuous evolutionary data.

[0057] This step generates a physically consistent "saturated soil baseline response database" covering various stress histories. With virtually zero experimental cost, it provides a massive amount of reliable training data that conforms to classical soil mechanics principles for subsequent machine learning models, fundamentally alleviating the constraint on model training caused by the scarcity of real experimental data for aerated clay.

[0058] Step 2 - Training the Deviation Prediction Model: Construct a machine learning model as the deviation learner. Use the simulated dataset from Step 1 and the measured test data of aerated clay as training samples. Use the response deviation of aerated clay relative to the saturated clay benchmark as the learning objective to train the deviation prediction model.

[0059] Among them, the measured mechanical response data of aerated clay samples during shearing was obtained by undrained triaxial compression tests, which constitutes the measured test data of aerated clay. This test data mainly includes data with different initial air contents. Axial strain under the condition Effective average stress eccentric stress and excess pore water pressure .

[0060] Furthermore, a random forest is preferred as the "biased learner" for constructing the machine learning model. Its input features are designed to include the current strain state, strain increment, stress history (OCR), initial conditions (including gas volume ψ, initial stress / pore pressure), and all mechanical states (q, p′, u) from the previous increment step. w A composite vector of η and η. The learning objective is set as the normalized rate of change of strain (including the rate of change of deviatoric stress dq / dε) of the air-bearing clay response relative to the saturated soil reference. a Effective average stress change rate dp' / dε a and the rate of change of excess pore water pressure du w / dε aA domain-weighted training strategy is adopted, assigning a base weight α to the physical baseline response trajectory generated by the modified Cambridge model and an additional weighting factor β to the air-bearing clay experimental data. By adjusting the weight ratio β / α (which is set to 20), the contributions of physical laws and experimental data are balanced to balance the constraints of physical laws and the learning of true bias. Preferably, three independent random forest regressors are used to predict three response rate targets respectively.

[0061] This step achieves a crucial transformation from "learning bias rate" to "generating a complete path." Its effects are mainly reflected in two aspects: First, it ensures the physical rationality of the predicted trajectory. Since the evolution at each step is guided by a model that has learned physical laws and is achieved through integration, the final path is inevitably continuous, smooth, and self-consistent, avoiding the non-physical interpretations that might arise from a purely data-driven model. Second, it achieves continuous prediction at arbitrary strain points, rather than just fitting discrete test points.

[0062] Step 3 - Reconstructing the full path response: Using the trained deviation prediction model, the full path undrained response trajectory of the aerated clay is reconstructed step by step from the initial state through rolling integration.

[0063] Specifically, from the known initial state of the experiment Initially, the axial strain loading process is discretized into hundreds of small incremental steps (e.g., K≥500). For the k-th step, the current state features are input into a trained "bias learner" to predict the response rate increment f for that step. θ (x k Subsequently, using the rolling integral formula... Perform recursive calculations to progressively accumulate the mechanical state (q) at each moment along the entire loading path. k ,p' k ,u w k This allows for the reconstruction of the complete response curve.

[0064] This step achieves a crucial transformation from "learning bias rate" to "generating a complete path." The effects are mainly reflected in two aspects: First, it ensures the physical rationality of the predicted trajectory. Since the evolution of each step is guided by a model that has learned physical laws and is achieved through integration, the final path is bound to be continuous, smooth, and self-consistent, avoiding the non-physical interpretations that may arise from a purely data-driven model. Second, it achieves continuous prediction at any strain point, rather than just fitting discrete test points.

[0065] Step 4 - Quantifying Prediction Uncertainty: Based on Monte Carlo simulation, the input parameters are perturbed to generate multiple prediction trajectories, and prediction intervals for quantifying prediction reliability are constructed through quantile statistics.

[0066] Specifically, key uncertainty input parameters affecting the prediction results are identified, including the overconsolidation ratio, initial pore pressure, and nominal initial gas content. Monte Carlo simulations are used to randomly sample these parameters extensively (e.g., n≥300 times) within their possible fluctuation range (typically described by a truncated normal distribution). The standard deviations of the overconsolidation ratio and initial pore pressure are preset constants, and the standard deviation of the nominal initial gas content is set as a nonlinear function of the nominal initial gas content. For each set of sampled parameters, steps 2-3 are repeated to generate a possible response trajectory.

[0067] Then, at specific axial strain locations (especially the endpoint), quantile statistics are performed on a certain response quantity for all trajectory samples, and these statistics are used to construct a prediction interval covering the entire strain range. Specifically, at a preset reference axial strain location, each response variable is calculated. The mean difference of n sample values 5th percentile and the 95th percentile And calculate the relative lower deviation relative to the mean according to the following formula. and relative deviation :

[0068] ;

[0069] For any experimental conditions, the corresponding terminal relative deviation can be obtained by piecewise linear interpolation based on the nominal initial gas content.

[0070] Use a monotonic, smooth transition function ,satisfy and The relative deviation at the terminal is smoothly expanded along the axial strain axis to construct the upper and lower bounds of the prediction interval across the entire strain range:

[0071] ;

[0072] in, The mean response trajectory predicted in step 4. Its terminal value;

[0073] The prediction intervals under different gas content conditions are output, and the prediction intervals are used to quantify the impact of key input parameter disturbances on the prediction of the undrained response of aerated clay.

[0074] Furthermore, the prediction intervals under different gas content conditions are constructed as follows: Monte Carlo simulations are performed when the nominal initial gas content is 0.02, 0.05, and 0.08, respectively, to obtain the prediction interval width corresponding to each gas content. For other gas content values, the corresponding prediction interval is determined by piecewise linear interpolation.

[0075] The core contribution of this step lies in providing a quantitative measure of uncertainty for machine learning predictions of aerated clay for the first time. This expands the prediction results from a single "mean trajectory" to a "reliable probability range." By using the prediction interval, the range of fluctuations in the prediction results caused by variations in input parameters (especially the gas content, which is difficult to measure precisely) can be intuitively assessed. This provides crucial reliability for engineering judgments and risk assessments, greatly enhancing the practical value and decision support capabilities of the prediction results.

[0076] These four steps are interconnected: Step 1 provides physical priors and a data foundation; Step 2 learns the specific biases caused by gas-bearing characteristics; Step 3 uses the biases to reconstruct a physically reasonable full path through integration; and Step 4 evaluates the impact of key parameter fluctuations on the final prediction. This entire methodology effectively solves the three major challenges of accuracy, physical reasonableness, and uncertainty quantification under limited data.

[0077] To enable those skilled in the art to better understand the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings.

[0078] Step 1: First, a physical baseline model of saturated clay is constructed. This model adopts the modified Cambridge model, and its specific material parameters and their physical meanings are shown in Table 1. These parameters represent the baseline mechanical behavior of saturated clay in the absence of gas influence.

[0079]

[0080] To simulate the effects of different stress histories, the overconsolidation ratio (OCR) was used as the primary variation parameter. For each numerical sample, the OCR value was randomly and uniformly sampled within the range of 1.0 to 1.3. This range corresponds to the normal consolidation to micro-overconsolidation state and is the region where the modified Cambridge model has good applicability. All samples were subjected to a uniform preconsolidation pressure. The initial confining pressure is then determined by Sure.

[0081] The undrained shear process is achieved through monotonic axial strain loading, with a constraint that the volumetric strain is zero. The maximum axial strain is set to This is sufficient to capture the complete response from the elastic segment to the elasto-plastic transition segment. The loading process is discretized into 500 incremental steps to ensure the continuity and smoothness of the evolution of stress, pore pressure, and effective stress paths. With the above settings, a total of 200 undrained response trajectories of saturated clay under different OCR conditions are generated, each trajectory containing the effective average stress. eccentric stress and excess pore water pressure With axial strain The entire path evolution data constitutes the physical baseline response trajectory for subsequent learning.

[0082] Step 2: Obtain a set of undrained triaxial test data for aerated clay samples. This set of data includes samples with different initial air contents. Axial strain under the condition Effective average stress eccentric stress and excess pore water pressure These experimental data represent the actual mechanical response of aerated clay and are used to revise and supplement the physical baseline model.

[0083] In this embodiment, four sets of undrained triaxial test data for aerated clay with different initial air contents (ψ=0.010, 0.024, 0.033, 0.055) were obtained for model construction and validation. To be concise and highlight the effects, the following section will use the two most representative sets of data (ψ=0.033 and ψ=0.055, corresponding to air contents of 3.3% and 5.5% respectively) as examples to demonstrate in detail the model prediction results, the influence of different domain weighting ratios, and the construction process of the uncertainty quantification interval. All four sets of data were used for model training and validation.

[0084] The main steps are as follows: to construct a data-driven bias learner to learn the response bias of aerated clay relative to saturated clay.

[0085] Step 21: Construct the input feature vector; for the k-th loading increment step, construct the input feature vector. It contains the following 12 features to comprehensively describe the current loading state and history. Feature Vector Represented as:

[0086] ;

[0087] in, Represented as current axial strain, For axial strain increment, For a high consolidation ratio, For the logarithmic value of the overconsolidated ratio, A g0 For initial gas content index parameters, Nominal initial gas content, p0 is the initial pore pressure, p0 is the initial average effective stress, and p0 is the deviatoric stress of the previous increment step. Effective average stress of the previous increment step The excess pore water pressure in the previous increment The stress ratio of the previous increment step can be expressed as:

[0088] ;

[0089] in, For the deviatoric stress of the previous increment step, This represents the effective average stress of the previous increment step.

[0090] Step 22: Determine the learning objective: Use the strain-normalized response rate as the learning objective, that is, predict the rate of change of each mechanical quantity with respect to axial strain under given input characteristics.

[0091] Specifically, the three learning objectives are: the rate of change of deviatoric stress with respect to axial strain. Rate of change of effective mean stress with respect to axial strain and the rate of change of axial strain with respect to excess pore water pressure .

[0092] Step 23: Construct a parallel random forest regressor; specifically, as follows... Figure 2 The diagram shows the training framework of the random forest biased learner, which uses three independent random forest regressors trained in parallel, each specifically responsible for predicting one of the three response rates. The random forest model was chosen for its robustness to feature scaling and nonlinear interactions. Comparative experiments yielded... Figures 3-5 The results show a comparison of the prediction performance of different candidate learning models.

[0093] Depend on Figure 3 It can be seen that, in terms of root mean square error (RMSE), the random forest model has a significant impact on the effective mean stress p′, deviatoric stress q, and excess pore water pressure u. w The prediction errors were 0.4646 kPa, 0.6549 kPa, and 0.3013 kPa, respectively, which were generally lower than those of the SVR and multi-output regression models; among them, the SVR model had the best prediction accuracy for q and u. w The RMSE values ​​were 8.729 kPa and 2.98 kPa, respectively, while the multi-output regression model reached 10.98 kPa and 7.894 kPa, respectively, indicating significantly larger errors.

[0094] Depend on Figure 4 It can be seen that, in terms of mean absolute error (MAE), the random forest model is effective for p′, q, and u. w The MAE values ​​were 0.1384 kPa, 0.2174 kPa, and 0.1142 kPa, respectively, remaining at a low level, indicating that they have relatively stable predictive ability for all three types of response variables.

[0095] Depend on Figure 5It can be seen that the peak error, critical state line error, and pore water pressure plateau error of the random forest model are 0.1698, 0.02466, and 0.09972, respectively, all of which are at a low level; at the same time, its computation time is 0.6359 s, which is significantly lower than that of LightGBM, SVR, and multi-output regression models.

[0096] The results above show that the random forest model not only has high accuracy in predicting effective mean stress, deviatoric stress and excess pore water pressure, but also balances computational efficiency and prediction accuracy of key mechanical state points. Therefore, it is suitable as the response prediction model in this method.

[0097] Step 24: Employ a domain-weighted training strategy; to address the issue of abundant synthetic data but scarce experimental data, assign basic weights to each sample in the physical baseline response trajectory (synthetic domain) generated in Step 1. A weighting factor is assigned to each sample in the air-bearing clay test data (experimental domain) obtained in step 2. By adjusting the weight ratio To balance the contributions of physical laws and experimental data.

[0098] Specifically, such as Figure 6-11 As shown, it illustrates the different domain weighting ratios. The impact on the predicted response. Among them, Figure 6 The figure shows a comparison between the predicted and experimental values ​​of the effective stress path under the operating condition (ψ=5.5%, OCR=1.12). As can be seen from the figure, under different domain weighting ratios, the predicted stress path generally maintains a good trend towards the critical state line, with the predicted result being closest to the experimental path when β / α=10. Figure 7 The figures show a comparison between predicted and experimental values ​​of the deviatoric stress-axial strain response under the same working condition. As can be seen from the figures, all predicted curves can effectively reflect the gradual stabilization of deviatoric stress with increasing axial strain. The larger domain weighting ratio more accurately reflects the development trend of deviatoric stress in the middle and later stages. Figure 8 The figure shows a comparison between the predicted and experimental values ​​of the pore water pressure-axial strain response under the same working condition. As can be seen from the figure, the pore water pressure gradually increases and tends to stabilize with the increase of axial strain. The predicted results can reflect the pore pressure development process well, and the results are closer to the experimental results when β / α = 10 to 20.

[0099] Figure 9 The figure shows a comparison between the predicted and experimental values ​​of the effective stress path under another working condition (ψ=3.3%, OCR=1.11). As can be seen from the figure, the overall trend of each predicted curve is consistent with the experimental path, and it can well reflect the coupled change process of the reduction of effective mean stress and the development of deviatoric stress. Figure 10The predicted results and experimental values ​​of the deviatoric stress-axial strain response under this working condition are shown. As can be seen from the figure, this method can predict the evolution of the deviatoric stress gradually stabilizing with increasing strain, and the predicted curve and the experimental curve are in good agreement overall. Figure 11 The figure shows a comparison between the predicted and experimental values ​​of the pore water pressure-axial strain response under this working condition. As can be seen from the figure, the predicted results can well reflect the trend of continuous increase of pore pressure with axial strain, while maintaining the continuity and smoothness of the response curve.

[0100] It can be seen that when When the strain is small, the predicted trajectory is generally close to the baseline of the modified Cambridge model for saturated clay, especially in the small strain stage, the deviatoric stress growth trend is basically consistent with the baseline model, and it fails to effectively reflect the response deviation caused by gas content; as As the strain increases, the predicted trajectory gradually approaches the experimental data curve. In the middle and late strain range, the evolution trend of deviatoric stress and pore water pressure is more consistent with the experimental results. At the same time, the stress path remains continuous and smooth and tends to the critical state line, indicating that the physical rationality of the mechanical response is maintained even when the influence of experimental data is introduced.

[0101] To determine the optimal weight ratio, this invention employs a sample validation method for quantitative evaluation. Specifically, a physical information-enhanced machine learning model is constructed and trained using each candidate weight ratio β / α value (1, 2, 5, 10, 20, 30). Subsequently, all models are used to predict on the same sample validation set (i.e., independent experimental data from four conditions ψ=0.010, 0.024, 0.033, 0.055, which do not participate in any model training). The model's response to various parameters (deviatoric stress q, effective mean stress p', excess pore water pressure u) is compared. w The predicted value y pred Its experimental observation value y true The root mean square error (RMSE) is calculated as the core performance indicator. The formula for calculating RMSE is:

[0102] ;

[0103] Where N is the number of samples in the validation set. The smaller the RMSE value, the higher the prediction accuracy of the model. The RMSE calculation results under different weight ratios are summarized in Table 2. Based on the data in Table 2, the model performance can be comprehensively evaluated. Regarding the prediction accuracy of q, as β / α increases from 1 to 5, the RMSE decreases significantly from 17.83 to 10.49, indicating that enhancing physical constraints greatly improves the prediction ability of deviatoric stress. Afterwards, in the range of 10-30, the RMSE stabilizes between 11.08 and 12.21, and the accuracy remains high. Regarding the prediction accuracy of p': it shows an overall downward trend, reaching the optimal value (59.76) when β / α=30, but the result (67.53) when β / α=20 is not much different and is at a good level. Regarding u w The prediction accuracy is most sensitive to the weight ratio. When β / α=1 (physical constraint weights are too low), the RMSE is as high as 306.75, indicating a complete prediction failure. As the physical constraint weights increase, the error decreases sharply, reaching lower levels of 56.53 and 57.60 when β / α=5 and 20, respectively. At that time, the model showed no predictive weaknesses in the prediction of any of the three response quantities, achieving "optimal" overall performance. However, when β / α continued to increase to 30, although it slightly improved p', it led to u w The prediction error increased significantly (from 57.60 to 75.51), disrupting the balance. Therefore, the overall evaluation should be... At that time, the model achieved an optimal balance in predicting the three response quantities, effectively learning the behavioral characteristics of aerated clay without excessively sacrificing physical consistency. Therefore, the optimal weight ratio was determined in this embodiment. .

[0104]

[0105] Step 3: Reconstruct the full-path undrained response trajectory of aerated clay using rolling integrals.

[0106] Specifically, using a trained bias learner, the complete response trajectory is reconstructed step by step through rolling integration; first, the initial response state is obtained. Effective average stress and excess pore water pressure; then discretize the loading path by adjusting the axial strain. The loading process is discretized into K incremental steps, and to ensure reconstruction accuracy, k=500 in this example. Then, rolling integral prediction is performed. For the k-th incremental step (k=1,2,…,K), the input feature vector for the current step is first constructed. Then, input this into the three random forest regressors trained in step 3 to predict the response rate vector for the current step:

[0107] ;

[0108] in This represents a supervised regression model. The complete undrained response trajectory is reconstructed using the rolling integral formula:

[0109] ;

[0110] Repeat the above process until all k incremental steps have been calculated, thus obtaining the complete undrained response trajectory of the aerated clay, i.e., the effective mean stress. eccentric stress and excess pore water pressure With axial strain The continuous evolution curve.

[0111] Step 4: Construction of prediction interval and uncertainty quantification based on Monte Carlo simulation; specifically, this step is used to quantify the impact of key input parameter perturbations on the prediction response and output prediction results with uncertainty boundaries; determine key input parameters and their nominal values, including the overconsolidation ratio. Initial pore pressure and nominal initial gas content .in It is the initial pore water pressure used to define the overpressure, satisfying... , Expressed as excess pore water pressure, This represents the current total pore water pressure.

[0112] In this example, a perturbation is applied to the key input parameters, and n sets of perturbed input samples are randomly sampled from a truncated normal distribution. n=300, and the sampling method is as follows:

[0113] ;

[0114] Among them, the standard deviation of the overconsolidation ratio and the standard deviation of initial pore pressure Set as a preset constant; initial gas content Standard deviation Set as nominal value Nonlinear functions.

[0115] For each set of perturbation input samples, repeat steps 2 and 3 to reconstruct a corresponding undrained response trajectory. A total of n sets of response trajectory samples are obtained, each trajectory containing... , and The continuous evolution of axial strain.

[0116] In this example, each response variable is calculated at the preset reference axial strain. The mean difference of n sample values 5th percentile and the 95th percentile And calculate the relative lower deviation relative to the mean according to the following formula. and relative deviation :

[0117]

[0118] For any experimental conditions, based on its nominal initial gas content The corresponding terminal relative deviation is obtained by piecewise linear interpolation. and In this example, it is pre-programmed in... Monte Carlo simulations were performed at three typical gas content levels of 0.02, 0.05, and 0.08 to obtain the corresponding relative deviations. For other gas contents, linear interpolation was used to determine the relative deviations.

[0119] Use a monotonic, smooth transition function The relative deviation at the terminal is smoothly expanded along the axial strain axis to construct the upper and lower bounds of the prediction interval across the entire strain range. The transition function satisfies and and has Continuity (using smoothstep polynomials):

[0120]

[0121] In this example, the upper and lower bounds of the prediction interval are:

[0122]

[0123] Where is the mean response trajectory predicted in step 4, and is its terminal value.

[0124] Finally, the prediction intervals under different gas content conditions are output, which are used to quantify the impact of key input parameter disturbances on the prediction of the undrained response of aerated clay.

[0125] In this example, to verify the effectiveness of the method of the present invention, a sample retention verification method is used. The four representative experimental conditions in step 2 ( The experimental data (e.g., 0.010, 0.024, 0.033, 0.055) were used as the validation set and were not used in model training. Figure 12-17 As shown, it illustrates the comparison between the predicted results of this method and the experimental values ​​under these four conditions, including the effective stress path ( Plane), deviatoric stress-strain ( ) and pore pressure-strain ( The figure shows that, under various working conditions, the predicted curves and the experimental curves are in good agreement on the overall trend and key characteristic points (such as the peak value of deviatoric stress and the development stage of pore pressure). They can accurately reproduce the process of stress path development towards the critical state line, indicating that this method can still achieve high-precision prediction of the undrained response of aerated clay under limited experimental data conditions.

[0126] in, Figure 12 The results of constructing the effective stress path prediction interval under the working condition (ψ=5.5%, OCR=1.12) are presented. As can be seen from the figure, the predicted mean trajectory generally maintains a trend towards the critical state line, and the upper and lower bounds of the prediction interval can well encompass the possible fluctuation range of the effective stress path, indicating that this method can effectively quantify the impact of key input parameter disturbances on the stress path prediction results. Figure 13 The results of constructing the prediction interval for the deviatoric stress-axial strain response under the same working condition are presented. As can be seen from the figure, the width of the prediction interval gradually increases with the increase of axial strain, indicating that the deviatoric stress response is more significantly affected by the disturbance of input parameters in the middle and later stages; at the same time, the mean prediction curve remains smooth and continuous overall, and can well reflect the evolution law of the deviatoric stress gradually stabilizing. Figure 14 The results of constructing the prediction interval for excess pore water pressure-axial strain response under the same working condition are presented. As can be seen from the figure, the prediction interval can well reflect the gradual development trend of pore water pressure with increasing axial strain, and the upper and lower boundaries are continuously and smoothly distributed, indicating that the method can stably quantify the uncertainty range of the pore pressure prediction results.

[0127] Figure 15 The results of constructing the effective stress path prediction interval are shown under another working condition (ψ=3.3%, OCR=1.11). As can be seen from the figure, the mean prediction trajectory and the upper and lower bounds of the prediction interval generally maintain a consistent evolution trend, and the prediction interval can cover the variation range of the effective stress path under different input disturbance conditions quite well. Figure 16 The results of constructing the prediction interval for the deviatoric stress-axial strain response under this working condition are presented. As can be seen from the figure, this method can well reflect the gradual development of deviatoric stress with increasing axial strain. At the same time, the boundary of the prediction interval is continuous and smooth, indicating that the model still has good stability and consistency under different working conditions. Figure 17 The results of constructing the prediction interval for the excess pore water pressure-axial strain response under this working condition are presented. As can be seen from the figure, the overall pore water pressure prediction results continuously increase with axial strain. The prediction interval can effectively characterize the response fluctuation range caused by the disturbance of key input parameters, demonstrating the effectiveness of this method in uncertainty quantification.

[0128] The above-described method for predicting the undrained behavior path and quantifying the uncertainty of aerated clay based on physical mechanisms and machine learning effectively solves the problem of predicting the undrained behavior of aerated clay under data-scarce conditions by constructing a hybrid framework of "physical baseline + machine learning correction" and combining it with Monte Carlo simulation. This provides a reliable technical means for related geotechnical engineering analysis and design.

[0129] The above are merely preferred embodiments of the present invention and do not limit the patent scope of the present invention. All equivalent structural transformations made using the contents of the present invention's specification and drawings under the inventive concept of the present invention, or direct / indirect applications in other related technical fields, are included within the patent protection scope of the present invention.

Claims

1. A method for predicting and quantifying the uncertainty of undrained shear response of aerated clay, characterized in that, The method constructs a hybrid framework that integrates physical mechanisms and data-driven approaches, specifically including the following steps: Step 1: Based on the constitutive model of saturated clay, generate a physically consistent undrained response simulation dataset; Step 2: Construct a machine learning model as a bias learner. Use the simulated dataset and the measured test data of aerated clay in Step 1 as training samples, and take the response deviation of aerated clay relative to the saturated clay benchmark as the learning objective to train and obtain a bias prediction model. Step 3: Using the trained bias prediction model, the full-path undrained response trajectory of the aerated clay is reconstructed step by step from the initial state through rolling integrals. Step 4: The input parameters are perturbed based on Monte Carlo simulation to generate multiple predicted trajectories, and a prediction interval is constructed using quantile statistics to quantify the reliability of the prediction.

2. The method according to claim 1, characterized in that, In step 1, the constitutive model of the saturated clay is a modified Cambridge model, whose material parameters include the compression index λ, the resilience index κ, the critical stress ratio M, the normal consolidation intercept N, and the Poisson's ratio v.

3. The method according to claim 1, characterized in that, In step 2, the bias learner is a random forest model, whose input feature vector includes current and historical mechanical state variables, strain increments, and initial condition parameters; the learning objective is the strain-normalized response rate of the air-bearing clay relative to the saturated soil benchmark, which includes the deviatoric stress change rate dq / dε. a Effective average stress change rate dp' / dε a and the rate of change of excess pore water pressure du w / dε a .

4. The method according to claim 3, characterized in that, Three independent random forest regressors were used to predict the rate of change of partial stress dq / dε. a Effective average stress change rate dp' / dε a and the rate of change of excess pore water pressure du w / dε a .

5. The method according to claim 1, characterized in that, In step 2, the deviation prediction model step adopts a domain-weighted training strategy.

6. The method according to claim 1, characterized in that, In step 3, the specific process of rolling integration is as follows: the loading path is discretized into K axial strain increment steps. For the k-th step, the current state characteristics are input into the deviation prediction model to obtain the predicted response rate value f for the k-th step. θ (x k Then, through the formula... Calculate the response state for the current step, where y k =[q k ,p' k ,u w k ] T Let Δε represent the response state vector at the k-th increment step. a k This represents the axial strain increment at the k-th increment step.

7. The method according to claim 1, characterized in that, Step 4 specifically includes: Step 41: Determine the key input parameters and their nominal values. The key input parameters include the overconsolidation ratio OCR0 and the initial pore pressure u. w0 and nominal initial gas content ψ0; Step 42: Randomly sample from the preset probability distribution of the key input parameters to generate n sets of perturbation input samples, where n≥300; Step 43: For each group of perturbation samples, perform steps 2-3 to obtain a predicted trajectory, and obtain a total of n groups of trajectory samples; Step 44: At the specified axial strain, calculate the mean, 5th percentile, and 95th percentile of the n sample values ​​for each response variable; Step 45: Based on the percentiles and mean, construct the upper and lower bounds of the prediction interval for the response variable across the entire strain range.

8. The method according to claim 1, characterized in that, In step 45, the upper and lower bounds of the prediction interval are constructed in the following way: First, calculate the relative lower deviation at the strain endpoint. and relative deviation : ； Then, an transition function is used. The relative deviation at the terminal is smoothly expanded along the axial strain axis to obtain the prediction interval of the weighted path: ； in, For the mean response trajectory, Its terminal value.

9. The method according to claim 7 or 8, characterized in that, In step 45, the prediction intervals for different nominal initial gas contents ψ0 are obtained by performing Monte Carlo simulations under multiple typical gas contents conditions to obtain the corresponding prediction interval widths.

10. A system for predicting and quantifying the uncertainty of undrained shear response in aerated clay, characterized in that, The system includes a memory, a processor, and a program stored in the memory and executable on the processor, which, when loaded and executed by the processor, implements the method for predicting and quantifying the undrained shear response of aerated clay as described in any one of claims 1 to 9.

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