A method for predicting nonlinear soil consolidation induced by groundwater based on a hard-constrained physical neural network

Abstract

This invention discloses a method for predicting consolidation of nonlinear soil layers induced by groundwater reduction based on a hard-constrained physical neural network. It establishes a nonlinear consolidation control equation applicable to any value of m, describing the process of groundwater level decline. A hard-constrained PINN model with an integrated output transformation function is constructed, where boundary and initial conditions are directly and rigorously embedded as "hard constraints" into the network's output layer. The network is trained using the residuals of the control equation as the sole loss function. Finally, by normalizing the input depth and time, high-precision consolidation degree can be output end-to-end without meshing. The hard-constraint mechanism of this invention ensures physical consistency and significantly improves accuracy compared to conventional PINN methods. It breaks through the theoretical limitations of traditional analytical solutions, handling arbitrary nonlinear soil parameters. The computational efficiency is 10-100 times higher than traditional numerical methods, providing an efficient and reliable new tool for real-time prediction and risk control of groundwater consolidation settlement in engineering projects such as foundation pit groundwater reduction.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of geotechnical engineering and artificial intelligence, specifically relating to a method for predicting nonlinear soil consolidation induced by groundwater reduction based on a hard-constrained physical neural network. Addressing the nonlinear consolidation settlement problem of the groundwater layer caused by engineering activities such as groundwater reduction in foundation pits, this method employs a hard-constrained physical-informed neural network (PINN) to achieve high-precision and high-efficiency prediction of soil consolidation. This method is particularly suitable for complex soil conditions where the ratio of the compression index to the permeability index, m = Cc / Ck, is an arbitrary positive real number. Background Technology

[0002] In engineering activities such as lowering the groundwater level in foundation pits and groundwater extraction, the decline in the groundwater level leads to an increase in the effective stress of the soil layer, triggering nonlinear consolidation settlement. For soft clay exhibiting significant nonlinearity in both permeability and compressibility, the consolidation process is jointly controlled by the compressibility index Cc and the permeability index Ck, and typically Cc ≠ Ck (i.e., m ≠ 1). Existing settlement prediction methods have significant limitations when dealing with such strongly nonlinear problems: (1) Limitations of analytical solutions: The analytical solutions of classical nonlinear consolidation theory only hold under the strong assumption that m=1, and cannot be applied to the common situation where Cc and Ck are not equal in real soil, leading to prediction bias.

[0003] (2) Inefficient numerical methods: Although numerical methods such as the finite element method and the finite difference method can handle nonlinear equations with m≠1, they require complex spatial-temporal grid division, iterative solution and convergence judgment, resulting in high computational costs and making it difficult to meet the needs of rapid prediction, parameter inversion and multi-condition comparative analysis in engineering design.

[0004] (3) Pure data-driven models have weak generalization ability: Machine learning models that rely entirely on historical monitoring data do not embed control equations describing physical processes. Their predictions heavily depend on the quality and coverage of training data. When data is scarce or extrapolated to engineering conditions that have not been experienced, they lack physical consistency guarantees and the prediction results are unreliable.

[0005] (4) Deficiencies of conventional physical information neural networks (PINN): As an emerging machine learning method based on physical laws, PINN constrains the neural network by adding the residuals of the control equations as a penalty term to the loss function. However, this "soft constraint" method is difficult to accurately satisfy complex boundary and initial conditions, which can easily lead to unstable training and large errors in the prediction results at the boundary (i.e., the "boundary leakage" problem), affecting the final accuracy and reliability.

[0006] Therefore, there is an urgent need for a new method for predicting consolidation settlement that can overcome the theoretical limitations of m=1, strictly guarantee physical constraints, and possess efficient computational capabilities, in order to meet the challenges of nonlinear consolidation analysis of unconfined layers in complex engineering environments. Summary of the Invention

[0007] This invention aims to overcome the shortcomings of the prior art and provide a method for predicting nonlinear soil consolidation induced by groundwater based on a hard-constrained physical neural network. The core innovation of this invention lies in designing a special output transformation function to directly and forcibly encode the upper boundary conditions and initial conditions as "hard constraints" into the structure of the neural network. This fundamentally solves the boundary non-satisfaction problem caused by "soft constraints" in conventional PINN, enabling fast and high-precision surrogate model solutions for nonlinear consolidation problems with arbitrary m values.

[0008] To achieve the above objectives, the present invention adopts the following technical solution: In a first aspect, the present invention provides a method for predicting the consolidation of groundwater-induced nonlinear soil layers based on a hard-constrained physical neural network, comprising the following steps: S1. Based on the theory of one-dimensional nonlinear consolidation induced by groundwater reduction, the nonlinear consolidation governing equations, boundary conditions, and initial conditions expressed in terms of pore water pressure u are established. This step establishes a complete physical and mathematical model of the problem.

[0009] S2. The governing equations and boundary conditions are dimensionlessly processed to obtain dimensionless governing equations and corresponding boundary and initial conditions with the compressibility-permeability index ratio m = Cc / Ck as the key parameter. This step simplifies the problem, highlights the core parameter m, and lays the foundation for meshless solution.

[0010] S3. Constructing a hard-constraint physical neural network model: A feedforward neural network (FNN) is used as an approximator, and the structure of the output transformation function is carefully designed so that the final output of the network can automatically and strictly satisfy the upper boundary conditions and initial conditions (i.e., "hard constraints"). For the impermeable condition of the lower boundary, a penalty term is added to the loss function to handle it (i.e., "soft constraints"). This is the core innovation of this invention.

[0011] S4. Model Training: For a given value of m, a large number of spatiotemporal configuration points are sampled in the normalized computational domain. The main body of the loss function is constructed using the residuals of the dimensionless governing equations obtained in step S2. Combined with the soft constraint term of the lower boundary, the network parameters are trained using an efficient optimization algorithm (such as Adam+L-BFGS) until the loss function converges.

[0012] S5. Model Prediction and Generalization: After training, the neural network becomes a high-precision surrogate model for a specific m-value condition. Inputting arbitrary normalized depth Z and time Tv, it can instantly and end-to-end output the corresponding degree of consolidation U. When changes in soil parameters cause a change in the m-value, simply repeating steps S3-S4 is sufficient to quickly obtain the surrogate model for the new condition, demonstrating excellent generalization and adaptability.

[0013] Furthermore, in step S1, the relationship between the soil void ratio and the effective stress is as follows:

[0014] In the formula, Effective stress; Porosity; The compression index (i.e.) The slope of the curve; This represents the initial effective stress of the soft soil layer. The void ratio of soft soil under initial effective stress.

[0015] Furthermore, in step S1, the relationship between the soil permeability coefficient and the effective stress is as follows:

[0016] In the formula, The penetration index (i.e.) The slope of the curve; is the initial permeability coefficient of the soft soil layer.

[0017] Furthermore, in step S1, the governing equation for the nonlinear consolidation of the soil is:

[0018] In the formula u is the excess pore water pressure. Let be the specific weight of water, and q be the total stress increment caused by the change in water level.

[0019]

[0020] In the formula, This represents the maximum increase in total stress caused by changes in water level. The maximum depth of precipitation, For the time it takes for precipitation to reach a steady state, γ and These are the natural and saturated unit weights of the soil, respectively.

[0021] Furthermore, in step S1, the boundary conditions and initial conditions of the equation are: hour:

[0022] hour:

[0023] hour:

[0024] In the formula .

[0025] Further, step S2 includes the following steps: S2.1 Order:

[0026] Therefore, the governing equations, boundary conditions, and initial conditions become:

[0027]

[0028]

[0029]

[0030] S2.2 Order:

[0031] Therefore, the governing equations, boundary conditions, and initial conditions become:

[0032]

[0033]

[0034]

[0035]

[0036] Furthermore, in step S3, the output transformation function is specifically:

[0037] in, A function to describe the piecewise load at the upper boundary. This is the raw output of the neural network. Through this transformation, it can be... It automatically satisfies the upper boundary conditions and initial conditions.

[0038] Furthermore, in step S4, the loss function is the weighted sum of the residuals of the PDE and the soft constraint boundary conditions:

[0039] in, The root mean square error of the residuals at the points within the domain for the dimensionless governing equations is given. The mean square error of the residuals at the boundary configuration points under the lower boundary conditions. and These are weighting coefficients, all set to 1.

[0040] In a second aspect, the present invention also provides a system for predicting nonlinear soil consolidation induced by groundwater, comprising: Memory and processor; The memory stores computer programs; When the processor executes the computer program, it implements the steps of the method for predicting the nonlinear soil consolidation caused by groundwater.

[0041] Thirdly, the present invention also provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, comprises the steps of the method for predicting the reduction of nonlinear soil consolidation induced by groundwater.

[0042] The beneficial effects of this invention are: (1) Strong physical consistency and high prediction accuracy: The "hard constraint" mechanism implemented through the output transformation function ensures that the critical boundaries and initial conditions are accurately satisfied, fundamentally eliminating the boundary leakage defect of conventional PINN. Figure 1 In the verification case shown, when m=1, the prediction curve of the method of this invention (hard-constrained PINN) almost completely coincides with the analytical solution, and the accuracy is significantly higher than that of the analytical solution. Figure 2 The example shown is a conventional soft constraint PINN.

[0043] (2) Breaking through the limitations of classical theory: It completely gets rid of the dependence of traditional analytical solutions on m=1, and can effectively solve nonlinear consolidation problems where m is any positive real number, greatly expanding the engineering application scope of the method and truly reflecting the nonlinear characteristics of different soils.

[0044] (3) Revolutionary improvement in computational efficiency: It adopts a meshless, end-to-end solution paradigm. After one offline training is completed, the online prediction only requires one forward propagation of the neural network, and the computation time is in the millisecond range. Compared with traditional numerical methods such as finite element and finite difference, the computational efficiency is improved by 10-100 times. It is particularly suitable for real-time prediction, parameter back analysis, reliability assessment and rapid comparison of a large number of working conditions.

[0045] (4) High practical value in engineering: The method and process are clear, and the input and output are directly connected to commonly used engineering parameters (depth, time, compression index, permeability index). It is easy to integrate into the existing geotechnical engineering investigation, design and monitoring analysis process, and provides a powerful tool for real-time prediction, risk assessment and intelligent control of settlement caused by foundation pit lowering and groundwater control projects. Attached Figure Description

[0046] 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 these drawings without creative effort.

[0047] Figure 1 A comparison curve of the prediction results and analytical solutions (m=1) of the hard-constrained physical neural network (PINNs-H) provided in the embodiments of the present invention.

[0048] Figure 2 A comparison graph showing the prediction results of conventional (soft-constrained) physical neural networks (PINNs) and the analytical solution (m=1) provided for the comparative example.

[0049] Figure 3 The degree of consolidation U predicted by the hard-constrained physical neural network under different m values ​​(m=0.5, 1, 2) provided in the embodiments of the present invention.

[0050] Figure 4 This is a physical schematic diagram illustrating the reduction of nonlinear consolidation induced by diving in one embodiment of the present invention.

[0051] Figure 5 This is a schematic diagram of the structure and output transformation principle of the hard-constrained physical neural network model used in this invention. Detailed Implementation

[0052] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. The following embodiments are for illustrative purposes only and do not constitute a limitation on the scope of protection of this invention.

[0053] A preferred embodiment of the present invention, taking a foundation pit lowering groundwater project in a soft soil area as an example, predicts the nonlinear consolidation settlement of the groundwater layer during the water level drop process (see...). Figure 4 The specific steps are as follows: S1. Establish the nonlinear consolidation control equations Consider a one-dimensional vertically consolidated problem with a soil layer of thickness H and an initial water level at ground level. The groundwater level is linearly lowered starting at t=0, and decreases by hc to a stable level at t=tc. This process induces a time-varying external load on the top of the soil layer.

[0054] The nonlinear stress-strain relationship and permeability of soil are described by the following model: Porosity e and effective stress Relationship (compressibility): (1) In the formula, Effective stress; Porosity; The compression index (i.e.) The slope of the curve; This represents the initial effective stress of the soft soil layer. The void ratio of soft soil under initial effective stress.

[0055] The relationship between void ratio e and permeability coefficient kv (permeability): (2) In the formula, The porosity corresponding to the initial effective stress. The initial permeability coefficient, This is the penetration index.

[0056] Based on Terzaghi's one-dimensional consolidation theory and the aforementioned nonlinear constitutive relations, the governing equation expressed in terms of excess pore water pressure u(z,t) can be derived: (3) in u is the excess pore water pressure. Let q be the specific weight of water, and q be the total stress increment caused by the change in water level.

[0057]

[0058] in, This represents the maximum increase in total stress caused by changes in water level. The maximum depth of precipitation, For the time it takes for precipitation to reach a steady state, γ and These are the natural and saturated unit weights of the soil, respectively.

[0059] The boundary conditions for the equation are: Upper boundary (z=0, permeable): (4) in ; Lower boundary (z=H, impermeable): (5) Initial conditions (t=0): (6) S2, Dimensionlessization of Governing Equations To simplify the equations and highlight the core parameter m=Cc / Ck, the following dimensionless processing is performed.

[0060] First, introduce variables. The governing equations (3) and the boundary and initial conditions (4)-(6) are transformed into equivalent forms in terms of μ.

[0061] Then, define the following dimensionless variables: Normalized depth:

[0062] Normalized time factor:

[0063] Normalized pore pressure / degree of consolidation:

[0064] Normalized effective stress:

[0065]

[0066] Normalized load parameters: , ,

[0067] Key parameter: m = Cc / Ck; Finally, we obtain the concise dimensionless governing equations and boundary value conditions: (7) Upper boundary (Z=0): (8) Lower boundary (Z=1): (9) Initial conditions (Tv=0): (10) Equation (7) is the core equation that needs to be solved in this invention, where m can take any positive real value.

[0068] S3. Construct a hard-constraint physical neural network model (see...) Figure 5 ) Construct a fully connected feedforward neural network (FNN) NN(Z, Tv; θ) as an approximator, where θ represents the network weights and bias parameters. The network input is a two-dimensional vector (Z, Tv).

[0069] The key to this invention lies in designing the following output transformation function to achieve hard constraints: (11) in, : Represents an embedded deep neural network, serving as the basic mapping unit for fitting the spatiotemporal evolution law; : Represents the training parameters of the neural network described above, which are iteratively updated by minimizing the physical information loss function; It is a piecewise function strictly defined according to the upper boundary condition (8): (12) The ingenious aspect of this design is: When Z = 0, The upper boundary condition (8) is automatically and accurately satisfied.

[0070] when When = 0, and Therefore 0, automatically and precisely satisfy the initial condition (10).

[0071] For the lower boundary condition (9), this invention employs a soft constraint, that is, adding a penalty term to the loss function. This is because the Neumann boundary condition can usually achieve satisfactory accuracy through soft constraints, while keeping the network structure relatively simple.

[0072] The network structure can be specifically set as follows: Input layer (2 nodes) -> 3 hidden layers (50 nodes each, using the tanh activation function) -> Output layer (1 node). The weights are initialized using a Glorot normal distribution.

[0073] S4, Model Training The computational domain is defined as Z∈[0, 1], Tv∈[0, 1]. Nf = 5000 points {Zi, Tvi} are randomly and uniformly sampled within the domain to compute the residuals of the governing equations, and Nb = 500 points are sampled at the lower boundary (Z=1) for soft constraints. The above number of points is the optimal result after convergence testing, achieving a good balance between ensuring accuracy and computational cost.

[0074] The loss function is constructed as follows: (13) in: The residual R of the governing equation (7) is the mean square error at all points in the domain. R is calculated using automatic differentiation techniques. For Z and The partial derivatives are obtained. This represents the total number of pre-defined physical information configuration points within the solution domain.

[0075] It is the mean square error of the residual of the lower boundary condition (9) at the boundary configuration point. This indicates the total number of pre-set physical information configuration points on the boundary.

[0076] and These are weighting coefficients, all set to 1.0 in this example to balance the two losses. The training employs a two-stage optimization strategy: 1. Adam pre-training: Using the Adam optimizer with a learning rate of 1e-3, iterate for about 20,000 steps to perform global and fast initial optimization.

[0077] 2. L-BFGS fine-tuning: Using the L-BFGS optimizer, the results of the previous stage are used as initial values ​​for fine-tuning until the loss function converges to a stable minimum (e.g., on the order of 1e-6).

[0078] To further improve accuracy, a periodic adaptive resampling strategy can be adopted during training. For example, every 3000 steps, some domain configuration points can be resampled based on the current residual size to more efficiently reduce the residuals in high error regions.

[0079] The periodic adaptive resampling strategy is as follows: every 3000 training steps, calculate the absolute value of the PDE residuals on all configuration points in the domain, discard the 20% of old points with the smallest residuals, and resample the same number of new points in the current region with larger residuals and add them to the training set.

[0080] S5, Prediction, Validation and Application After training converges, the network This refers to a high-precision surrogate model trained for a specific value of m.

[0081] Verification: To verify the superiority of this invention (hard-constrained PINN), a special case of m=1 is selected (in which case a classical analytical solution exists). Under the same parameters and training settings, the prediction results of this invention are compared with those of conventional PINN (no output transformation, all boundaries and initial conditions are softly constrained by the loss function). Figure 1As shown, the predicted curve of this invention almost completely overlaps with the analytical solution curve, with a calculated root mean square error (RMSE) of 1e-3, a mean square error (MAE) of 1e-3, and an R² of 0.99996. Figure 2 The results show that the predictions of conventional PINN, especially in the near-boundary region, deviate noticeably from the analytical solutions, with a root mean square error (RMSE) of 1e-2, a mean square error (MAE) of 4e-3, and an R² of 0.998, significantly lower than that of this invention. This comparison strongly demonstrates the crucial role of hard constraint mechanisms in improving the accuracy of PINN solutions to boundary value problems.

[0082] Application: For practical engineering projects, Cc and Ck of the upper layer of the site are first determined through indoor experiments, and the m value is calculated. Then, the method of this invention is used to train a consolidation field surrogate model corresponding to the m value. In actual prediction, engineers only need to input the normalized depth Z and time Tv of interest, and the model can output the consolidation distribution U(Z, Tv) of the entire soil layer within milliseconds, thereby quickly calculating the settlement over time and providing immediate and reliable data for construction progress control, risk assessment, and decision support.

[0083] When site soil parameters change or the impact of different m values ​​needs to be analyzed, simply repeat the training process of steps S3-S4 with the new m value as the target to quickly obtain a new surrogate model. This demonstrates good flexibility and engineering applicability. A comparison of consolidation conditions for different m values ​​can be found in [reference needed]. Figure 3 .

[0084] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the appended claims.

Claims

1. A method for predicting groundwater-induced nonlinear soil consolidation based on a hard-constrained physical neural network, characterized in that, Includes the following steps: S1 is based on the theory of reducing one-dimensional nonlinear consolidation induced by groundwater, and establishes the nonlinear consolidation control equation and its boundary conditions and initial conditions expressed in terms of pore water pressure u. S2 performs dimensionless processing on the control equations and boundary conditions to obtain dimensionless control equations with the compression-permeability index ratio m as the key parameter, as well as the corresponding boundary conditions and initial conditions. S3 constructs a hard-constraint physical neural network model: It adopts a feedforward neural network and designs the structure of the output transformation function so that the network output automatically and strictly satisfies the upper boundary conditions and initial conditions; soft constraints are used for the lower boundary conditions. S4 model training: For a specified value of m, sample spatiotemporal configuration points in the computational domain, construct a loss function using the residuals of the aforementioned dimensionless control equations, and train the network using an optimization algorithm until the loss converges; S5 Model Prediction and Generalization: After training, inputting depth and time will directly output the corresponding degree of consolidation U; for different m values, repeating steps S3-S4 will yield a high-precision solution for the corresponding working condition.

2. The method according to claim 1, characterized in that, In step S1, the relationship between the soil void ratio and the effective stress is as follows: In the formula, Effective stress; Porosity; The compression index (i.e.) The slope of the curve; This represents the initial effective stress of the soft soil layer. The void ratio of soft soil under initial effective stress.

3. The method according to claim 1, characterized in that, In step S1, the relationship between the soil permeability coefficient and the effective stress is as follows: In the formula, The penetration index (i.e.) The slope of the curve; is the initial permeability coefficient of the soft soil layer.

4. The method according to claim 1, characterized in that, In step S1, the governing equation for the nonlinear consolidation of the soil is: In the formula u is the excess pore water pressure. Let be the specific weight of water, and q be the total stress increment caused by the change in water level. In the formula, This represents the maximum increase in total stress caused by changes in water level. The maximum depth of precipitation, For the time it takes for precipitation to reach a steady state, γ and These are the natural and saturated unit weights of the soil, respectively.

5. The method according to claim 1, characterized in that, In step S1, the boundary conditions and initial conditions of the equation are as follows: hour: hour: hour: In the formula .

6. The method according to claim 1, characterized in that, Step S2 includes the following steps: S2.1 Order: Therefore, the governing equations, boundary conditions, and initial conditions become: S2.2 Order: Therefore, the governing equations, boundary conditions, and initial conditions become: 。 7. The method according to claim 1, characterized in that, In step S3, the output transformation function is specifically: in, A function to describe the piecewise load at the upper boundary. This is the raw output of the neural network. Through this transformation, it can be... It automatically satisfies the upper boundary conditions and initial conditions.

8. The method according to claim 1, characterized in that, In step S4, the loss function is the weighted sum of the residuals of the PDE and the soft constraint boundary conditions: in, The root mean square error of the residuals at the points within the domain for the dimensionless governing equations is given. The mean square error of the residuals at the boundary configuration points under the lower boundary conditions. and These are weighting coefficients, all set to 1.

9. A system for predicting nonlinear soil consolidation induced by groundwater, characterized in that, include: Memory and processor; The memory stores computer programs; When the processor executes the computer program, it implements the steps of the method for predicting nonlinear soil consolidation induced by groundwater as described in any one of claims 1 to 8.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the steps of the method for predicting nonlinear soil consolidation induced by groundwater as described in any one of claims 1 to 8.

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