Soil hydraulic coupling simulation method and system based on physical information neural network

Abstract

The invention discloses a soil hydraulic coupling simulation method and system based on a physical information neural network. The method comprises the following steps: firstly, constructing two improved full-connection neural networks for respectively predicting pore water pressure and deformation response in a soil body; according to a soil hydraulic coupling partial differential equation set, non-dimensionalization control equation loss items of a liquid phase and a solid phase are established based on a fixed stress splitting method, and a total loss function of the liquid phase and the solid phase is established in combination with an initial boundary value condition; and finally, alternately training loss functions of a liquid phase and a solid phase based on a sequential training strategy and a gradient normalization self-adaptive loss balance strategy. According to the method, a sequential training strategy, an improved full-connection neural network and a self-adaptive loss balance scheme are adopted to improve a traditional physical information neural network, and prediction precision and training stability are improved. The trained physical information neural network can be used as an agent model to predict pore water pressure and deformation, and has great application potential.

Description

Technical Field

[0001] The present invention relates to the technical field of physical neural networks, and in particular to a method and system for simulating soil hydraulic coupling based on a physics-informed neural network. Background Art

[0002] The hydraulic coupling characteristics of soil are an important physical process in geotechnical engineering, which affects the safety of natural or artificial structures such as slopes, foundation pits, and dams. Simulating the water - force coupling behavior of soil based on numerical methods has been widely studied. These traditional numerical methods are based on physical laws described by partial differential equations. However, due to the inconsistency between input parameters and actual site conditions, they often have difficulty in capturing the real soil response. The latest progress in machine learning technology has introduced data - driven surrogate models, which can establish direct non - linear mappings to provide practical predictions without any prior conditions. However, without sufficient big data, the prediction performance and generalization of pure data - driven models cannot be guaranteed. In addition, as black - box data - driven models usually cannot reveal potential physical mechanisms, resulting in weak physical interpretability. Considering the sparse characteristics of on - site monitoring data, developing a hybrid method of physical information and data - driven is the best choice.

[0003] Physics - informed neural networks provide a promising paradigm for developing hybrid methods of physical information and data - driven. This method uses neural networks to approximate the unknown solution that satisfies physical laws and monitoring information. Physics - informed neural networks can be used as pure partial differential equation solvers for forward simulation or as tools for inverse analysis to estimate unknown parameters. However, the training of physics - informed neural networks is very difficult, especially for multi - physical - field and multi - scale physical problems. The soil hydraulic coupling problem is a typical multi - physical - field problem, and the existing physics - informed neural network methods for predicting soil hydraulic coupling behavior often present problems of low prediction accuracy and unstable training. Summary of the Invention

[0004] The purpose of the present invention is to solve the problem that the existing physics - informed neural network methods are difficult to accurately simulate the process of soil hydraulic coupling behavior, and to provide a method and system for simulating soil hydraulic coupling based on a physics - informed neural network.

[0005] The specific technical solution adopted by the present invention is as follows: In the first aspect, the present invention provides a method for simulating soil hydraulic coupling based on a physics - informed neural network, which includes: S1: For the target site that needs to simulate soil hydraulic coupling, obtain the spatial domain, time domain, initial - boundary value conditions, and soil parameters of the simulation; S2. Construct a liquid-phase physics-informed neural network for predicting pore water pressure in soil, and set the weighted sum of the loss terms of the dimensionless control equation of the liquid phase, the loss terms of the liquid-phase boundary conditions, and the loss terms of the liquid-phase initial conditions as the corresponding total loss function; construct a solid-phase physics-informed neural network for predicting the deformation in soil, and set the weighted sum of the loss terms of the dimensionless control equation of the solid phase, the loss terms of the solid-phase boundary conditions, and the loss terms of the solid-phase initial conditions as the corresponding total loss function; wherein, the loss terms of the dimensionless control equation of the liquid phase and the loss terms of the dimensionless control equation of the solid phase are constructed in a dimensionless manner after converting the soil hydro-mechanical coupled partial differential equations by the fixed stress splitting method. S3. Based on the sequential training strategy, alternately train the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network iteratively until the iteration termination condition is reached, and use the two trained networks to predict the pore water pressure and deformation of the soil; in each iteration process, alternately freeze the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network, and sample the collocation points from the spatial domain and the temporal domain and input them into the two physics-informed neural networks to obtain the predicted values of the pore water pressure and deformation. Finally, calculate the value of the total loss function corresponding to the unfrozen physics-informed neural network, and update the network parameters of the unfrozen physics-informed neural network by backpropagation according to the value of the total loss function.

[0006] Preferably, as in the first aspect above, both the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network adopt the same fully connected network with an attention mechanism. The network input is the collocation points composed of the spatial points collected from the spatial domain and the temporal points collected from the temporal domain. The collocation points are respectively mapped to the first attention feature and the second attention feature through two additional fully connected layers, and then the collocation points, the first attention feature, and the second attention feature are input into the multi-layer sequential cascaded network layers. The first network layer directly takes the collocation points as the input and obtains the output feature through the fully connected layer. In the remaining network layers, the output feature of the previous network layer is first passed through the fully connected layer to obtain the intermediate feature H, and then the first attention feature and the second attention feature are used to weight 1 - H and H respectively and then added together to obtain the output feature of the current network layer.

[0007] Preferably, as in the first aspect above, in the total loss function of the liquid-phase physics-informed neural network, The loss term of the dimensionless control equation of the liquid phase is the deviation between the dimensionless liquid-phase continuity balance equation after conversion by the fixed stress splitting method and zero; the method of conversion by the fixed stress splitting method is to replace the derivative term of the volumetric strain with respect to time in the liquid-phase continuity balance equation of the soil hydro-mechanical coupled partial differential equations with the derivative term of the mean volumetric stress with respect to time. The loss term of the liquid-phase initial condition is the deviation between the predicted value and the true value of the pore water pressure at the initial time point. The loss term of the liquid-phase boundary condition is composed of the deviation between the predicted value and the true value of the pore water pressure on the Dirichlet boundary and the deviation between the predicted value and the true value of the flow velocity on the Neumann boundary condition.

[0008] As a preference of the first aspect above, in the total loss function of the solid-phase physics-informed neural network, The loss term of the dimensionless control equation for the solid phase is: the deviation between the dimensionless stress-deformation equilibrium equation in the soil hydraulic coupling partial differential equation system and zero; The loss term of the solid-phase initial condition is the deviation between the predicted value and the true value of the soil deformation at the initial time point; The loss term of the solid-phase boundary condition is composed of the deviation between the predicted value and the true value of the soil deformation on the Dirichlet boundary and the deviation between the predicted value and the true value of the stress on the Robin mixed boundary condition.

[0009] As a preference of the first aspect above, when alternately freezing the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network and optimizing the other unfrozen physics-informed neural network, it is necessary to first sample a first configuration point sample composed of spatial points and time points from the complete spatial domain and time domain, sample time points from the complete time domain and form a second configuration point sample with the spatial points at the simulated space boundary, sample spatial points from the complete spatial domain and form a third configuration point sample with the initial time point corresponding to the simulated initial moment, input the three types of samples obtained by sampling into the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network respectively to obtain the predicted value of the soil pore water pressure and the predicted value of the soil deformation; then according to the total loss function corresponding to the unfrozen physics-informed neural network, calculate the loss term of the dimensionless control equation in the total loss function from the predicted value of the first configuration point sample, calculate the loss term of the boundary condition in the total loss function from the predicted value of the second configuration point sample, calculate the loss term of the initial condition in the total loss function from the predicted value of the third configuration point sample, and finally obtain the total loss function value by weighting and update the network parameters of the unfrozen physics-informed neural network using the gradient descent optimization algorithm.

[0010] As a preference of the first aspect above, a gradient normalization adaptive loss balance strategy is introduced in each iteration process to adaptively adjust the weighting weights of the three loss terms in the total loss function.

[0011] In the second aspect, the present invention provides a soil hydraulic coupling simulation system based on a physics-informed neural network, which includes: An initial parameter setting module, which is used to obtain the simulated spatial domain, time domain, initial and boundary conditions, and soil parameters for the target site that needs to perform soil hydraulic coupling simulation; The model and loss construction module is used to construct a liquid-phase physics-informed neural network for predicting pore water pressure in soil, and set the weighted sum of the liquid-phase non-dimensionalized control equation loss term, the liquid-phase boundary condition loss term, and the liquid-phase initial condition loss term as the corresponding total loss function; construct a solid-phase physics-informed neural network for predicting deformation in soil, and set the weighted sum of the solid-phase non-dimensionalized control equation loss term, the solid-phase boundary condition loss term, and the solid-phase initial condition loss term as the corresponding total loss function; wherein, the liquid-phase non-dimensionalized control equation loss term and the solid-phase non-dimensionalized control equation loss term are constructed after transforming the soil hydro-mechanical coupling partial differential equation set based on the fixed stress splitting method. The model training and prediction module is used to alternately train the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network iteratively based on the sequential training strategy until the iteration termination condition is reached, and use the two trained networks to predict the pore water pressure and deformation of the soil; in each round of iteration, alternately freeze the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network, sample configuration points from the spatial domain and the temporal domain and input them into the two physics-informed neural networks to obtain the predicted values of the pore water pressure and deformation, and finally calculate the total loss function value corresponding to the unfrozen physics-informed neural network, and update the network parameters of the unfrozen physics-informed neural network by backpropagation according to the total loss function value.

[0012] In a third aspect, the present invention provides a computer program product, including computer programs / instructions, which when executed by a processor, can implement the soil hydro-mechanical coupling simulation method based on a physics-informed neural network as described in any one of the above first aspect solutions.

[0013] In a fourth aspect, the present invention provides a computer-readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, it can implement the soil hydro-mechanical coupling simulation method based on a physics-informed neural network as described in any one of the above first aspect solutions.

[0014] In a fifth aspect, the present invention provides a computer electronic device, which includes a memory and a processor; The memory is used to store a computer program; The processor is used to, when executing the computer program, implement the soil hydro-mechanical coupling simulation method based on a physics-informed neural network as described in any one of the above first aspect solutions.

[0015] Compared with the prior art, the beneficial effects of the present invention are as follows: 1) The present invention provides a method for simulating soil hydro-mechanical coupling based on a physics-informed neural network, which improves the traditional physics-informed neural network by adopting a sequential training strategy and a fully connected neural network with an attention mechanism, thereby enhancing the prediction accuracy and training stability. The trained physics-informed neural network can be used as a surrogate model for pore water pressure and deformation prediction. Compared with the traditional finite element method, the present invention has the advantages of being meshless and data-driven, and has physical interpretability compared with a pure data-driven black box model. According to this method, inverse analysis and surrogate models can be further established to estimate unknown soil parameters and constitutive models, and used as a hybrid model integrating physical information and data-driven to predict soil responses in unexplored spatio-temporal domains, showing great application potential.

[0016] 2) The sequential training strategy introduced in the present invention avoids the simultaneous training of the solid phase and liquid phase networks, and approaches the true solution in an iterative training manner. This training method reduces the difficulty of network optimization, makes it easier to find the optimal solution, and thus has higher accuracy. In contrast, the traditional co-training method simultaneously optimizes the two networks of the solid phase and liquid phase, inevitably increasing the training difficulty and resulting in a worse accuracy of the obtained solution.

[0017] 3) Inspired by the neural attention mechanism commonly used in computer vision, the present invention introduces two additional mapping networks into the traditional fully connected neural network. These two additional networks map the input variables into a high-dimensional feature space as attention weight features, and improve the forward propagation process within the hidden layer of the traditional fully connected neural network, reducing the maximum eigenvalue of the Hessian matrix during the training process, and thus enhancing the training stability.

[0018] 4) The present invention can also introduce an adaptive loss balancing strategy during the training process, enabling the model to more evenly consider each loss term during training, achieving the balance of the backpropagation gradients of each loss term, and thus improving the training accuracy and generalization ability. Description of the Drawings

[0019] Figure 1 is the flowchart of the steps of the method for simulating soil hydro-mechanical coupling based on a physics-informed neural network; Figure 2 is the schematic diagram of the sequential training strategy Figure 3 is the schematic diagram of the module composition of the system for simulating soil hydro-mechanical coupling based on a physics-informed neural network; Figure 4 is the schematic diagram of the structure of a computer electronic device; Figure 5 is the schematic diagram of a one-dimensional saturated soil model under the action of overlying load. Detailed Embodiments

[0020] To make the above objects, features, and advantages of the present invention more apparent and understandable, the following provides a detailed description of the specific embodiments of the present invention in conjunction with the accompanying drawings. Many specific details are set forth in the following description to facilitate a full understanding of the present invention. However, the present invention can be implemented in many other ways different from those described herein, and those skilled in the art can make similar improvements without departing from the connotation of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed below. The technical features in each embodiment of the present invention can be combined correspondingly without conflict.

[0021] As Figure 1 shown, in a preferred embodiment of the present invention, a method for simulating soil-water coupling based on a physics-informed neural network is provided, and the specific steps are as follows: S1: For the target site that needs to simulate soil-water coupling, obtain the spatial domain, time domain, initial and boundary conditions, and soil parameters of the simulation.

[0022] It should be noted that the target site in the present invention refers to the object that needs to simulate soil-water coupling, such as natural or artificial structures such as slopes, foundation pits, and dams, and the specific object is not limited. In addition, the spatial domain, time domain, and initial and boundary conditions of the simulation can all be determined according to the simulation scheme, and the soil parameters are determined according to the actual situation of the site. The spatial domain refers to the three-dimensional spatial range of the simulation object, the time domain refers to the time range of the simulation, that is, the time interval from the initial moment to the termination moment. Both the spatial domain and the time domain can be discretized into a series of spatial points and time points, and a configuration point is composed of a spatial point and a time point. The initial and boundary conditions refer to the initial conditions and boundary conditions, and the specific initial conditions and boundary conditions need to be determined according to the actual simulation conditions.

[0023] S2. Construct a liquid-phase physics-informed neural network for predicting the pore water pressure in the soil, and set the weighted sum of the liquid-phase dimensionless control equation loss term, the liquid-phase boundary condition loss term, and the liquid-phase initial condition loss term as the corresponding total loss function; construct a solid-phase physics-informed neural network for predicting the deformation in the soil, and set the weighted sum of the solid-phase dimensionless control equation loss term, the solid-phase boundary condition loss term, and the solid-phase initial condition loss term as the corresponding total loss function; wherein, the liquid-phase dimensionless control equation loss term and the solid-phase dimensionless control equation loss term are constructed in a dimensionless manner after converting the soil-water coupling partial differential equation set by the fixed stress splitting method.

[0024] In an embodiment of the present invention, both the liquid-phase physical information neural network and the solid-phase physical information neural network adopt the same improved fully connected network with an attention mechanism. The improved fully connected neural network is inspired by the neural attention mechanism commonly used in computer vision, and two additional fully connected layers are introduced into the traditional fully connected neural network as mapping networks. These two additional mapping networks map the input variables into a high-dimensional feature space and improve the forward propagation process within the hidden layer of the traditional fully connected neural network.

[0025] The specific network structure of the improved fully connected network is described below. In this improved fully connected network, the network input is the spatial points collected in the spatial domain x and the temporal points collected in the temporal domain t that form the configuration points. The configuration points are mapped into the first attention feature and the second attention feature respectively through two additional introduced fully connected layers, and then the configuration points, the first attention feature, and the second attention feature are input into the multi-layer sequential cascaded network layers. Among them, the first network layer directly takes the configuration points as the input and obtains the output feature through the fully connected layer. In the remaining network layers, first, the output feature of the previous network layer is passed through the fully connected layer to obtain the intermediate feature H, and then the first attention feature and the second attention feature are used to weight 1 - H and H respectively and then added together to obtain the output feature of the current network layer. For the sake of distinction, the variable l is used to represent the number of the network layer, and the intermediate feature H corresponding to the l th network layer is denoted as . The specific forward propagation process can be expressed by the formula:

[0026] In the formula, x and t are the input spatial point coordinates and temporal point coordinates, represents the concatenation operation on x and t ; represents the output feature of the lth network layer; W and b are the weights and biases, and their superscripts are used to distinguish different network layers; is the activation function, and the tanh activation function is preferably adopted in the embodiment of the present invention; ⊙ represents element-wise multiplication. This improved fully connected neural network can reduce the maximum eigenvalue of the Hessian matrix in the training process compared with the traditional fully connected neural network, thereby improving the training stability.

[0027] In addition, it should be noted that both the liquid-phase physical information neural network and the solid-phase physical information neural network need to set the total loss function required for network training. When training each physical information neural network, the corresponding total loss function needs to be called to calculate the total loss value. The loss functions of the two physical information neural networks are similar in form and are obtained by weighting a total of three loss terms: the dimensionless control equation loss term, the boundary condition loss term, and the initial condition loss term. However, since one of the two physical information neural networks predicts for the solid phase (deformation) and the other predicts for the liquid phase (pore water), there are differences in their control equations, boundary conditions, and initial conditions.

[0028] In the embodiment of the present invention, the total loss function for training the liquid-phase physical information neural network is expressed as:

[0029] Where: 、 、 are the dimensionless control equation loss term of the liquid phase, the boundary condition loss term of the liquid phase, and the initial condition loss term of the liquid phase, respectively. All three are related to and ; 、 、 are the weights of the corresponding loss terms, respectively, and are the network parameters of the solid-phase physical information neural network and the liquid-phase physical information neural network, respectively.

[0030] In the embodiment of the present invention, the total loss function for training the solid-phase physical information neural network is expressed as:

[0031] Where, 、 、 are the dimensionless control equation loss term of the solid phase, the boundary condition loss term of the solid phase, and the initial condition loss term of the solid phase, respectively. All three are related to and ; 、 、 are the weights of the corresponding loss terms, respectively, and are the network parameters of the solid-phase physical information neural network and the liquid-phase physical information neural network, respectively.

[0032] The specific forms of the above two total loss functions of the present invention are related to the selected partial differential equations of soil hydraulic coupling and the corresponding initial and boundary conditions. It is necessary to establish the loss terms of the dimensionless control equations for the liquid phase and the solid phase respectively based on the fixed stress splitting method according to the selected partial differential equations of soil hydraulic coupling, and then combine the boundary conditions and initial conditions in the initial and boundary conditions to establish the other two loss terms for the liquid phase and the solid phase, and further construct the total loss function.

[0033] In the embodiment of the present invention, the total loss function of the above liquid-phase physical information neural network The specific calculation methods of the three loss terms in it are as follows: Loss term of the dimensionless control equation for the liquid phase It is: the deviation between the dimensionless liquid-phase continuity balance equation after conversion by the fixed stress splitting method and zero; the conversion method of the fixed stress splitting method is to replace the derivative term of the volumetric strain with respect to time in the liquid-phase continuity balance equation of the partial differential equations of soil hydraulic coupling with the derivative term of the mean volumetric stress with respect to time; Loss term of the liquid-phase initial condition It is the deviation between the predicted value and the true value of the pore water pressure at the initial time point; Loss term of the liquid-phase boundary condition It is composed of the sum of the deviation between the predicted value and the true value of the pore water pressure on the Dirichlet boundary and the deviation between the predicted value and the true value of the flow velocity on the Neumann boundary condition.

[0034] Similarly, the total loss function of the solid-phase physical information neural network The specific calculation methods of the three loss terms in it are as follows: Loss term of the dimensionless control equation for the solid phase It is: the deviation between the dimensionless stress-deformation balance equation in the partial differential equations of soil hydraulic coupling and zero; Loss term of the solid-phase initial condition It is the deviation between the predicted value and the true value of the soil deformation at the initial time point; Loss term of the solid-phase boundary condition It is composed of the sum of the deviation between the predicted value and the true value of the soil deformation on the Dirichlet boundary and the deviation between the predicted value and the true value of the stress on the Robin mixed boundary condition.

[0035] To better understand the specific forms of the above total loss functions, the following specifically gives the method of converting and dimensionlessizing the equations using the fixed stress splitting method according to the selected partial differential equations of soil hydraulic coupling (including the liquid-phase continuity balance equation and the stress-deformation balance equation ).

[0036] In the embodiments of the present invention, the fixed stress splitting method refers to replacing the derivative term of the volumetric strain with respect to time in the liquid-phase continuity equilibrium equation with the derivative term of the mean total stress with respect to time. , without considering the compression of water, water sources, and body forces, and using the liquid-phase continuity equilibrium equation after conversion by the fixed stress splitting method and the stress-deformation equilibrium equation The expressions are as follows:

[0037] In the formula, K is the soil permeability coefficient tensor, P is the pore water pressure, is the unit weight of water, h is the elevation head, n is the porosity, S w is the degree of saturation, c is the hydro-mechanical coupling coefficient, is the three-dimensional drainage bulk modulus, v is the mean total stress, " " and " " are the gradient and divergence operators, and are the Lame constants, u i and x i are the deformations and spatial point positions in the i direction (i.e., the x, y, z directions in three-dimensional space), is the volumetric strain, is the Laplace operator.

[0038] According to the liquid-phase continuity equilibrium principle and the stress-deformation equilibrium principle, the values of equation and equation should theoretically be 0, that is, satisfying:

[0039] To avoid the influence of the dimensional differences of different parameters on the final model training, a dimensionless factor is introduced to dimensionless the two equilibrium equations, so that the input and output of the neural network are within the same order of magnitude, thereby improving the training effect and stability of the physics-based neural network. The liquid-phase continuity equilibrium equation and the stress-deformation equilibrium equation after equation dimensionless ( and are denoted as and respectively after dimensionless) are specifically represented as follows:

[0040] In the formula, after adding a marked horizontal line "—" to any parameter A at the top, it represents the dimensionless form of the parameter A (here A is the general expression form of the parameter symbol in the formula, not representing a specific parameter, the same below). Specifically: 、 、 represent the gradient operator, divergence operator, and Laplace operator in dimensionless form respectively, 、 represent the pore water pressure and time in dimensionless form respectively (i.e., the simulated time point, determined according to the input time point), represents the elevation head in dimensionless form, represents the average total stress in dimensionless form, and are the deformation and spatial point position in dimensionless form in the i direction respectively, represents the volumetric strain in dimensionless form; and are the characteristic length and characteristic pressure respectively, with the units of meter and pascal. and m are intermediate parameters, and their values are:

[0041] In the formula E and v are the elastic modulus and Poisson's ratio respectively.

[0042] Since and are actually the dimensionless forms of the above liquid-phase continuity equilibrium equation and the stress-deformation equilibrium equation , and the theoretical values of the liquid-phase continuity equilibrium equation and the stress-deformation equilibrium equation are 0. Therefore, when optimizing and training the model, it should be ensured that the values of and after dimensionlessization are as close to 0 as possible. Thus, according to the established dimensionless control equations for the liquid phase and solid phase, the loss terms of the dimensionless control equation for the liquid phase and the loss term of the dimensionless control equation for the solid phase used in the training of two physical information neural networks can be expressed as:

[0043] In the formula N gis the number of collocation points for calculating the loss term of the control equation; after adding an upper triangular mark on top of any parameter A, represents the predicted value of the parameter in the physics-informed neural network, that is, and are the pore water pressure predicted by combining the physics-informed neural network of the liquid phase and the deformation predicted by the physics-informed neural network of the solid phase, respectively, calculated according to their respective equilibrium equations and .

[0044] In addition, in the embodiments of the present invention, the initial and boundary conditions include initial conditions and boundary conditions. Different initial and boundary conditions can be set for the liquid phase and the solid phase respectively, and then different initial condition loss terms and boundary condition loss terms are constructed.

[0045] In the embodiments of the present invention, the initial condition of the liquid phase is the constraint condition of the pore water pressure, and the boundary condition includes the Dirichlet boundary involving the pore water pressure and the Neumann boundary condition involving the flow velocity. The initial condition of the solid phase is the constraint condition of the deformation, and the solid phase boundary condition includes the Dirichlet boundary involving the deformation and the Robin mixed boundary condition involving the stress. The expressions of the initial and boundary conditions for the solid phase and the liquid phase are as follows:

[0046]

[0047] In the formula, N ini , N b are the numbers of collocation points corresponding to the calculation of the initial condition and the boundary condition respectively; i represents the stress in the i direction; after adding a tilde mark on top of any parameter A, represents the true value (groundtruth) of the parameter. Specifically: is the true value of the pore water pressure P , is the true value of the deformation i in the u i direction, is the true value of the stress i in the i direction. Correspondingly, is the predicted value of the pore water pressure P , is the deformation iDeformation in the u i predicted value, is the i stress in the i predicted value.

[0048] S3. Based on the sequential training strategy, alternately train the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network in an iterative manner until the iterative termination condition is reached, and use the two trained networks to predict the pore water pressure and deformation of the soil body; in each round of iteration, alternately freeze the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network, and sample the collocation points from the spatial domain and the temporal domain and input them into the two physics-informed neural networks to obtain the predicted values of the pore water pressure and deformation. Finally, calculate the total loss function value corresponding to the unfrozen physics-informed neural network, and update the network parameters of the unfrozen physics-informed neural network by backpropagation according to the total loss function value.

[0049] It should be noted that the sequential training strategy is adopted to alternately train the liquid-phase and solid-phase physics-informed neural networks, which means first freezing the parameters of the solid-phase neural network to train the liquid-phase physics-informed neural network, then freezing the parameters of the liquid-phase neural network, and starting to train the solid-phase physics-informed neural network. Finally, alternately train for several rounds until the loss converges and stabilizes. This process can be seen in Figure 2 shown, where the liquid-phase modified fully connected neural network , the solid-phase modified fully connected neural network are the two physics-informed neural networks for the liquid phase and the solid phase respectively. The training method of the present invention is not limited, and the stochastic gradient descent optimization algorithm can be adopted.

[0050] In the embodiments of the present invention, since the information required to calculate the three loss terms in the total loss function is different, different configuration points need to be sampled to calculate their respective loss terms. Specifically, when alternately freezing the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network and optimizing the other unfrozen physics-informed neural network, it is necessary to first sample a first configuration point sample composed of spatial points and time points from the complete spatial domain and time domain, sample time points from the complete time domain and combine them with the spatial points at the simulated spatial boundary to form a second configuration point sample, sample spatial points from the complete spatial domain and combine them with the initial time point corresponding to the simulated initial moment to form a third configuration point sample, and input the three types of samples obtained by sampling into the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network respectively to obtain the predicted values of the soil pore water pressure and the soil deformation; then, according to the total loss function corresponding to the unfrozen physics-informed neural network, calculate the non-dimensional control equation loss term in the total loss function from the predicted values of the first configuration point sample, calculate the boundary condition loss term in the total loss function from the predicted values of the second configuration point sample, calculate the initial condition loss term in the total loss function from the predicted values of the third configuration point sample, and finally obtain the total loss function value by weighting and update the network parameters of the unfrozen physics-informed neural network using the gradient descent optimization algorithm.

[0051] In addition, in the embodiments of the present invention, a gradient normalization adaptive loss balancing strategy can also be introduced in each iteration process to adaptively adjust the weighting weights of the three loss terms in the total loss function. The use of the gradient normalization adaptive loss balancing strategy means that the weights corresponding to each loss term are automatically updated during the training process. The gradient normalization adaptive loss weight update method mainly updates the weights by calculating the gradient vectors L 2 of each loss term with respect to the network parameters, and calculating the relative training degree of each loss term. The gradient normalization adaptive loss balancing strategy belongs to the prior art. In the embodiments of the present invention, the loss weight calculation method of gradient normalization can be implemented according to the following steps: (1) Denote the current loss weight ( k = 1, 2, 3 respectively correspond to k in the aforementioned liquid-phase total loss function, , , , or , , in the solid-phase total loss function) of any th loss term in the total loss function as , and calculate the L 2 norm of the gradient vector of each loss term in the total loss function with respect to the network parameter , where is the k norm corresponding to the L 2 th loss term, is the k loss value of the k th loss term, and

[0052] L 2 is the average value of the norms of all loss terms .

[0053] (3) Calculate the training degree of each loss term, that is, where k represents the loss value at the initial training of the

[0054] .

[0055] (4) Calculate the relative training degree of each loss term, that is,

[0055] . (5) Adjust the loss weights so that the

[0056] of each loss term approaches an average level , that is, the weight value used for the next round of total loss calculation after adjustment is: where

[0057] This adaptive loss balancing strategy for gradient normalization is applicable to the gradient balancing of multiple loss terms. Compared with fixed loss weights, this method can improve the prediction accuracy.

[0058] In summary, the method of the present invention uses a sequential training strategy, an improved fully connected neural network, and an adaptive loss balancing scheme to improve the traditional physics-informed neural network, thereby enhancing the prediction accuracy and training stability. Meanwhile, compared with the traditional finite element method, this method has the advantages of meshless and data-driven, and has physical interpretability compared with pure data-driven black-box models. The trained physics-informed neural network of the present invention can be used for pore water pressure and deformation prediction. The hydro-mechanical coupling behavior of soil is mainly generated by two boundary conditions. The first is the rainfall boundary condition, and the second is the load boundary condition. The present invention can be applied to both of these two boundary conditions to explore the seepage and deformation behavior of soil under rainfall or overlying load. According to this method, an inverse analysis and a surrogate model can be further established to estimate unknown soil parameters and constitutive models, and used as a hybrid model integrating physics information and data-driven to predict the soil response in the unexplored spatio-temporal domain, showing great application potential.

[0059] It should be noted that the method steps shown in S1 to S3 above can essentially be implemented in the form of a computer program.

[0060] Therefore, based on the same inventive concept, as Figure 3 shown, the present invention also provides a soil hydro-mechanical coupling simulation system based on a physics-informed neural network corresponding to the soil hydro-mechanical coupling simulation method based on a physics-informed neural network provided in the above embodiment, which includes: An initial parameter setting module, configured to obtain the spatial domain, time domain, initial and boundary conditions, and soil parameters of the simulation for a target site where soil hydro-mechanical coupling simulation is to be performed; A model and loss construction module, configured to construct a liquid-phase physics-informed neural network for predicting the pore water pressure in soil, and set the weighted sum of the liquid-phase dimensionless governing equation loss term, liquid-phase boundary condition loss term, and liquid-phase initial condition loss term as the corresponding total loss function; construct a solid-phase physics-informed neural network for predicting the deformation in soil, and set the weighted sum of the solid-phase dimensionless governing equation loss term, solid-phase boundary condition loss term, and solid-phase initial condition loss term as the corresponding total loss function; wherein, the liquid-phase dimensionless governing equation loss term and the solid-phase dimensionless governing equation loss term are constructed after transforming the soil hydro-mechanical coupling partial differential equation set based on the fixed stress splitting method; The model training and prediction module is used to alternately train the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network in an iterative manner based on the sequential training strategy until the iterative termination condition is reached, and use the two trained networks to predict the pore water pressure and deformation of the soil mass; in each round of iteration, alternately freeze the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network, and sample configuration points from the spatial domain and the temporal domain and input them into the two physics-informed neural networks to obtain the predicted values of the pore water pressure and deformation. Finally, calculate the total loss function value corresponding to the unfrozen physics-informed neural network, and update the network parameters of the unfrozen physics-informed neural network by backpropagation according to the total loss function value.

[0061] In addition, based on the same inventive concept, as Figure 4 shown, the present invention also provides a computer electronic device corresponding to a soil hydraulic coupling simulation method based on a physics-informed neural network provided in the above embodiment, which includes a memory and a processor; The memory is used to store computer programs; The processor is used to implement the soil hydraulic coupling simulation method based on a physics-informed neural network as described above when executing the computer program; Furthermore, when the logical instructions in the above-mentioned memory are implemented in the form of software function units and sold or used as an independent product, they can be stored in a computer-readable storage medium. Based on such an understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or a part of this technical solution, can be embodied in the form of a software product. The computer software product is stored in a storage medium and includes several instructions for causing a computer device (which may be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the methods described in various embodiments of the present invention.

[0062] Therefore, based on the same inventive concept, the present invention provides a computer-readable storage medium corresponding to a soil hydraulic coupling simulation method based on a physics-informed neural network. A computer program is stored on the storage medium, and when the computer program is executed by a processor, it can implement the soil hydraulic coupling simulation method based on a physics-informed neural network as described above.

[0063] Therefore, based on the same inventive concept, the present invention provides a computer program product, including computer programs / instructions, which can implement the soil hydraulic coupling simulation method based on a physics-informed neural network as described above when the computer programs / instructions are executed by a processor.

[0064] Specifically, in the computer-readable storage media of the above three embodiments, the stored computer programs are executed by a processor, and the steps of the foregoing S1~S3 can be executed.

[0065] It is understandable that the above storage medium may include a Random Access Memory (RAM), and may also include a Non-Volatile Memory (NVM), such as at least one disk memory. At the same time, the storage medium may also be various media that can store program codes, such as a USB flash drive, a mobile hard disk, a magnetic disk, or an optical disc.

[0066] It is understandable that the above processor may be a general-purpose processor, including a Central Processing Unit (CPU), a Network Processor (NP), etc.; it may also be a Digital Signal Processor (DSP), an Application Specific Integrated Circuit (ASIC), a Field-Programmable Gate Array (FPGA), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components.

[0067] In addition, it should be noted that those skilled in the art can clearly understand that for the convenience and simplicity of description, the specific working process of the above-described system can refer to the corresponding process in the foregoing method embodiment, and will not be repeated here. In each embodiment provided in the present application, the division of steps or modules in the system and method is only a logical function division, and there may be other division methods in actual implementation. For example, multiple modules or steps can be combined or integrated together, and a module or step can also be split.

[0068] Next, the present invention will further use a specific embodiment to demonstrate the detailed implementation process and technical effects of the soil hydraulic coupling simulation method based on the physics-informed neural network shown in the above S1~S3 steps in a specific example scenario, so as to facilitate the understanding of the essence of the present invention.

[0069] Embodiment The steps of this embodiment are the same as those of the soil hydraulic coupling simulation method based on the physics-informed neural network shown in the foregoing S1~S3 steps, and will not be completely repeated here. The key steps, specific simulation scenarios, partial specific parameter settings, and implementation results of this embodiment will be mainly demonstrated.

[0070] In this embodiment, taking the overlying load boundary condition of a one-dimensional saturated soil body as an example, the simulated structure is a soil column, and its three-dimensional sketch is shown in Figure 5This soil mass satisfies the lateral constraint condition, and only the top surface can permeate water. The overlying pressure is 100 kPa. The soil mass parameters are as follows: a c = 1.0, E = 1.9 10 4 Kpa, v = 0.3, w = 9.8 kN / m 3 , and K sat = 3.47 10 -5 m / s. The parameters used to train the physics neural network are as follows: 4000 first configuration points are sampled to optimize the residual of the control equation, that is, N g is 4000. The second and third configuration points used to optimize the boundary condition loss term and the initial condition loss term are 80 and 500 respectively, that is, N b is 80, N ini is 500. These points are randomly sampled in each round.

[0071] In addition, both the liquid-phase physics-informed neural network and the solid-phase physics-informed neural network in this embodiment adopt an improved fully connected network with an attention mechanism. In order to compare and show the effects between the improved fully connected network and the traditional fully connected network, and at the same time to prove the advantages of the adaptive loss balance strategy of the gradient normalization method for model training, a total of four models are set in this embodiment for comparison, as shown in Table 1: The first model M1 adopts the traditional fully connected neural network and the method of not updating the loss weights. The second model M2 adopts the traditional fully connected neural network and the adaptive loss balance strategy of the gradient normalization method. The third model M3 adopts the improved fully connected neural network and the method of not updating the loss weights. The fourth model M4 adopts the improved fully connected neural network and the adaptive loss balance strategy of the gradient normalization method. By calculating the relative L 2 error between the predicted value and the analytical solution to determine the performance of the model. The smaller the relative L2 error, the more accurate the model. Under different random seeds, each model is calculated 10 times.

[0072] Table 1

[0073] The traditional fully connected neural network uses 5 fully connected hidden layers, with 128 neurons in each layer. The improved fully connected neural network uses the same fully connected hidden layers (a total of 5 layers, with 128 neurons in each layer), and includes two additional fully connected layers as mapping networks. Each mapping network consists of a fully connected hidden layer with 128 neurons, which is used to map the original configuration points into the first attention feature and the second attention feature. The Tanh function is used as the activation function for the fully connected hidden layers in all network models. In the sequential training strategy, the number of internal training rounds of the physical information neural networks for the aqueous and solid phases is the same, both being 100,000. Since the total stress of the entire compressed column remains constant within a one-dimensional range, the number of external rounds is set to 1. The optimizer for network parameter optimization is Adam.

[0074] Table 2 presents the means and standard deviations of the relative L 2 errors of the pore water pressure and vertical displacement for four comparison models after the training is completed. The models M2 and M4 using the adaptive loss balancing strategy have lower L 2 errors, indicating an improvement in the prediction accuracy. For M3 and M4 using the improved fully connected neural network, compared with models M1 and M2, the L 2 standard deviation of the error is reduced, indicating an improvement in the training stability. Finally, combining the improved fully connected neural network and the adaptive loss balancing strategy achieves the optimal prediction performance.

[0075] Table 2

[0076] In addition, in this embodiment, the model accuracies of the sequential training strategy and the traditional training strategy are also compared. The traditional training strategy means not using the fixed stress splitting method to process the fluid-solid coupling equilibrium equation, and at the same time using the co-training method to train the two networks of the solid phase and the liquid phase. In this comparison method, neither of the two models uses the improved fully connected neural network and the gradient normalization balancing method. By comparing the sequential training method using the fixed stress splitting method and the traditional co-training method, the results shown in Table 3 indicate that for the prediction of pore water pressure, the means of the L2 error accuracies are 0.0365 and 0.0405 respectively, and the standard deviations are 0.00179 and 0.00256 respectively. For the prediction of vertical deformation, the means of the L2 error accuracies are 0.0027 and 0.0045 respectively, and the standard deviations are 0.00077 and 0.00121 respectively. It can be seen that the sequential training strategy using the fixed stress splitting method adopted in the present invention has smaller errors, higher prediction accuracy, and stronger stability than the traditional co-training method.

[0077] Table 3

[0078] It should be particularly noted that although the sequential training strategy can be carried out based on the fixed strain splitting method and the fixed stress splitting method. When the liquid phase continuity equilibrium equation in the fluid-solid coupling equilibrium equation does not change with the fixed stress splitting method, that is, the volumetric strain term is retained in the equation, it is applicable to the fixed strain splitting method. However, the reason why the present invention is based on the sequential training strategy of the fixed stress splitting method is that the fixed strain splitting method is extremely unstable in solving and is prone to trivial solutions and incorrect solutions. And retaining the volumetric strain term in the liquid phase continuity equilibrium equation of the fluid-solid coupling equation is applicable to the collaborative training strategy, that is, training the solid phase and the liquid limit network simultaneously, so that the correct solution can be obtained. This way belongs to a traditional training method. The sequential training strategy of the present invention based on the fixed stress splitting method can avoid the simultaneous training of the solid phase and the liquid phase network, and approach the true solution in an iterative training manner. This training method reduces the difficulty of network optimization, is easier to find the optimal solution, and thus has higher accuracy. On the contrary, the traditional collaborative training method optimizes the solid phase and the liquid phase networks simultaneously, inevitably increasing the training difficulty and having a worse accuracy of the obtained solution.

Claims

1. A soil hydraulic coupling simulation method based on physical information neural network, characterized in that: include: S1: For the target site where soil-hydraulic coupling simulation is required, obtain the simulation space domain, time domain, initial and boundary conditions and soil parameters; S2. Construct a liquid-phase physical information neural network for predicting pore water pressure in soil, and set the weighted sum of the liquid-phase dimensionless control equation loss term, the liquid-phase boundary condition loss term, and the liquid-phase initial condition loss term as the corresponding total loss function; construct a solid-phase physical information neural network for predicting deformation in soil, and set the weighted sum of the solid-phase dimensionless control equation loss term, the solid-phase boundary condition loss term, and the solid-phase initial condition loss term as the corresponding total loss function; wherein the liquid-phase dimensionless control equation loss term and the solid-phase dimensionless control equation loss term are constructed in a dimensionless manner after converting the soil hydraulic coupling partial differential equation group by the fixed stress splitting method; S3. Based on the sequential training strategy, the liquid phase physical information neural network and the solid phase physical information neural network are alternately trained in an iterative manner until the iteration termination condition is reached, and the two trained networks are used to predict the pore water pressure and deformation of the soil; in each round of iteration, the liquid phase physical information neural network and the solid phase physical information neural network are alternately frozen, and the sampled configuration points in the spatial domain and the time domain are input into the two physical information neural networks to obtain the predicted values ​​of pore water pressure and deformation, and finally the total loss function value corresponding to the unfrozen physical information neural network is calculated, and the network parameters of the unfrozen physical information neural network are updated by back propagation according to the total loss function value.

2. The soil hydraulic coupling simulation method based on physical information neural network according to claim 1, characterized in that: The liquid-phase physical information neural network and the solid-phase physical information neural network both use the same fully connected network with an attention mechanism. The network inputs are configuration points composed of spatial points collected from the spatial domain and time points collected from the time domain. The configuration points are mapped to first attention features and second attention features respectively through two additional fully connected layers. Then, the configuration points, the first attention features and the second attention features are input into multiple layers of sequentially cascaded network layers. The first network layer directly uses the configuration points as input to obtain output features through the fully connected layer. In the remaining network layers, the output features of the previous network layer are first passed through the fully connected layer to obtain the intermediate feature H. Then, the first attention feature and the second attention feature are used to weight 1-H and H respectively and then added to obtain the output features of the current network layer.

3. The soil hydraulic coupling simulation method based on physical information neural network according to claim 1, characterized in that: In the total loss function of the liquid phase physical information neural network, The loss term of the dimensionless liquid phase control equation is: the deviation of the dimensionless liquid phase continuity balance equation after the fixed stress splitting method conversion from zero; the fixed stress splitting method conversion method is to replace the time derivative term of the volume strain in the liquid phase continuity balance equation of the soil hydraulic coupling partial differential equation group with the time derivative term of the average volume stress; The liquid phase initial condition loss term is the deviation between the predicted value and the true value of the pore water pressure at the initial time point; The liquid phase boundary condition loss term is the sum of the deviation between the predicted and true pore water pressure on the Dirichlet boundary and the deviation between the predicted and true flow velocity on the Neumann boundary condition.

4. The soil hydraulic coupling simulation method based on physical information neural network according to claim 1, characterized in that: In the total loss function of the solid-phase physical information neural network, The loss term of the dimensionless control equation of the solid phase is: the deviation of the dimensionless stress-deformation equilibrium equation in the soil hydraulic coupled partial differential equation group from zero; The solid phase initial condition loss term is the deviation between the predicted value and the true value of soil deformation at the initial time point; The solid boundary condition loss term is the sum of the deviation between the predicted value and the true value of soil deformation on the Dirichlet boundary and the deviation between the predicted value and the true value of stress on the Robin mixed boundary condition.

5. The soil hydraulic coupling simulation method based on physical information neural network according to claim 1, characterized in that: When alternately freezing the liquid phase physical information neural network and the solid phase physical information neural network and optimizing another unfrozen physical information neural network, it is necessary to first sample the first configuration point sample consisting of space points and time points from the complete space domain and time domain, sample time points from the complete time domain and form the second configuration point sample with the space points at the boundary of the simulation space, sample space points from the complete space domain and form the third configuration point sample with the initial time points corresponding to the initial moment of the simulation, and input the three types of samples obtained by sampling into the liquid phase physical information neural network and the solid phase physical information neural network respectively to obtain the predicted value of the soil pore water pressure and the predicted value of the soil deformation; then, according to the total loss function corresponding to the unfrozen physical information neural network, the dimensionless control equation loss term in the total loss function is calculated by the predicted value of the first configuration point sample, the boundary condition loss term in the total loss function is calculated by the predicted value of the second configuration point sample, and the initial condition loss term in the total loss function is calculated by the predicted value of the third configuration point sample, and finally the total loss function value is obtained by weighting and the network parameters of the unfrozen physical information neural network are updated by the gradient descent optimization algorithm.

6. The soil hydraulic coupling simulation method based on physical information neural network according to claim 1, characterized in that: In each round of iteration, a gradient normalization adaptive loss balance strategy is introduced to adaptively adjust the weights of the three loss terms in the total loss function.

7. A soil hydraulic coupling simulation system based on physical information neural network, characterized in that: include: The initial parameter setting module is used to obtain the simulation space domain, time domain, initial boundary conditions and soil parameters for the target site where soil-hydraulic coupling simulation is required; A model and loss construction module is used to construct a liquid phase physical information neural network for predicting pore water pressure in soil, and set the weighted sum of the liquid phase dimensionless control equation loss term, the liquid phase boundary condition loss term and the liquid phase initial condition loss term as the corresponding total loss function; construct a solid phase physical information neural network for predicting deformation in soil, and set the weighted sum of the solid phase dimensionless control equation loss term, the solid phase boundary condition loss term and the solid phase initial condition loss term as the corresponding total loss function; wherein the liquid phase dimensionless control equation loss term and the solid phase dimensionless control equation loss term are constructed by converting the soil hydraulic coupling partial differential equation group based on the fixed stress splitting method; The model training and prediction module is used to alternately train the liquid phase physical information neural network and the solid phase physical information neural network in an iterative manner based on a sequential training strategy until the iteration termination condition is reached, and use the two trained networks to predict the pore water pressure and deformation of the soil; in each round of iteration, the liquid phase physical information neural network and the solid phase physical information neural network are alternately frozen, and the sampled configuration points in the spatial domain and the time domain are input into the two physical information neural networks to obtain the predicted values ​​of pore water pressure and deformation, and finally the total loss function value corresponding to the unfrozen physical information neural network is calculated, and the network parameters of the unfrozen physical information neural network are updated by back propagation according to the total loss function value.

8. A computer program product comprising a computer program / instructions, characterized in that When the computer program / instruction is executed by a processor, it can implement the soil-hydraulic coupling simulation method based on physical information neural network as described in any one of claims 1 to 7.

9. A computer-readable storage medium, characterized in that: The storage medium stores a computer program, and when the computer program is executed by the processor, the soil-hydraulic coupling simulation method based on physical information neural network as described in any one of claims 1 to 7 is implemented.

10. A computer electronic device, characterized in that: including memory and processor; The memory is used to store computer programs; The processor is used to implement the soil-hydraulic coupling simulation method based on physical information neural network as described in any one of claims 1 to 7 when executing the computer program.

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