Seepage Parameterization Learning Method, Device, Equipment and Storage Medium

Abstract

The present application provides a seepage parameterization learning method, device, equipment and storage medium, which relates to the technical field of seepage numerical simulation. The method includes: defining a seepage parameterization learning mathematical model; building a Fourier neural operator network structure with time-frequency double filtering; defining a physical constraint loss function based on finite volume discretization; using sequential Gaussian simulation for sampling to generate different permeability fields, and dividing them into a training set and a test set; training a Fourier neural operator network proxy model on the training set and testing the inference effect on the test set. The present application can achieve training with unlabeled data, and can quickly predict the results under different permeability distributions through the trained model, with significantly improved prediction accuracy.

Description

Technical Field

[0001] The present application relates to the technical field of seepage numerical simulation, and in particular to a seepage parameterized learning method, apparatus, device and storage medium. Background Art

[0002] Physical information neural network is a useful paradigm that can perform forward and backward deduction of physical processes with limited or no labeled data. It is also used to analyze flow processes in porous media, such as parameterized learning of mathematical models of seepage and predicting the distribution of physical fields. However, these methods are trained for specific conditions, such as the physical parameters of the characteristics and the setting of boundary conditions. When any change occurs in the input parameters, they must be retrained, and their ability to handle variable parameter problems is limited. The neural operator network agent model is a paradigm with good generalization and can learn the mapping between parameter space and variable space. However, this method usually adopts a label-based data-driven approach for training, which requires the preparation of samples and labeled data in advance. The model performance is constrained by the amount and quality of data, and there is a lack of physical consistency in model reasoning.

[0003] To address these challenges, neural operator surrogate models based on physics-driven training have been widely studied and are beginning to be used in engineering to replace repetitive numerical simulations for rapid prediction of physical fields, balancing computational efficiency and accuracy. This approach establishes an end-to-end prediction framework based on the working mode of the surrogate model, enabling the trained model to directly perform rapid predictions without the need for additional computation. It also incorporates partial differential equations defining the physical processes to establish physical consistency assessment criteria, enabling the model to better learn the underlying physical laws. Furthermore, neural operator learning strategies are used to extend finite-dimensional mappings to infinite-dimensional mappings, improving the model's generalization performance and avoiding frequent retraining on new problems. However, current approaches primarily rely on automatic differentiation to construct physical constraints, making it difficult to address the derivative order of the physical quantities being processed, particularly for approximating the derivatives of spatially inhomogeneous physical fields. Furthermore, current neural operator networks still struggle to extract low-frequency modes and infer inaccurately under varying physical parameter field distributions. Summary of the invention

[0004] The present application provides a seepage parameterization learning method, device, equipment and storage medium, designs and constructs a brand-new Fourier neural operator network structure with time-frequency double filtering, pre-screens purer features through time-domain filtering, and then performs low-pass filtering in the frequency domain to enhance the main modes, so as to accurately capture the general generalization mode; combines the finite volume method to replace the automatic differentiation method for spatial derivative calculation, strictly defines the flux continuity between adjacent units with different properties, reduces the required derivative order of calculation, and avoids the automatic differentiation calculation required for constructing the loss term. Under the same parameter settings and computing resources, the present application significantly improves the training convergence of the physical-driven parameterization learning of the surrogate model (Fourier neural operator network structure), and has higher prediction accuracy than the data-driven surrogate model and the physical-driven surrogate model before improvement.

[0005] In a first aspect, the present application provides a seepage parameterization learning method, including:

[0006] Establish a seepage parameterization learning mathematical model; wherein, the seepage parameterization learning mathematical model is a single-phase flow model in heterogeneous porous media based on Darcy's equation;

[0007] Build a Fourier neural operator network structure with time-frequency double filtering; the Fourier neural operator network structure is used to learn the mapping from the permeability field and spatial coordinates to the pressure field;

[0008] Design a physical constraint loss function based on finite volume discretization;

[0009] Based on the seepage parameterization learning mathematical model, use sequential Gaussian simulation for sampling, generate different permeability fields, and divide them into a training set and a test set;

[0010] Based on the physical constraint loss function, train the Fourier neural operator network structure on the training set and test the inference effect on the test set.

[0011] In a possible design, the control equation of the seepage parameterization learning mathematical model is:

[0012] ,

[0013] wherein, is the flow velocity; is the dynamic viscosity; is the permeability field of the porous medium; is the fluid pressure; is the source-sink term; represents the gradient operator; represents the divergence operator.

[0014] The boundary conditions of the seepage parameterization learning mathematical model are defined as:

[0015] ,

[0016] wherein is the preset fixed pressure value of the Dirichlet boundary; the preset fixed pressure value of the Neumann boundary; is the preset fixed velocity value of the Neumann boundary; the preset fixed velocity value of the Neumann boundary; is the normal vector of the boundary position; on Dirichlet boundary means on the Dirichlet boundary, and on Neumann boundary means on the Neumann boundary.

[0017] In a possible design, the Fourier neural operator network structure includes an input fully connected layer, a first gated Fourier layer, a second gated Fourier layer, a third gated Fourier layer, and an output fully connected layer; wherein, the input fully connected layer is located at the forefront of the neural network, the first gated Fourier layer is located after the input fully connected layer, the second gated Fourier layer is located after the first gated Fourier layer, the third gated Fourier layer is located after the second gated Fourier layer, and the output fully connected layer is located after the third gated Fourier layer; the first gated Fourier layer, the second gated Fourier layer, and the third gated Fourier layer all include two parts: time-domain feature selection and frequency-domain Fourier filtering.

[0018] In a possible design, the process of the Fourier neural operator network structure learning the mapping from the permeability field and spatial coordinates to the pressure field is as follows:

[0019] Taking the permeability matrix and the spatial coordinate matrix as the input matrices of the input fully connected layer, the mapping function related to the input fully connected layer is as follows:

[0020] ,

[0021] wherein and are the weights and biases of the input layer; is the pressure matrix after dimension expansion; is the permeability field; and are the spatial coordinates;

[0022] The mapping of the pressure includes the step-by-step refinement of three gated Fourier layers, and the complete mapping process is as follows:

[0023] ,

[0024] wherein, is the input of the neural network; is the result of time-domain feature selection; is the output of the gated Fourier layer; is the result of the second-layer time-domain feature selection; is the output of the second-layer gated Fourier layer; is the output of the neural network;

[0025] The mapping function of the adaptive feature selection part in the gated Fourier layer is:

[0026] ,

[0027] where is batch normalization, which is used to stabilize the feature distribution in the mapping; is the learnable weight; is pointwise multiplication; is the non-linear rectifier unit activation function, which is used to sparsely activate the neurons that contribute the most to pattern recognition and realize the pre-screening of features;

[0028] The mapping function of the frequency-domain Fourier filtering part is as follows:

[0029] ,

[0030] where is the fast Fourier transform; is the low-pass filtering in the frequency domain; is the inverse Fourier transform; is the convolutional layer; is the number of low-frequency modes extracted; and are respectively x and y the sequence numbers in the is the maximum number of modes, is x the maximum sequence number in the is y the maximum sequence number in the

[0031] The output layer maps the features back to the low-dimensional original space, and the mapping function is:

[0032] ,

[0033] where is the weight of the output layer; is the bias of the output layer; is the pressure matrix of the final output of the neural network.

[0034] In a possible design, the physical constraint loss function is expressed as:

[0035] ,

[0036] in is the number of units used to calculate the residual loss value; is the number of cells used to compute the Dirichlet frontier loss; R i It is i The physical constraint loss value of each unit; L is the loss value; is the first i The pressure of the grid, is the first i The pressure of the grid.

[0037] In one possible design, based on the parametric learning mathematical model of seepage, sequential Gaussian simulation is used for sampling to generate different permeability fields, which are divided into training sets and test sets, including:

[0038] The size parameters of the seepage parameterized learning mathematical model are set, and the grid is divided into Cartesian grids, and the size parameters of each Cartesian grid are the same; the upper boundary and the lower boundary are defined as closed boundaries; the Dirichlet boundary condition is defined as a left boundary pressure of 1 MPa and a right boundary pressure of 0 MPa; sequential Gaussian simulation is performed to sample different permeability field distributions, and the base value, nugget value and permeability range are set. The mean, standard deviation and range of the permeability are adjusted to generate permeability fields representing different heterogeneous environments, and the permeability range is 0.2 mD to 200 mD.

[0039] In one possible design, based on the physical constraint loss function, the Fourier neural operator network structure is trained on a training set, and the reasoning effect is tested on a test set, including:

[0040] In the training of the Fourier neural operator network structure, the weights of the neural network are uniformly initialized, the biases are uniformly initialized to 0, and the optimizer is used to perform physical-driven training of unlabeled data through a small batch gradient descent strategy. The neural network parameter update formula is:

[0041] ,

[0042] in is a small batch size; is a trainable parameter in the neural network; is the learning rate; is the serial number of the sample in the small batch; is the gradient operator of the learnable parameters; is the complete neural network mapping function; is the loss value calculated based on the output result of the neural network;

[0043] Based on the training set, train the Fourier neural operator network structure. Optimize the neural network parameters by continuously performing forward propagation, backward propagation, and gradient descent, including inputting the permeability matrix and the spatial coordinate matrix into the neural network for forward propagation to output the predicted pressure matrix, and calculating the loss value through the physical constraint loss function; calculate the gradient value through the neural network parameter update formula during backward propagation, and update the neural network parameters to complete one training, and perform a set number of gradient descent trainings in total;

[0044] Test the inference ability of the Fourier neural operator network structure of the test set. Input the permeability fields with different distributions in the test set into the trained model to quickly predict the pressure field under this permeability distribution; evaluate the inference effect through two indicators: mean absolute error and goodness of fit.

[0045] In a second aspect, the present application provides a seepage parameterization learning device, and the device includes:

[0046] A model construction module configured to establish a seepage parameterization learning mathematical model; wherein, the seepage parameterization learning mathematical model is a single-phase flow model in heterogeneous porous media based on Darcy's equation;

[0047] A network structure building module configured to build a Fourier neural operator network structure with time-frequency double filtering; the Fourier neural operator network structure is used to learn the mapping from the permeability field and spatial coordinates to the pressure field;

[0048] A loss function design module configured to design a physical constraint loss function based on finite volume discretization;

[0049] A data set generation module configured to sample based on the seepage parameterization learning mathematical model using sequential Gaussian simulation to generate different permeability fields, and divide them into a training set and a test set;

[0050] A model training and testing module configured to train the Fourier neural operator network structure on the training set and test the inference effect on the test set based on the physical constraint loss function.

[0051] In a third aspect, an embodiment of the present application provides an electronic device, including: at least one processor and a memory; the memory stores computer execution instructions; the at least one processor executes the computer execution instructions stored in the memory, so that the at least one processor executes the seepage parameterization learning method described in the first aspect above and various possible designs of the first aspect.

[0052] Fourthly, an embodiment of the present application provides a computer-readable storage medium storing computer-executable instructions, which, when executed by a processor, implement the seepage parameterization learning method as described in the first aspect above and various possible designs of the first aspect.

[0053] Fifthly, an embodiment of the present application provides a computer program product including a computer program, which, when executed by a processor, implements the seepage parameterization learning method as described in the first aspect above and various possible designs of the first aspect.

[0054] The seepage parameterization learning method, device, equipment, and storage medium provided by the present application have at least the following beneficial effects:

[0055] The present application can achieve parameterization learning of the seepage mathematical model and quickly and accurately predict the pressure field under different permeability distributions without retraining the neural network. Under the same parameter settings and computing resources, the present application significantly improves the physical-driven training convergence of the surrogate model and has a higher prediction accuracy than the data-driven surrogate model and the physical-driven surrogate model before improvement. Description of the Drawings

[0056] The drawings here are incorporated into the specification and constitute a part of this specification, showing embodiments consistent with the present application and used together with the specification to explain the principles of the present application.

[0057] Figure 1 It is a flowchart of a seepage parameterization learning method provided by an embodiment of the present application;

[0058] Figure 2 It is a schematic diagram of a defined numerical simulation model provided by an embodiment of the present application. Among them, (a) is the numerical model size and boundary conditions, and (b) is the mesh division diagram of the numerical model;

[0059] Figure 3 It is a schematic diagram of samples with different permeability distributions obtained by sampling based on the SGeMS software provided by an embodiment of the present application;

[0060] Figure 4 It is a schematic diagram of the time-frequency dual-filtered Fourier neural operator network structure provided by an embodiment of the present application;

[0061] Figure 5 It is a loss decline curve of the time-frequency dual-filtered Fourier neural operator network during training on the training set provided by an embodiment of the present application;

[0062] Figure 6 It is a curve of the change in the mean absolute error of the time-frequency dual-filtered Fourier neural operator network during training on the test set provided by an embodiment of the present application;

[0063] Figure 7 This is the curve of the goodness of fit in the test set during the training process of the time-frequency double-filtered Fourier neural operator network provided by the embodiments of the present application.

[0064] Figure 8 This is a schematic diagram of the predicted pressure in the test set after the time-frequency double-filtered Fourier neural operator network provided by the embodiments of the present application is trained; wherein, (a) is the prediction result of the neural network, (b) is the reference pressure field, and (c) is the error between the prediction result of the neural network and the reference value.

[0065] Figure 9 This is a distribution diagram of the mean absolute error between the prediction results of all samples in the test set and the reference values after the time-frequency double-filtered Fourier neural operator network provided by the embodiments of the present application is trained.

[0066] Figure 10 This is a distribution diagram of the goodness of fit between the prediction results of all samples in the test set and the reference values after the time-frequency double-filtered Fourier neural operator network provided by the embodiments of the present application is trained.

[0067] Figure 11 This is a structural diagram of the seepage parameterization learning device provided by the embodiments of the present application.

[0068] Through the above-mentioned drawings, specific embodiments of the present application have been shown, and there will be more detailed descriptions hereinafter. These drawings and textual descriptions are not intended to limit the scope of the concept of the present application in any way, but to illustrate the concept of the present application to those skilled in the art by referring to specific embodiments. Detailed Description of the Embodiments

[0069] Here, the exemplary embodiments will be described in detail, and the examples are shown in the drawings. When the following description refers to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with the present application. On the contrary, they are merely examples of devices and methods consistent with some aspects of the present application as detailed in the appended claims.

[0070] In the technical solution of the present application, the collection, storage, use, processing, transmission, provision, and disclosure of information such as financial data or user data comply with the provisions of relevant laws and regulations and do not violate public order and good customs.

[0071] It should be noted that in the embodiments of the present application, some industry-existing solutions such as certain software, components, models, etc. may be mentioned. They should be regarded as exemplary, and their purpose is only to illustrate the feasibility in the implementation of the technical solutions of the present application, but it does not mean that the applicant has already or necessarily used this solution.

[0072] The following uses specific embodiments to elaborate in detail on the technical solutions of the present application and how the technical solutions of the present application solve the above technical problems. These several specific embodiments below can be combined with each other, and the same or similar concepts or processes may not be repeated in some embodiments. The embodiments of the present application will be described below in conjunction with the accompanying drawings.

[0073] The embodiments of the present application provide a seepage parameterization learning method. As Figure 1 shown, it is the flowchart of the seepage parameterization learning method provided by the embodiments of the present application. This seepage parameterization learning method includes the following steps S100 - S500.

[0074] S100: Establish a seepage parameterization learning mathematical model; wherein, the seepage parameterization learning mathematical model is a single - phase flow model in heterogeneous porous media based on Darcy's equation.

[0075] In this embodiment, the purpose of step S100 is to define the seepage parameterization mathematical model to be solved, and prepare the physical constraint criteria based on the mathematical model for the training of the neural network, so as to achieve training without data labels. In this embodiment, the seepage parameterization learning mathematical model is defined as single - phase flow in heterogeneous porous media based on Darcy's equation, where the closed - boundary condition is the Neumann boundary condition with zero flow velocity, and the Dirichlet boundary condition is the constant - pressure boundary condition.

[0076] In some embodiments, the control equation of the seepage parameterization learning mathematical model is:

[0077] (1)

[0078] (2)

[0079] Wherein, is the flow velocity; is the dynamic viscosity; is the permeability field of the porous medium; is the fluid pressure; is the source - sink term; represents the gradient operator; represents the divergence operator.

[0080] The boundary conditions of the seepage parameterization learning mathematical model are defined as:

[0081] (3)

[0082] (4)

[0083] wherein is the preset fixed pressure value of the Dirichlet boundary ; is the preset fixed velocity value of the Neumann boundary ; is the normal vector of the boundary position; on Dirichlet boundary represents on the Dirichlet boundary, and on Neumann boundary represents on the Neumann boundary.

[0084] S200: Construct a time - frequency double - filtering Fourier neural operator network structure; the Fourier neural operator network structure is used to learn the mapping from the permeability field and spatial coordinates to the pressure field.

[0085] In this embodiment, the time - frequency double - filtering Fourier neural operator network structure established in step S200, by innovatively introducing an adaptive feature selection process in the time domain, together with the low - frequency extraction process in the frequency domain, constitutes a time - frequency double - filtering structure, and extracts a purer general low - frequency mode.

[0086] In some embodiments, the time - frequency double - filtering Fourier neural operator network structure includes an input fully - connected layer, a first gated Fourier layer, a second gated Fourier layer, a third gated Fourier layer, and an output fully - connected layer; wherein, the input fully - connected layer is located at the forefront of the neural network, the first gated Fourier layer is located after the input fully - connected layer, the second gated Fourier layer is located after the first gated Fourier layer, the third gated Fourier layer is located after the second gated Fourier layer, and the output fully - connected layer is located after the third gated Fourier layer; the first gated Fourier layer, the second gated Fourier layer, and the third gated Fourier layer each include two parts: time - domain feature selection and frequency - domain Fourier filtering. First, the features are pre - screened in the time domain, then low - pass filtered through Fourier transform to the frequency domain, and finally Fourier inverse transform is performed back to the time domain; the Fourier neural operator network proxy model learns the mapping from the permeability field and spatial coordinates to the pressure field.

[0087] In some embodiments, regarding the specific mapping function of each layer in the time - frequency double - filtering Fourier neural operator network model, a series of network structures are innovatively designed to enhance the ability of the neural network to extract low - frequency modes and enhance the inference generalization ability of the neural network. In this embodiment, the specific mapping function of each layer of the constructed time - frequency double - filtering Fourier neural operator network is introduced in detail as follows.

[0088] Taking the permeability matrix and the spatial coordinate matrix as the input matrices of the input fully connected layer, the mapping function of the input fully connected layer is as follows:

[0089] (5)

[0090] where and are the weights and biases of the input layer; is the pressure matrix for dimension expansion; is the permeability field; and are the spatial coordinates; the mapping of pressure includes the step-by-step refinement of three gated Fourier layers. The complete mapping process is as follows:

[0091] (6)

[0092] Taking the mapping of in the first gated Fourier layer as an example, the mapping function of the adaptive feature selection part in the gated Fourier layer is:

[0093] (7)

[0094] where is batch normalization, which is used to stabilize the feature distribution in the mapping; is the learnable weight; is the pointwise multiplication; is the non-linear rectifier unit activation function, which is used to sparsely activate the neurons that contribute the most to pattern recognition and achieve pre-selection of features; is the time-domain feature selection result; the mapping function of the frequency-domain Fourier filtering part is as follows:

[0095] (8)

[0096] (9)

[0097] where is the fast Fourier transform; is the low-pass filtering in the frequency domain; is the inverse Fourier transform; is the convolutional layer; is the output of the gated Fourier layer; is the number of low-frequency modes extracted; i and j are respectively x and y the sequence numbers of the directions; is the maximum number of modes, isx The maximum serial number in the direction is y the maximum serial number in the direction;

[0098] Finally, the output layer maps the features back to the low-dimensional original space, and the mapping function is:

[0099] (10)

[0100] where are the weights of the output layer; is the bias of the output layer; is the pressure matrix finally output by the neural network.

[0101] In some embodiments, in step S200, setting the parameters of the neural network model (Fourier neural operator network structure) includes: setting the input layer to map the permeability and 3 channels of spatial coordinates to 10 channels; setting the Fourier layer to continue mapping 10 channels to 10 channels in the high-dimensional space, setting the number of modes extracted to 10, setting the number of cascaded superpositions of the Fourier layer to 3; setting the output layer to perform mapping from 10 channels to a single channel; the design of the neural network structure and parameter setting are completed.

[0102] S300: Design a physical constraint loss function based on finite volume discretization.

[0103] In this embodiment, in step S300, the physical constraints defined based on the mathematical model are discretized by the finite volume method, a physical consistency criterion based on flux conservation is constructed, the order of the derivative of the physical quantity to be processed is solved, and the approximation of the derivative of the physical field with spatial inhomogeneity is improved.

[0104] In some embodiments, the physical constraint loss function based on finite volume discretization is designed in the following manner:

[0105] The finite volume method discretization process for single-phase flow in heterogeneous porous media based on Darcy's equation includes: defining a loss function including a residual term and a boundary condition term, defining a calculation method for the inter-grid conductivity coefficient, defining a smoothing method for the loss function, and defining a weighting method for different loss terms; defining a general physical constraint loss function as:

[0106] (11)

[0107] where is the loss function; are the neural network parameters; is the residual term of the loss function; is the residual term; is the initial condition term; is the boundary condition term; this method considers the steady-state underground flow problem, and the initial condition term in the loss function can be ignored; the closed boundary in the model is considered as the Neumann boundary, so the closed boundary is automatically constrained in the residual term, and formula (11) is further simplified to:

[0108] (12)

[0109] where is the Dirichlet boundary condition term; combining formula (1) and formula (2), the finite difference method is designed to discretize the residual term, and the discretized form is:

[0110] (13)

[0111] where i and j are the discrete element numbers; G i is the i th adjacent element of the th discrete element; i is the j th and the

[0112] (14)

[0113] where is the i th and the j th contact area of the elements; and are the normal distances from the centers of the i th and the j th elements to the contact surface respectively; is the harmonic mean of the permeabilities of the i th and the j th elements; combining formula (14), the physical constraint loss value of the i th element is defined as:

[0114] (15)

[0115] where p * is the pressure matrix output by the neural network; finally, the physical constraint loss function based on the finite volume method and the mean square error is defined as:

[0116] (16)

[0117] where is the number of elements used to calculate the residual loss value; is the number of units for calculating the Dirichlet boundary loss; is the pressure of the i th grid in the neural network output pressure matrix, is the pressure of the i th grid in the reference solution pressure matrix.

[0118] S400: Based on the percolation parameterized learning mathematical model, use sequential Gaussian simulation for sampling, generate different permeability fields, and divide them into training sets and test sets.

[0119] In some embodiments, step S400 can be performed using sequential Gaussian simulation based on the SGeMS software. Define the model size parameters as 500m × 500m, the grid is divided into a 100×100 Cartesian grid, and each grid size parameter is 5m × 5m; the upper and lower boundaries are defined as closed boundaries; the Dirichlet boundary condition is defined as the left boundary pressure of 1MPa and the right boundary pressure of 0MPa; use the SGeMS software to perform sequential Gaussian simulation to sample different permeability field distributions, set the sill value to 1, set the nugget value to 0, set the permeability range to 30 grids, adjust the mean, standard deviation, and range of the controlled permeability, generate permeability fields representing different heterogeneous environments, and the permeability range is 0.2mD to 200mD.

[0120] S500: Based on the physical constraint loss function, train the Fourier neural operator network structure on the training set and test the inference effect on the test set.

[0121] In this embodiment, in the training of the Fourier neural operator network, the weights of the neural network are uniformly initialized using Xavier, the biases are uniformly initialized to 0, and the Adam optimizer is used to perform physical-driven training of unlabeled data through the mini-batch gradient descent strategy. The neural network parameter update formula is:

[0122] (17)

[0123] where is the mini-batch size, set to 4; the mini-batch sampling method is set to the Shuffle mode; the learning rate is set to 0.001; are the trainable parameters in the neural network; is the learning rate; is the serial number of the samples in the mini-batch; is the gradient operator of the learnable parameters; is the complete neural network mapping function; represents the loss value calculated based on the neural network output result.

[0124] Use the training set sampled in step S400 for the training of the neural network. Continuously perform forward propagation, backward propagation, and gradient descent to optimize the neural network parameters, including inputting the permeability matrix and the spatial coordinate matrix into the neural network for forward propagation, outputting the predicted pressure matrix, and calculating the loss value through formula (16); calculating the gradient value through formula (17) in the backward propagation, and updating the neural network parameters to complete one training. Set a total of 10,000 generations of gradient descent training to be executed.

[0125] Use the test set sampled in step S400 for the inference ability test of the neural network. Input the permeability fields with different distributions in the test set into the trained model, and the pressure field under this permeability distribution can be quickly predicted; analyze the inference effect through two indicators, the mean absolute error and the goodness of fit. The calculation formulas of the indicators are:

[0126] (18)

[0127] (19)

[0128] where MAE is the mean square error; is the prediction result of the neural network; is the reference field; is the goodness of fit; is the mean value of the reference field.

[0129] In some embodiments, 100 samples accounting for 10% are used for training, and the remaining 900 samples accounting for 90% are used for testing. Thus, the definition, sample preparation, training, and testing of the seepage parameterized learning surrogate model are completed.

[0130] Therefore, the seepage parameterized learning surrogate model provided by this embodiment realizes the parameterized learning of the seepage mathematical model and the fast prediction based on the surrogate model, and can be used in engineering to replace the repeated numerical simulation process, taking into account both computational efficiency and prediction accuracy. This method establishes an end-to-end prediction framework by referring to the working mode of the surrogate model, enabling the trained model to directly perform fast prediction without additional calculations; combines the partial differential equation defining the physical process to establish a physical consistency evaluation criterion, enabling the model to better learn the potential physical laws; combines the neural operator learning strategy to extend the finite-dimensional mapping to an infinite-dimensional mapping, improving the generalization performance of the model and avoiding frequent retraining on new problems. Innovatively, this method designs and constructs a brand-new time-frequency double-filtered Fourier neural operator network structure, which pre-screens purer features through time-domain filtering and then enhances the main modes through low-pass filtering in the frequency domain to accurately capture the general generalization mode; combines the finite volume method to replace the automatic differentiation method for spatial derivative calculation, strictly defines the flux continuity between adjacent units with different properties, reduces the order of derivatives required for calculation, and avoids the automatic differentiation calculation required for constructing the loss term. Under the same parameter settings and computing resources, the method proposed in this application significantly improves the physical-driven training convergence of the surrogate model and has higher prediction accuracy than the data-driven surrogate model and the physical-driven surrogate model before improvement.

[0131] To prove the feasibility of the present invention, a specific example of single-phase flow in a heterogeneous porous medium based on Darcy's equation is given. The specific process of parameterized learning to construct a surrogate model in the example is as follows:

[0132] Step 1: Construct the numerical model size and grid as Figure 2 shown. The model size parameters are 500m×500m, and the grid is divided into a 100×100 Cartesian grid, with each grid size parameter being 5m×5m; the upper and lower boundaries are defined as closed boundaries; the Dirichlet boundary condition is defined as the pressure on the left boundary being 1MPa and the pressure on the right boundary being 0Mpa.

[0133] Step 2: The collected permeability distribution example is as Figure 3 shown. The sill value is 1, the nugget value is set to 0, the permeability range is set to 30 grids, and the mean, standard deviation, and range of the permeability are adjusted to generate permeability fields representing different heterogeneous environments, with the permeability range from 0.2mD to 200mD; a total of 1000 permeability samples with different distributions are collected, and 100 of them are taken as the training set, and the remaining 900 permeability samples are taken as the test set.

[0134] Step 3: Construct the neural network structure as Figure 4As shown, the input layer is set to map the permeability and the 3 channels of spatial coordinates to 10 channels; the Fourier layer is set to continue the mapping of 10 channels to 10 channels in the high-dimensional space, and the number of modes extracted is set to 10, and the number of cascaded superpositions of the Fourier layer is set to 3; the output layer is set to perform the mapping of 10 channels to a single channel.

[0135] Step 4: Perform neural network training on the training set, set the optimizer to Adam and the learning rate to 0.001; set the mini-batch size to 4 and the sampling mode to Shuffle; set the permeability field and spatial coordinates as multi-channel data to input into the neural network, calculate the finite volume residual using the output permeability field, and perform backpropagation to update the neural network; a total of 10,000 generations of gradient descent training are executed; the loss reduction image of the model is as Figure 5 shown; during the training process, the change of the mean absolute error index during synchronous inference on the test set is as Figure 6 shown, and the change of the goodness-of-fit index during synchronous inference on the test set is as Figure 7 shown.

[0136] In this embodiment, the comparison result between the pressure field inferred by the seepage parameterized learning agent model of the present invention on different permeability distributions and the reference solution pressure field is as Figure 8 shown; the average absolute error distribution result of all agent model prediction results compared with the reference solution for all 900 test samples is as Figure 9 shown; the goodness-of-fit distribution result of all agent model prediction results compared with the reference solution for all 900 test samples is as Figure 10 shown.

[0137] The embodiment of the present application also provides a seepage parameterized learning device, as Figure 11 shown, and this seepage parameterized learning device includes:

[0138] A model construction module 1101, configured to establish a seepage parameterized learning mathematical model; wherein, the seepage parameterized learning mathematical model is a single-phase flow model in heterogeneous porous media based on Darcy's equation;

[0139] A network structure building module 1102, configured to build a Fourier neural operator network structure with time-frequency double filtering; the Fourier neural operator network structure is used to learn the mapping from the permeability field and spatial coordinates to the pressure field;

[0140] A loss function design module 1103, configured to design a physical constraint loss function based on finite volume discretization;

[0141] The data set generation module 1104 is configured to perform sampling based on the seepage parameterized learning mathematical model using sequential Gaussian simulation to generate different permeability fields and divide them into a training set and a test set;

[0142] The model training and testing module 1105 is configured to train the Fourier neural operator network structure on the training set based on the physical constraint loss function, and test the inference effect on the test set.

[0143] An embodiment of the present application provides an electronic device, which may include a processor and a memory, wherein the processor and the memory can communicate with each other; illustratively, the processor and the memory communicate with each other via a communication bus.

[0144] The processor executes the computer-executable instructions stored in the memory, so that the processor implements the solutions in the above embodiments. The processor can be a general-purpose processor, including a central processing unit (CPU), a network processor (NP), etc.; it can 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, or discrete hardware components.

[0145] The communication bus can be a Peripheral Component Interconnect (PCI) bus or an Extended Industry Standard Architecture (EISA) bus. System buses can be categorized as address buses, data buses, and control buses. Transceivers facilitate communication between the database access device and other computers (e.g., clients, read-write libraries, and read-only libraries). Memory may include random access memory (RAM) or non-volatile memory.

[0146] The electronic device provided in the embodiment of the present application may be the terminal device of the above embodiment.

[0147] An embodiment of the present application further provides a computer-readable storage medium, in which computer instructions are stored. When the computer instructions are executed on a computer, the computer executes the technical solution of the percolation parameterized learning method of the above embodiment.

[0148] An embodiment of the present application also provides a computer program product. The computer program product includes a computer program stored in a computer-readable storage medium. At least one processor can read the computer program from the computer-readable storage medium, and when the at least one processor executes the computer program, the technical solution of the seepage parameterization learning method in the above embodiment can be implemented.

[0149] In several embodiments provided by the present application, it should be understood that the disclosed devices and methods can be implemented in other ways. For example, the device embodiments described above are merely illustrative. For example, the division of modules is only a logical function division. In actual implementation, there may be other division methods. For example, multiple modules can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the displayed or discussed coupling or direct coupling or communication connection to each other can be through some interfaces, and the indirect coupling or communication connection of devices or modules can be in an electrical, mechanical or other form.

[0150] The modules described as separate components may or may not be physically separated, and the components displayed as modules may or may not be physical units, that is, they may be located in one place, or may be distributed to multiple network units. Some or all of the modules can be selected according to actual needs to implement the solution of this embodiment.

[0151] In addition, in each embodiment of the present application, the functional modules can be integrated in a processing unit, or each module can exist physically alone, or two or more modules can be integrated in a unit. The unit formed by the above modules can be implemented in the form of hardware, or in the form of a hardware plus software functional unit.

[0152] The above integrated modules implemented in the form of software functional modules can be stored in a computer-readable storage medium. The above software functional modules are stored in a storage medium, including several instructions for causing a computer device (which can be a personal computer, a server, or a network device, etc.) or a processor to execute some steps of the methods in each embodiment of the present application.

[0153] It should be understood that the above-mentioned processor can be a Central Processing Unit (CPU), or other general-purpose processors, Digital Signal Processors (DSPs), Application Specific Integrated Circuits (ASICs), etc. The general-purpose processor can be a microprocessor or any conventional processor, etc. The steps of the method disclosed in combination with the invention can be directly implemented by a hardware processor, or by a combination of hardware and software modules in the processor.

[0154] The memory may include high-speed RAM memory, and may also include non-volatile storage NVM, such as at least one disk memory, and can also be a USB flash drive, a mobile hard disk, a read-only memory, a magnetic disk, or an optical disc, etc.

[0155] The bus can be an Industry Standard Architecture (ISA) bus, a Peripheral Component Interconnect (PCI) bus, an Extended Industry Standard Architecture (EISA) bus, etc. The bus can be divided into an address bus, a data bus, a control bus, etc.

[0156] The above storage medium can be implemented by any type of volatile or non-volatile storage device or a combination thereof, such as Static Random Access Memory (SRAM), Electrically Erasable Programmable Read-Only Memory (EEPROM), Erasable Programmable Read-Only Memory (EPROM), Programmable Read-Only Memory (PROM), Read-Only Memory (ROM), magnetic memory, flash memory, a magnetic disk, or an optical disc. The storage medium can be any available medium that can be accessed by a general or special-purpose computer.

[0157] An exemplary storage medium is coupled to the processor, enabling the processor to read information from the storage medium and write information to the storage medium. Of course, the storage medium can also be a component of the processor. The processor and the storage medium can be located in an Application Specific Integrated Circuit (ASIC). Of course, the processor and the storage medium can also exist as discrete components in an electronic control unit or a master control device.

[0158] Those of ordinary skill in the art will understand that all or part of the steps of implementing the above method embodiments can be completed by hardware related to program instructions. The aforementioned program can be stored in a computer-readable storage medium. When the program is executed, it performs the steps of the above method embodiments; and the aforementioned storage medium includes: various media such as ROM, RAM, magnetic disks, or optical discs that can store program codes.

[0159] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present application, rather than to limit them; although the present application has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some or all of the technical features; and these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present application.

Claims

1. A seepage parameterization learning method, characterized in that The method includes: Establishing a mathematical model for parametric learning of seepage; wherein, the mathematical model for parametric learning of seepage is a single-phase flow model in heterogeneous porous media based on Darcy's equation; Constructing a Fourier neural operator network structure with time-frequency double filtering; the Fourier neural operator network structure is used to learn the mapping from the permeability field and spatial coordinates to the pressure field; Designing a physical constraint loss function based on finite volume discretization; Based on the mathematical model for parametric learning of seepage, using sequential Gaussian simulation for sampling to generate different permeability fields, and dividing them into a training set and a test set; Based on the physical constraint loss function, training the Fourier neural operator network structure on the training set and testing the inference effect on the test set; The governing equation of the mathematical model for parametric learning of seepage is: ， wherein, is the flow velocity; is the dynamic viscosity; is the permeability field of the porous medium; is the fluid pressure; is the source-sink term; denotes the gradient operator; denotes the divergence operator; The boundary conditions of the mathematical model for parametric learning of seepage are defined as: ， where is the preset fixed pressure value of the Dirichlet boundary ; is the preset fixed velocity value of the Neumann boundary ; is the normal vector of the boundary position; on Dirichlet boundary means on the Dirichlet boundary, and on Neumann boundary means on the Neumann boundary; The process by which the Fourier neural operator network structure learns the mapping from the permeability field and spatial coordinates to the pressure field is: Taking the permeability matrix and the spatial coordinate matrix as the input matrices of the input fully connected layer, and the mapping function involving the input fully connected layer is as follows: ， wherein and are the weights and biases of the input layer; is the pressure matrix for dimension expansion; is the permeability field; and are the spatial coordinates; The mapping of pressure includes successive refinement by three gated Fourier layers, and the complete mapping process is as follows: ， Among them, is the neural network input; is the time-domain feature selection result; is the output of the gated Fourier layer; is the second-layer time-domain feature selection result; is the output of the second-layer gated Fourier layer; is the neural network output; Adaptive Feature Selection Part in the Gated Fourier Layer The mapping function is as follows: ， Among them is batch normalization, which is used to stabilize the feature distribution in the mapping; is the learnable weight; is the pointwise multiplication; is the non-linear rectifier unit activation function, which is used to sparsely activate the neurons that contribute the most to pattern recognition and realize the pre-screening of features; Frequency-domain Fourier filtering section The mapping function is as follows: ， wherein is the fast Fourier transform; is the low-pass filtering in the frequency domain; is the inverse Fourier transform; is the convolutional layer; is the number of the extracted low-frequency modes; and are respectively x and y the sequence numbers in the directions; is the maximum number of modes, is x the maximum sequence number in the is y the maximum sequence number in the The output layer maps the features back to the low-dimensional original space, and the mapping function is: ， where are the weights of the output layer; is the bias of the output layer; is the pressure matrix of the final output of the neural network.

2. The seepage parameterization learning method according to claim 1, characterized in that The Fourier neural operator network structure includes an input fully connected layer, a first gated Fourier layer, a second gated Fourier layer, a third gated Fourier layer, and an output fully connected layer; wherein, the input fully connected layer is located at the front end of the neural network, the first gated Fourier layer is located after the input fully connected layer, the second gated Fourier layer is located after the first gated Fourier layer, the third gated Fourier layer is located after the second gated Fourier layer, and the output fully connected layer is located after the third gated Fourier layer; the first gated Fourier layer, the second gated Fourier layer, and the third gated Fourier layer all include two parts: time-domain feature selection and frequency-domain Fourier filtering.

3. The seepage parameterization learning method according to claim 1, characterized in that The physical constraint loss function is expressed as: ， where is the number of units for calculating the residual loss value; is the number of units for calculating the Dirichlet boundary loss; R i is the physical constraint loss value of the i -th unit; L is the loss value; is the pressure of the i -th grid in the neural network output pressure matrix, is the pressure of the i -th grid in the reference solution pressure matrix.

4. The seepage parameterization learning method according to claim 1, characterized in that Based on the mathematical model for parametric learning of seepage, using sequential Gaussian simulation for sampling to generate different permeability fields, and dividing them into a training set and a test set, including: Setting the size parameters of the mathematical model for parametric learning of seepage, dividing the grid into Cartesian grids with the same size parameters for each Cartesian grid; defining the upper and lower boundaries as closed boundaries; defining the Dirichlet boundary condition as the pressure on the left boundary being 1 MPa and the pressure on the right boundary being 0 MPa; performing sequential Gaussian simulation to sample different permeability field distributions, setting the sill value, nugget value, and permeability range, and adjusting the mean, standard deviation, and range of the controlled permeability to generate permeability fields representing different heterogeneous environments, with the permeability range being 0.2 mD to 200 mD.

5. The seepage parameterization learning method according to claim 1, wherein Based on the physical constraint loss function, training the Fourier neural operator network structure on the training set and testing the inference effect on the test set, including: In the training of the Fourier neural operator network structure, the weights of the neural network are uniformly initialized, and the biases are uniformly initialized to 0. An optimizer is used to perform physics-driven training on unlabeled data through the mini-batch gradient descent strategy. The neural network parameter update formula is as follows: ， wherein is a small batch size; are trainable parameters in the neural network; is the learning rate; is the serial number of the samples in the small batch; is the gradient operator of the learnable parameters; is the complete neural network mapping function; is the loss value calculated based on the output result of the neural network; Based on the training set, train the Fourier neural operator network structure. Optimize the neural network parameters by continuously performing forward propagation, backward propagation, and gradient descent. This includes inputting the permeability matrix and the spatial coordinate matrix into the neural network for forward propagation to output the predicted pressure matrix, and calculating the loss value through the physical constraint loss function. In the backward propagation, calculate the gradient value through the neural network parameter update formula and update the neural network parameters to complete one training. A total of a set number of gradient descent trainings are performed. Test the inference ability of the Fourier neural operator network structure on the test set. Input permeability fields with different distributions in the test set into the trained model to quickly predict the pressure field under this permeability distribution. Evaluate the inference effect through two metrics: the mean absolute error and the goodness of fit.

6. A seepage parameterization learning device, characterized in that The device includes: A model construction module configured to establish a mathematical model for seepage parameterization learning. Among them, the mathematical model for seepage parameterization learning is a single-phase flow model in heterogeneous porous media based on Darcy's equation. A network structure building module configured to build a Fourier neural operator network structure with time-frequency double filtering. The Fourier neural operator network structure is used to learn the mapping from the permeability field and spatial coordinates to the pressure field. A loss function design module configured to design a physical constraint loss function based on finite volume discretization. A data set generation module configured to sample based on the mathematical model for seepage parameterization learning using sequential Gaussian simulation to generate different permeability fields, and divide them into a training set and a test set. A model training and testing module configured to train the Fourier neural operator network structure on the training set and test the inference effect on the test set based on the physical constraint loss function. The governing equation of the mathematical model for seepage parameterization learning is: ， wherein, is the flow velocity; is the dynamic viscosity; is the permeability field of the porous medium; is the fluid pressure; is the source-sink term; represents the gradient operator; represents the divergence operator; The boundary conditions of the mathematical model for seepage parameterization learning are defined as: ， where is the preset fixed pressure value for the Dirichlet boundary ; is the preset fixed velocity value for the Neumann boundary ; is the normal vector of the boundary position; on Dirichlet boundary means on the Dirichlet boundary, and on Neumann boundary means on the Neumann boundary; The process by which the Fourier neural operator network structure learns the mapping from the permeability field and spatial coordinates to the pressure field is: Take the permeability matrix and the spatial coordinate matrix as the input matrices of the fully connected layer. The mapping function involving the input fully connected layer is as follows: ， where and are the weights and biases of the input layer; is the pressure matrix for dimensional expansion; is the permeability field; associated with is the spatial coordinate; The mapping of pressure includes successive refinement through three gated Fourier layers. The complete mapping process is as follows: ， Among them, is the neural network input; is the time-domain feature selection result; is the output of the gated Fourier layer; is the second-layer time-domain feature selection result; is the output of the second-layer gated Fourier layer; is the neural network output; Adaptive Feature Selection Part in the Gated Fourier Layer The mapping function is as follows: ， Among them is batch normalization, which is used to stabilize the feature distribution in the mapping; is a learnable weight; is pointwise multiplication; is a non-linear rectifier unit activation function, which is used to sparsely activate the neurons that contribute the most to pattern recognition and realize the pre-screening of features; Frequency-domain Fourier filtering part The mapping function is as follows: ， where is the fast Fourier transform; is low-pass filtering in the frequency domain; is the inverse Fourier transform; is the convolutional layer; is the number of low-frequency modes extracted; and are respectively x and y the serial numbers of the directions; is the maximum number of modes, is x the maximum number of serial numbers in the is y the maximum number of serial numbers in the The output layer maps the features back to the low-dimensional original space. The mapping function is: ， wherein are the weights of the output layer; is the bias of the output layer; is the pressure matrix finally output by the neural network.

7. An electronic device, characterized in that, Includes: A processor, and a memory communicatively connected to the processor; The memory stores computer-executable instructions; The processor executes the computer-executable instructions stored in the memory to implement the seepage parameterization learning method according to any one of claims 1-5.

8. A computer-readable storage medium, characterized in that, Computer-executable instructions are stored in the computer-readable storage medium. When the computer-executable instructions are executed by the processor, they are used to implement the seepage parameterization learning method according to any one of claims 1-5.

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