Seepage parameterization learning method and device, equipment and storage medium

Abstract

The invention provides a seepage parameterization learning method and device, equipment and a storage medium. Relates to the technical field of seepage numerical simulation. The method comprises the following steps: defining a seepage parameterization learning mathematical model; building a Fourier neural operator network structure of time-frequency dual filtering; defining a physical constraint loss function based on finite volume discretization; sampling by using sequential Gaussian simulation to generate different permeability fields, and dividing the permeability fields into a training set and a test set; and training the Fourier neural operator network agent model on the training set, and testing the reasoning effect on the test set. According to the method, the training of label-free data can be realized, the results under different permeability distributions can be quickly predicted through the trained model, and the prediction precision is remarkably improved.

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, device, equipment and storage medium. Background Art

[0002] Physical information neural network is a useful paradigm that can perform forward and reverse 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 proxy 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 training method, which requires the preparation of samples and label 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] In order to solve the above problems, neural operator proxy models based on physical information-driven training have begun to be widely studied and used in engineering to replace repeated numerical simulation processes for rapid prediction of physical fields, taking into account both computational efficiency and prediction accuracy. This method establishes an end-to-end prediction framework based on the working mode of the proxy model, so that the trained model can directly make rapid predictions without additional calculations; combines the partial differential equations that define the physical process to establish physical consistency evaluation criteria, so that the model can better learn the underlying physical laws; combines the neural operator learning strategy to expand the finite-dimensional mapping to an infinite-dimensional mapping, improve the generalization performance of the model, and avoid frequent retraining on new problems. However, the current physical constraints are mainly constructed based on automatic differentiation, which makes it difficult to solve the derivative order of the physical quantity to be processed, especially the derivative approximation of the physical field with spatial inhomogeneity. In addition, the current neural operator network still has insufficient ability to extract low-frequency modes, and the reasoning accuracy is insufficient under different physical parameter field distributions. Summary of the invention

[0004] The present application provides a seepage parameterization learning method, apparatus, device, and storage medium, and designs and constructs a brand-new Fourier neural operator network structure with time-frequency double filtering. By pre-screening purer features through time-domain filtering and then enhancing the main modes through low-pass filtering in the frequency domain, it realizes the accurate capture of general generalization patterns. Combining the finite volume method to replace the automatic differentiation method for spatial derivative calculation, strictly defining the flux continuity between adjacent units with different properties, reduces the required derivative order for 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: 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; 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; Design a physical constraint loss function based on finite volume discretization; Based on the seepage parameterization learning mathematical model, use sequential Gaussian simulation for sampling to generate different permeability fields, and divide them into a training set and a test set; 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.

[0006] In a possible design, the control equation of the seepage parameterization learning mathematical model 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.

[0007] The boundary conditions of the seepage parameterization learning mathematical model are defined as: , where is the Dirichlet boundary is the preset fixed pressure value; is the Neumann boundary is the preset fixed speed value; 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.

[0008] 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; among them, 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.

[0009] In a possible design, 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 as follows: 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: , where 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; The mapping of the pressure includes successive refinement by three layers of gated Fourier layers, and the complete mapping process is as follows: , where is the neural network input; is the result of time-domain feature selection; is the output of the gated Fourier layer; is the result of the second layer of time-domain feature selection; is the output of the second layer of gated Fourier layer; is the neural network output; The adaptive feature selection part in the gated Fourier layer has a mapping function as: , 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 rectification unit activation function, which is used to sparsely activate the neurons that contribute the most to pattern recognition and achieve pre-screening of features; Frequency domain Fourier filtering part The mapping function is as follows: , 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 serial numbers in the directions; is the maximum number of modes, is x the maximum number of serial numbers in the direction, y is the maximum number of serial numbers in the , where is the weight of the output layer; is the bias of the output layer; is the pressure matrix finally output by the neural network.

[0010] In a possible design, the physical constraint loss function is expressed as: , where is the number of units used to calculate the residual loss value; is the number of units used to calculate 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.

[0011] In a possible design, based on the percolation parameterized learning mathematical model, sequential Gaussian simulation is used for sampling to generate different permeability fields, which are divided into a training set and a test set, including: Set the size parameters of the percolation parameterized learning mathematical model, divide the grid into Cartesian grids with the same size parameters for each Cartesian grid; define the upper and lower boundaries as closed boundaries; define the Dirichlet boundary condition as the pressure on the left boundary being 1 MPa and the pressure on the right boundary being 0 MPa; perform sequential Gaussian simulation to sample different permeability field distributions, set the sill value, nugget value, and permeability range, adjust and control the mean, standard deviation, and range of permeability, and generate permeability fields representing different heterogeneous environments with the permeability range from 0.2 mD to 200 mD.

[0012] In a possible design, based on the physical constraint loss function, the Fourier neural operator network structure is trained on the training set and the inference effect is tested on the test set, including: In the training of the Fourier neural operator network structure, set the weights of the neural network to be uniformly initialized and the biases to be uniformly initialized to 0. Use an optimizer to perform physics-driven training on unlabeled data through the mini-batch gradient descent strategy. The neural network parameter update formula is: , where is the mini-batch size; 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; is the loss value calculated based on the output result of the neural network; Train the Fourier neural operator network structure based on the training set, and optimize the neural network parameters by continuously performing forward, 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; calculating the gradient value through the neural network parameter update formula in the backward propagation, and updating the neural network parameters to complete one training, and performing a set number of gradient descent trainings in total; Test the inference ability of the Fourier neural operator network structure on the test set, input the permeability fields with different distributions in the test set into the trained model, and perform rapid prediction of the pressure field under this permeability distribution; evaluate the inference effect through two indicators: mean absolute error and goodness of fit.

[0013] Second aspect, the present application provides a seepage parameterization learning device, the device includes: 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; 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 perform sampling using sequential Gaussian simulation based on the seepage parameterization learning mathematical model, 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.

[0014] 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.

[0015] Fourth aspect, an embodiment of the present application provides a computer-readable storage medium, in which computer execution instructions are stored, and when the processor executes the computer execution instructions, the seepage parameterization learning method described in the first aspect above and various possible designs of the first aspect are implemented.

[0016] Fifth aspect, an embodiment of the present application provides a computer program product, including a computer program, and when the computer program is executed by a processor, the seepage parameterization learning method described in the first aspect above and various possible designs of the first aspect are implemented.

[0017] The seepage parameterization learning method, device, equipment and storage medium provided by the present application have at least the following beneficial effects: The present application can realize the 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 higher prediction accuracy than the data-driven surrogate model and the physical-driven surrogate model before improvement. Description of the Drawings

[0018] The accompanying drawings here are incorporated into the specification and form a part of this specification, showing embodiments consistent with this application, and are used together with the specification to explain the principles of this application.

[0019] Figure 1 It is a flowchart of a seepage parameterization learning method provided for an embodiment of this application; Figure 2 It is a schematic diagram of defining a numerical simulation model provided for an embodiment of this application. Among them, (a) is the numerical model size and boundary conditions, and (b) is the mesh division diagram of the numerical model; Figure 3 It is a schematic diagram of samples with different permeability distributions obtained by sampling based on the SGeMS software provided for an embodiment of this application; Figure 4 It is a schematic diagram of the Fourier neural operator network structure of time-frequency double filtering provided for an embodiment of this application; Figure 5 It is the loss decline curve of the Fourier neural operator network of time-frequency double filtering in the training set during the training process provided for an embodiment of this application; Figure 6 It is the average absolute error change curve of the Fourier neural operator network of time-frequency double filtering in the test set during the training process provided for an embodiment of this application; Figure 7 It is the goodness-of-fit change curve of the Fourier neural operator network of time-frequency double filtering in the test set during the training process provided for an embodiment of this application; Figure 8 It is a schematic diagram of the predicted pressure of the Fourier neural operator network of time-frequency double filtering in the test set after training provided for an embodiment of this application; among them, (a) is the neural network prediction result, (b) is the reference pressure field, and (c) is the error between the neural network prediction result and the reference value; Figure 9 It is a distribution diagram of the average absolute error between the predicted results of all samples in the test set and the reference value of the Fourier neural operator network of time-frequency double filtering after training provided for an embodiment of this application; Figure 10 It is a distribution diagram of the goodness-of-fit between the predicted results of all samples in the test set and the reference value of the Fourier neural operator network of time-frequency double filtering after training provided for an embodiment of this application; Figure 11 It is a structural diagram of a seepage parameterization learning device provided for an embodiment of this application.

[0020] Through the above-mentioned accompanying drawings, specific embodiments of the present application have been shown, and will be described in more detail 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

[0021] Here, exemplary embodiments will be described in detail, and examples are shown in the accompanying drawings. When the following description refers to the accompanying drawings, unless otherwise indicated, the same numerals 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.

[0022] In the technical solution of the present application, the processing of information such as financial data or user data, including collection, storage, use, processing, transmission, provision, and disclosure, complies with the provisions of relevant laws and regulations and does not violate public order and good customs.

[0023] 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 considered exemplary, and their purpose is only to illustrate the feasibility in the implementation of the technical solution of the present application, but it does not mean that the applicant has already or necessarily used this solution.

[0024] The following uses specific embodiments to describe in detail the technical solution of the present application and how the technical solution of the present application solves 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 with reference to the accompanying drawings.

[0025] The embodiment of the present application provides a seepage parameterization learning method. As Figure 1 shown, it is a flowchart of the seepage parameterization learning method provided by the embodiment of the present application. The seepage parameterization learning method includes the following steps S100 - S500.

[0026] 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.

[0027] In this embodiment, the purpose of step S100 is to define the seepage parameterized 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 parameterized learning mathematical model is defined as the 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.

[0028] In some embodiments, the governing equation of the seepage parameterized learning mathematical model is: (1) (2) Where 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.

[0029] The boundary conditions of the seepage parameterized learning mathematical model are defined as: (3) (4) Where is the Dirichlet boundary is the preset fixed pressure value; is the Neumann boundary is the preset fixed velocity value; is the normal vector at the boundary position; on Dirichlet boundary means on the Dirichlet boundary, and on Neumann boundary means on the Neumann boundary.

[0030] S200: Build a time-frequency dual-filtered 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.

[0031] In this embodiment, the time-frequency dual-filtered Fourier neural operator network structure established in step S200 constitutes a time-frequency dual-filtering structure together with the adaptive feature selection process innovatively introduced in the time domain and the low-frequency extraction process in the frequency domain, and extracts purer general low-frequency modes.

[0032] In some embodiments, the Fourier neural operator network structure with time-frequency double filtering 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, features are pre-screened in the time domain, then low-pass filtered in the frequency domain through Fourier transform, and finally inverse Fourier transformed 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.

[0033] In some embodiments, for the specific mapping function of each layer in the Fourier neural operator network model involving time-frequency double filtering, a series of network structures are innovatively designed to improve 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 Fourier neural operator network with time-frequency double filtering is introduced in detail as follows.

[0034] Taking the permeability matrix and the spatial coordinate matrix as the input matrices of the input fully connected layer, the mapping function involving the input fully connected layer is as follows: (5) where 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; the mapping of pressure includes step-by-step refinement of three gated Fourier layers, and the complete mapping process is as follows: (6) Taking the mapping of the first gated Fourier layer as an example, the mapping function of the adaptive feature selection part in the gated Fourier layer is: (7) where is batch normalization, which is used to stabilize the feature distribution in the mapping; is the learnable weight; is pointwise multiplication; It is a non - linear rectification unit activation function, which is used to sparsely activate neurons that contribute most to pattern recognition and achieve pre - screening of features; It is the result of time - domain feature selection; the frequency - domain Fourier filtering part The mapping function is as follows: (8) (9) Among them 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 serial numbers in the 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 Finally, the output layer maps the features back to the low - dimensional original space, and the mapping function is: (10) Among them is the weight of the output layer; is the bias of the output layer; is the pressure matrix finally output by the neural network.

[0035] 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 - channel spatial coordinates to 10 channels; setting the Fourier layer to continue the mapping of 10 channels to 10 channels in the high - dimensional space, setting the number of extracted modes to 10, setting the number of cascaded superpositions of the Fourier layer to 3; setting the output layer to perform the mapping of 10 channels to a single channel; the design of the neural network structure and the setting of parameters are completed.

[0036] S300: Design a physical - constraint loss function based on finite - volume discretization.

[0037] In this embodiment, step S300 discretizes the physical constraints defined based on the mathematical model by the finite - volume method, constructs a physical - consistency criterion based on flux conservation, solves the derivative order of the physical quantity to be processed, and improves the approximation of the physical - field derivative of spatial heterogeneity.

[0038] In some embodiments, the physical constraint loss function based on finite volume discretization is designed as follows: 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: (11) 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 a Neumann boundary, so the closed boundary is automatically constrained in the residual term, and formula (11) is further simplified to: (12) where is the Dirichlet boundary condition term; Combining formula (1) and formula (2), design to use the finite difference method to discretize the residual term, and the discretized form is: (13) 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 (14) where is the i th and j th contact area between the elements; and are the i th and j th normal distances from the centers of the elements to the contact surface respectively; is the i th and j th harmonic mean of the permeabilities of the elements; Combining formula (14), define the physical constraint loss value of the i th element as: (15) 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: (16) where is the number of units for calculating the residual loss value; is the number of units for calculating the Dirichlet boundary loss; is the i -th grid pressure in the pressure matrix output by the neural network, is the i -th grid pressure in the reference solution pressure matrix.

[0039] S400: Based on the percolation parameterized learning mathematical model, use sequential Gaussian simulation for sampling, generate different permeability fields, and divide them into a training set and a test set.

[0040] In some embodiments, step S400 can be performed using sequential Gaussian simulation based on the SGeMS software. Define the model size parameter 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 is 1MPa and the right boundary pressure is 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.

[0041] 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.

[0042] In this embodiment, in the training of the Fourier neural operator network, set the weights of the neural network to be uniformly initialized by Xavier, the biases to be uniformly initialized to 0, and use the Adam optimizer to perform physical-driven training of unlabeled data through the mini-batch gradient descent strategy. The neural network parameter update formula is: (17) 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 a small batch; is the gradient operator of the learnable parameters; is the complete neural network mapping function; represents the loss value calculated based on the output result of the neural network.

[0043] Use the training set sampled in step S400 for the training of the neural network. By continuously performing forward propagation, backward propagation, and gradient descent, 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); in the backward propagation, calculate the gradient value through formula (17), and update the neural network parameters to complete one training. Set a total of 10,000 generations of gradient descent training to be executed; 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: mean absolute error and goodness of fit. The calculation formulas of the indicators are: (18) (19) 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.

[0044] 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 proxy model are completed.

[0045] Therefore, the seepage parameterized learning surrogate model provided by this embodiment realizes the parameterized learning of the seepage mathematical model and 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 partial differential equations defining physical processes to establish a physical consistency evaluation criterion, enabling the model to better learn 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 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 achieve precise capture of general generalization patterns; combines the finite volume method to replace the automatic differentiation method for spatial derivative calculation, strictly defines the flux continuity between adjacent cells with different properties, reduces the required derivative order for calculation, and avoids the automatic differentiation calculation required for constructing the loss term. Under the same parameter settings and computational resources, the method proposed in this application significantly improves the physical-driven training convergence of the surrogate model and has higher prediction accuracy than data-driven surrogate models and the physical-driven surrogate model before improvement.

[0046] 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 constructing a surrogate model through parameterized learning in the example is as follows: Step 1: Construct the numerical model size and grid as Figure 2 shown. The model size parameter is 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.

[0047] Step 2: The example of the collected permeability distribution 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 being 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.

[0048] 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, 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.

[0049] Step 4: Conduct 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 be 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 in the mean absolute error index during synchronous inference on the test set is as Figure 6 shown, and the change in the goodness-of-fit index during synchronous inference on the test set is as Figure 7 shown.

[0050] In this embodiment, the comparison results between the pressure fields inferred by the seepage parameterized learning proxy model of the present invention on different permeability distributions and the reference solution pressure field are as Figure 8 shown; the average absolute error distribution results of all proxy model prediction results compared with the reference solution for all 900 test samples are as Figure 9 shown; the goodness-of-fit distribution results of all proxy model prediction results compared with the reference solution for all 900 test samples are as Figure 10 shown.

[0051] The embodiment of the present application also provides a seepage parameterized learning device, as Figure 11 shown. The seepage parameterized learning device includes: 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; 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; A loss function design module 1103, configured to design a physical constraint loss function based on finite volume discretization; A data set generation module 1104, configured to sample using sequential Gaussian simulation based on the seepage parameterized learning mathematical model to generate different permeability fields, and divide them into a training set and a test set; 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 reasoning effect on the test set.

[0052] An embodiment of the present application provides an electronic device, which may include: a processor and a memory, wherein the processor and the memory may communicate with each other; illustratively, the processor and the memory communicate with each other via a communication bus.

[0053] The processor executes the computer execution instructions stored in the memory, so that the processor executes the scheme in the above embodiment. 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 gates or transistor logic devices, and discrete hardware components.

[0054] The communication bus can be a peripheral component interconnect (PCI) bus or an extended industry standard architecture (EISA) bus. The system bus can be divided into an address bus, a data bus, a control bus, etc. The transceiver is used to implement communication between the database access device and other computers (such as clients, read-write libraries, and read-only libraries). The memory may include random access memory (RAM) and may also include non-volatile memory.

[0055] The electronic device provided in the embodiment of the present application may be the terminal device of the above embodiment.

[0056] The embodiment of the present application also 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 parameterization learning method of the above embodiment.

[0057] An embodiment of the present application also provides a computer program product, which 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. When at least one processor executes the computer program, it can implement the technical solution of the percolation parameterization learning method in the above embodiment.

[0058] 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, direct coupling, or communication connection between each other can be through some interfaces, and the indirect coupling or communication connection of devices or modules can be in electrical, mechanical, or other forms.

[0059] The modules described as separate components may or may not be physically separated. The components displayed as modules may or may not be physical units, that is, they may be located in one place or 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.

[0060] 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 one 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.

[0061] The 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 to enable 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.

[0062] It should be understood that the above processor can be a central processing unit (Central Processing Unit, abbreviated as CPU), and can also be other general-purpose processors, digital signal processors (Digital Signal Processor, abbreviated as DSP), application specific integrated circuits (Application Specific Integrated Circuit, abbreviated as ASIC), etc. The general-purpose processor can be a microprocessor or the processor can also be any conventional processor, etc. The steps of the method disclosed in combination with the invention can be directly embodied as being executed by the hardware processor, or executed by the combination of hardware and software modules in the processor.

[0063] 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 disk, or an optical disc, etc.

[0064] 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.

[0065] 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 accessible by a general-purpose or special-purpose computer.

[0066] 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 Circuits (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.

[0067] Those of ordinary skill in the art can understand that all or part of the steps of implementing the above method embodiments can be completed by hardware related to program instructions. The foregoing program can be stored in a computer-readable storage medium. When the program is executed, it executes the steps including the above method embodiments; and the foregoing storage medium includes: various media such as ROM, RAM, magnetic disks or optical discs that can store program codes.

[0068] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present application and are not intended 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 on 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 percolation parametric learning; wherein, the mathematical model for percolation parametric learning 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 percolation parametric learning, using sequential Gaussian simulation for sampling, generating 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.

2. The seepage parameterization learning method according to claim 1, characterized in that, The governing equation of the mathematical model for percolation parametric learning is: ， Among them, 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 percolation parametric learning are defined as: ， 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 means on the Dirichlet boundary, and on Neumann boundary means on the Neumann boundary.

3. 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 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.

4. The seepage parameterization learning method according to claim 3, characterized in that, 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 dimensional 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 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 pre-screening of features; Frequency-domain Fourier filtering part The mapping function is as follows: ， Among them 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 sequence numbers in the directions; is the maximum number of modes, is x the maximum sequence number in the direction, is y the maximum sequence number in the direction; The output layer maps the features back to the low-dimensional original space, and the mapping function is: ， Among them 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.

5. The seepage parameterization learning method according to claim 1, characterized in that, The physical constraint loss function is expressed as: ， Among them is the number of units used to calculate the residual loss value; is the number of units used to calculate 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.

6. The seepage parameterization learning method according to claim 1, characterized in that,Based on the mathematical model for percolation parametric learning, using sequential Gaussian simulation for sampling, generating different permeability fields, and dividing them into a training set and a test set, including: Setting the size parameters of the mathematical model for percolation parametric learning, 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, adjusting the mean, standard deviation, and range of the controlled permeability, generating permeability fields representing different heterogeneous environments, and the permeability range is from 0.2 mD to 200 mD.

7. The seepage parameterization learning method according to claim 1, characterized in that, 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 of unlabeled data through the mini-batch gradient descent strategy. The neural network parameter update formula is as follows: ， wherein is the small batch size; are the trainable parameters 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 parameter; 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, the Fourier neural operator network structure is trained. By continuously performing forward and backward propagation and gradient descent, the neural network parameters are optimized, 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; in the backward propagation, the gradient value is calculated through the neural network parameter update formula, and the neural network parameters are updated to complete one training. A total of a set number of gradient descent trainings are performed; The inference ability of the Fourier neural operator network structure of the test set is tested. The permeability fields with different distributions in the test set are input into the trained model to quickly predict the pressure field under this permeability distribution; the inference effect is evaluated through two indicators: the mean absolute error and the goodness of fit.

8. A seepage parameterization learning device, characterized in that, The device includes: A model construction module configured to establish a mathematical model for percolation parameterization learning; wherein, the mathematical model for percolation 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 percolation parameterization learning mathematical model 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.

9. An electronic device, characterized in that, Including: 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 percolation parameterization learning method according to any one of claims 1-7.

10. A computer-readable storage medium, characterized in that, Computer-executable instructions are stored in the computer-readable storage medium, and when the computer-executable instructions are executed by the processor, they are used to implement the percolation parameterization learning method according to any one of claims 1-7.

Images