A physical information-guided deep learning geoelectric model electromagnetic forward simulation method

Abstract

The present invention relates to the fields of geophysics and artificial intelligence, and specifically discloses a method for electromagnetic forward modeling of a deep learning geoelectric model guided by physical information, the method comprising: S1, preparing neural network input data; S2, building a fully connected deep neural network; S3, adding physical information constraints, adding the Helmholtz equation satisfied by the electromagnetic field of the geoelectric model and the boundary conditions as physical information constraints to the loss function; S4, performing network training, using automatic differentiation to obtain the partial derivative of the network output to the input, and obtaining a trained fully connected deep neural network; S5, performing result prediction, inputting the spatial coordinates of any point into the training network that has saved the neural network model parameters, and obtaining the electromagnetic response of the point in the geoelectric model. The present invention realizes the solution of the electromagnetic response at any position in the geoelectric model, and provides an intelligent solution for the calculation of electromagnetic response in electromagnetic exploration.

Description

Technical Field

[0001] The present invention relates to the field of geophysics and artificial intelligence technology, and specifically to a physical information-guided deep learning geoelectric model electromagnetic forward simulation method. Background Art

[0002] Electromagnetic exploration has a wide range of application value in the exploration and prospecting of groundwater resources, mineral resources, geothermal resources, oil and gas resources, and deep structures. Electromagnetic exploration uses the large electrical difference between the underground target anomaly and the surrounding medium to delineate the resistivity anomaly area, thereby achieving the characterization of the spatial position and occurrence form of the underground target anomaly. The physical mapping relationship between the resistivity distribution and the electromagnetic response of the geoelectric model is that the electromagnetic response of any point in the geoelectric model satisfies Maxwell's equations.

[0003] Due to the large computational space and high model complexity, in order to balance computational accuracy and efficiency, the traditional approach is to divide the model space area into many grids given a geoelectric model, calculate the electromagnetic fields at these grid nodes using numerical simulation methods such as finite differences and finite elements, and then approximate the electromagnetic response of the entire model space through interpolation fitting. However, although the meshing interpolation technology simplifies the solution of the computational space to a certain extent, it essentially sacrifices a certain degree of computational accuracy in exchange for higher computational efficiency. In addition, although the meshing interpolation technology has very good technical transferability, in the era of big data, simple and repetitive large-scale matrix solutions are somewhat weak and inefficient when dealing with massive model calculations.

[0004] In recent years, in the era of big data, with the advancement of artificial intelligence technology, artificial neural networks have provided many intelligent solutions for human social life and new ideas and methods for geophysics. Neural networks achieve the mapping between network input and output by superimposing the weights of neurons, and have very powerful nonlinear expression capabilities. However, there are relatively mature physical mechanisms and mathematical logics behind solving geophysical problems, especially the electromagnetic response must strictly satisfy Maxwell's electromagnetic theory, while traditional neural networks are overly dependent on data and lack the support of physical laws, which seriously restricts their generalization and prediction accuracy. Therefore, only by adding physical constraints to the data features extracted by the neural network can the advantages of the neural network be maximized to achieve intelligent solutions to the electromagnetic response of the geoelectric model. Summary of the invention

[0005] In view of the above-mentioned problems in the prior art, the present invention provides a physical information-guided deep learning geoelectric model electromagnetic forward simulation method, which can avoid the accuracy loss caused by the grid subdivision and interpolation technology in the traditional numerical method, and realize the solution of the electromagnetic response of the geoelectric model driven by both data and physical models, providing an intelligent solution for the calculation of electromagnetic responses in electromagnetic exploration.

[0006] The technical solution adopted by the present invention is as follows:

[0007] A physical information-guided deep learning geoelectric model electromagnetic forward simulation method, the steps are as follows:

[0008] S1. Prepare neural network input data, sample points from the calculation area of ​​the geoelectric model to be solved and obtain their corresponding spatial coordinates. The prediction point can be any point in the calculation area, and its spatial coordinates are also obtained by sampling;

[0009] S2. Build a fully connected deep neural network, add an activation function in the hidden layer, input the normalized sampling point coordinates, and output the real part of the electric field E after normalization of the surface electric field value. r and the imaginary part E i ;

[0010] S3, adding physical information constraints, taking the residual between the output obtained after inputting the data point coordinates into the neural network and its true electromagnetic response as the data error, and adding the Helmholtz equation satisfied by the electromagnetic field of the geoelectric model and the boundary conditions as physical information constraints into the loss function;

[0011] S4. Perform network training, use automatic differentiation to obtain the partial derivative of the network output with respect to the input, i.e., the partial differential term in the Helmholtz equation, to obtain a trained fully connected deep neural network. Specifically, use a suitable optimizer and learning rate, and use error back propagation to iteratively update the network parameters, so that the loss function is minimized and converges;

[0012] S5. Predict the results, input the spatial coordinates of any point into the training network that has saved the neural network model parameters, and obtain the electromagnetic response of the point in the geoelectric model.

[0013] Preferably, in S1, the electromagnetic response of the geoelectric model includes magnetotelluric of natural field sources and controllable source electromagnetic of artificial field sources, and includes two polarization modes: TE / TM.

[0014] Preferably, in S2, the deep neural network includes multiple hidden layers, each hidden layer includes multiple neurons; the activation function is a sine or tanh nonlinear activation function; the coordinate normalization method is Min-Max normalization and is normalized to the interval [0,1].

[0015] Preferably, in S3, the data residual is calculated using three calculation methods: MAE, MSE, and RMSE. The Helmholtz equation includes two polarization modes: TE / TM. The Helmholtz equation satisfied is:

[0016]

[0017] Among them, E x is the electric field value, ω=2πf is the angular frequency, μ=4π×10 -7 is the magnetic permeability, σ is the electrical conductivity; then the partial differential equation loss satisfied is:

[0018]

[0019] Among them, Loss pde represents the partial differential equation loss, N pde represents the number of configuration points that satisfy the loss;

[0020] When the data residual strictly satisfies the Helmholtz equation, the error is zero. If it does not satisfy the Helmholtz equation, the error is caused by the network output not strictly satisfying the Helmholtz equation.

[0021] Preferably, in S3, the constraint of the Helmholtz equation is decomposed into the real part pde r , imaginary part pde i The two parts are expressed as:

[0022]

[0023] Preferably, in S3, the boundary conditions include Dirichlet (first type) boundary conditions, Neumann (second type) boundary conditions, and Robin (third type) boundary conditions; the upper boundary conditions are given in the form of strong constraints, and the assumptions for constructing the neural network solution are:

[0024] E r =1-tanh(αz * ) r ;

[0025] E i =tanh(αz * ) i ;

[0026] where u r ,u i is the neural network output, E r ,E i For E x The real and imaginary parts of , tanh is the hyperbolic tangent function, α is a trainable parameter, and its initial value is 1;

[0027] The lower boundary condition of the geoelectric model is given by the third type of boundary condition, and the lower boundary loss is defined as:

[0028]

[0029] Where N lb represents the number of lower boundary configuration points, z N is the coordinate of the maximum depth point.

[0030] Preferably, in S3, the loss function is a weighted sum of the partial differential equation error and the boundary condition error, and the final composite loss function Loss total for:

[0031] Loss total =Loss pde +Loss lbc .

[0032] Preferably, in S4, the partial differential terms are the second-order partial derivatives of the electromagnetic field in the Helmholtz equation and the first-order partial derivatives of the electromagnetic field in the lower boundary condition; the optimizer includes gradient descent, adaptive and quasi-Newton optimizers; the learning rate update strategy includes step size adjustment, exponential decay, adaptive adjustment and periodic adjustment strategies.

[0033] Preferably, in S4, the automatic differentiation is implemented by the built-in differentiation engine torch.autograd in PyTorch, and the network output E is obtained by chain derivation method. r ,E i The partial derivative with respect to z is The minimization of the loss function is achieved by cumulative gradient descent during back-propagation.

[0034] Preferably, in S5, in step S4, Then, the magnetic field response of the point in the geoelectric model is predicted based on the relationship between the electric field and the magnetic field. The expression is:

[0035]

[0036] Where Hy is the magnetic field response of the geoelectric model under TE polarization mode;

[0037] The neural network is a deep neural network that adds physical constraints to the loss function; the spatial coordinates of any point are normalized coordinates.

[0038] The present invention has the following characteristics and advantages:

[0039] The method proposed in the present invention utilizes a physical information neural network to solve the electromagnetic response of the geoelectric model, avoiding the accuracy loss caused by the grid subdivision and interpolation technology in the traditional numerical method. At the same time, it also adds the Helmholtz equation and boundary condition physical constraints on the basis of the traditional neural network, and realizes the solution of the electromagnetic response of the geoelectric model driven by both data and physical models, providing an intelligent solution for the calculation of electromagnetic response in electromagnetic exploration.

[0040] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments, but the embodiments should not be construed as limiting the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 A schematic diagram of a physical information-guided deep learning geoelectric model electromagnetic forward simulation method of the present invention;

[0042] Figure 2 A flow chart of a physical information-guided deep learning geoelectric model electromagnetic forward simulation method of the present invention;

[0043] Figure 3 A fully connected neural network provided for an embodiment of a physical information guided deep learning geoelectric model electromagnetic forward simulation method of the present invention;

[0044] Figure 4 A geoelectric model provided for an embodiment of a physical information-guided deep learning geoelectric model electromagnetic forward simulation method of the present invention;

[0045] Figure 5 A schematic diagram of the error between the electric field response neural network prediction result and the true value provided in an embodiment of the electromagnetic forward simulation method of a deep learning geoelectric model guided by physical information of the present invention;

[0046] Figure 6 A schematic diagram of the error between the magnetic field response neural network prediction result and the true value provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0047] The technical solution of the present invention is further described below through the accompanying drawings and embodiments.

[0048] The technical solution of the present invention will be described clearly and completely below in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0049] Figure 1It is a schematic diagram of a physical information-guided deep learning geoelectric model electromagnetic forward simulation method provided according to an embodiment of the present invention. In the embodiment of the present invention, the electromagnetic response of the geoelectric model includes magnetotelluric (MT) of natural field sources and controlled source electromagnetic (CSEM) of artificial field sources, and includes two polarization modes TE / TM. The embodiment of the present invention takes one-dimensional magnetotelluric TE polarization as an example, and the geoelectric model is a one-dimensional layered resistivity model. All codes are written based on the PyTorch deep learning framework. Figure 2 As shown, the method specifically comprises the following steps:

[0050] S1. Prepare neural network input data, sample points from the calculation area of ​​the geoelectric model to be solved and obtain their corresponding spatial coordinates. The prediction point can be any point in the calculation area, and its spatial coordinates are also obtained by sampling;

[0051] In S1, the sampling points are a certain number of random sampling points in space. In the embodiment of the present invention, a uniform sampling method is used to uniformly sample 10001 sampling points at 1 meter intervals in the interval [0,10000] in the z direction, and normalize them to the range [0,1]; the prediction points are 10000 random points in the range [0,1]. After the coordinate z is normalized to [0,1] by Min-Max:

[0052]

[0053] S2. Build a fully connected deep neural network, add an activation function in the hidden layer, input the normalized sampling point coordinates, and output the real and imaginary parts of the electric field normalized by the surface electric field value;

[0054] In this embodiment of the present invention, the fully connected neural network in S2 is as follows: Figure 3 As shown in the figure, the input layer size is 1, the output layer size is 2, the middle hidden layer contains 5 fully connected layers, each layer contains 50 neurons, and the activation function uses the tanh activation function. Input z * ∈[0,1], the outputs are E x The real part E r , imaginary part E i .

[0055] S3, adding physical information constraints, taking the residual between the output obtained after inputting the data point coordinates into the neural network and its true electromagnetic response as the data error, and adding the Helmholtz equation satisfied by the electromagnetic field of the geoelectric model and the boundary conditions as physical information constraints into the loss function;

[0056] In the embodiment of the present invention, the network residual RMSE calculation method in S3 is calculated according to the Helmholtz equation: Among them, E x is the electric field value, ω=2πf is the angular frequency, μ=4π×10 -7 is the magnetic permeability, σ is the electrical conductivity;

[0057] make,

[0058] Among them, Loss pde represents the partial differential equation loss, N pde represents the number of configuration points that satisfy the loss;

[0059] When the Helmholtz equation is strictly satisfied, this error is zero. Otherwise, the error of this calculation is the error caused by the network output not strictly satisfying the Helmholtz equation. It is worth noting that in the embodiment of the present invention, since the network needs to calculate and output the real and imaginary parts of the electromagnetic field separately, the constraints of the Helmholtz equation also need to be decomposed into the real part pde r , imaginary part pde i Two parts:

[0060]

[0061] In the present invention, the upper boundary condition of TE polarization is given in the form of a strong constraint, that is, the assumption for constructing the neural network solution:

[0062] E r =1-tanh(αz * ) r ;

[0063] E i =tanh(αz * ) i ;

[0064] Among them, u r ,u i is the neural network output, E r ,E i For E x The real and imaginary parts of , tanh is the hyperbolic tangent function, α is a trainable parameter, and its initial value is 1;

[0065] In the embodiment of the present invention, the lower boundary condition of the geoelectric model is given by the third type of boundary condition, and the lower boundary loss is defined as:

[0066]

[0067] Where N lb represents the number of lower boundary configuration points, z N is the coordinate of the maximum depth point;

[0068] Based on the above two losses, the final composite loss function Loss is obtainedtotal :

[0069] Loss total =Loss pde +Loss lbc ;

[0070] S4. Perform network training, use automatic differentiation to obtain the partial derivative of the network output with respect to the input, i.e., the partial differential term in the Helmholtz equation, to obtain a trained fully connected deep neural network. Specifically, use a suitable optimizer and learning rate, and use error back propagation to iteratively update the network parameters, so that the loss function is minimized and converges;

[0071] In the embodiment of the present invention, in S4, automatic differentiation is implemented based on the built-in differentiation engine torch.autograd in PyTorch, and the network output E is obtained by chain derivation method. r ,E i Partial derivative with respect to z

[0072] In the embodiment of the present invention, the optimizer is an Adam optimizer, and the training cycle of Adam is set to 50,000 rounds, the initial learning rate is 0.005, and an adaptive adjustment strategy is adopted. When the loss does not decrease for 1,500 consecutive rounds, the learning rate decays to half of the original value, and the minimum learning rate is 10 -5 .

[0073] In the embodiment of the present invention, minimization of the loss function is achieved by cumulative gradient descent during the back-propagation process.

[0074] S5. Predict the results, input the spatial coordinates of any point into the training network that has saved the neural network model parameters, and obtain the electromagnetic response of the point in the geoelectric model.

[0075] At the same time, we can obtain After that, the magnetic field response of the point in the geoelectric model can be quickly predicted based on the relationship between the electric field and the magnetic field. The expression is:

[0076]

[0077] Where Hy is the magnetic field response of the geoelectric model under the TE polarization mode.

[0078] The following is a set of actual results of the complete implementation of the method of the present invention. The geoelectric model corresponding to the results is as follows: Figure 4As shown, the model background resistivity is 100Ω·m, and the model depth is 10000m, wherein the model can be divided into a low resistance model, a uniform half-space model, and a high resistance model according to the different resistivities in the depth range of 1000m to 2000m set to 1 / 100 / 10000Ω·m. For different detection frequencies of 1Hz, 10Hz, 100Hz, and 1000Hz, after a series of steps in the above embodiment, the comparison results of the electromagnetic field response prediction results (PINN prediction real part PINN-Real, imaginary part PINN-Imag) of the neural network corresponding to the three models and the real value (finite difference result real part FD-Real, imaginary part FD-Imag) are as follows: Figure 5 and Figure 6 As shown, the accuracy of the prediction results is very consistent with the true value, which verifies the effectiveness of the method of the present invention.

[0079] Therefore, the method proposed in the present invention avoids the accuracy loss caused by the grid subdivision and interpolation technology in the traditional numerical method, and at the same time realizes the solution of the electromagnetic response of the geoelectric model driven by both data and physical models, thereby improving the calculation efficiency.

[0080] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A physical information-guided deep learning geoelectric model electromagnetic forward simulation method, characterized in that: Here are the steps: S1. Prepare neural network input data, sample points from the calculation area of ​​the geoelectric model to be solved and obtain their corresponding spatial coordinates. The prediction point can be any point in the calculation area, and its spatial coordinates are also obtained by sampling; S2. Build a fully connected deep neural network, add an activation function in the hidden layer, input the normalized sampling point coordinates, and output the real part of the electric field normalized by the surface electric field value. and the imaginary part ; S3, adding physical information constraints, taking the residual between the output obtained after inputting the data point coordinates into the neural network and its true electromagnetic response as the data error, and adding the Helmholtz equation satisfied by the electromagnetic field of the geoelectric model and the boundary conditions as physical information constraints into the loss function; In S3, the data residual is calculated using three calculation methods: MAE, MSE, and RMSE; the Helmholtz equation includes two polarization modes, TE / TM, and the satisfied Helmholtz equation is: ； in, is the electric field value, , is the angular frequency, is the magnetic permeability, is the conductivity, then the partial differential loss is: ； in, represents the partial differential equation loss, represents the number of configuration points that satisfy the loss, Used to traverse from 1 to For each element in are the spatial vertical coordinates of these configuration points; When the data residual strictly satisfies the Helmholtz equation, the error is zero. If it does not, the error is caused by the network output not strictly satisfying the Helmholtz equation. In S3, the constraints of the Helmholtz equation are decomposed into the real part , imaginary part The two parts are expressed as: ； ； in, is the symbol of partial derivative, for The real and imaginary parts of In S3, the boundary conditions include the first-type Dirichlet boundary conditions, the second-type Neumann boundary conditions, and the third-type Robin boundary conditions; the upper boundary conditions of the geoelectric model are given in the form of strong constraints, and the assumptions for constructing the neural network solution are: ； ； in, is the neural network output, is the hyperbolic tangent function, is a trainable parameter, is the normalized vertical coordinate in space; The lower boundary condition of the geoelectric model is given by the third type of boundary condition, and the lower boundary loss is defined as: ； In the formula, Indicates the number of lower boundary configuration points, Used to traverse from 1 to For each element in is the coordinate of the maximum depth point; S4. Perform network training, use automatic differentiation to obtain the partial derivative of the network output with respect to the input, i.e., the partial differential term in the Helmholtz equation, to obtain a trained fully connected deep neural network. Specifically, use a suitable optimizer and learning rate, and use error back propagation to iteratively update the network parameters, so that the loss function is minimized and converges; S5. Predict the results, input the spatial coordinates of any point into the training network that has saved the neural network model parameters, and obtain the electromagnetic response of the point in the geoelectric model.

2. According to a physical information guided deep learning geoelectric model electromagnetic forward simulation method according to claim 1, it is characterized in that: In S1, the electromagnetic response of the geoelectric model includes the magnetotelluric of natural field sources and the controllable source electromagnetic of artificial field sources, and includes two polarization modes: TE / TM.

3. According to the physical information guided deep learning geoelectric model electromagnetic forward simulation method of claim 1, it is characterized in that: In S2, the deep neural network includes multiple hidden layers, each hidden layer includes multiple neurons; the activation function is a nonlinear activation function such as sine and tanh; the coordinate normalization method is Min-Max normalization and normalized to interval.

4. According to a physical information guided deep learning geoelectric model electromagnetic forward simulation method according to claim 1, it is characterized in that: In S3, the loss function is the weighted sum of the partial differential equation error and the boundary condition error, and the final composite loss function is for: 。 5. According to a physical information guided deep learning geoelectric model electromagnetic forward simulation method according to claim 1, it is characterized in that: In S4, the partial differential terms are the second-order partial derivatives of the electromagnetic field in the Helmholtz equation and the first-order partial derivatives of the electromagnetic field in the lower boundary condition; the optimizer includes gradient descent, adaptive and quasi-Newton optimizers; the learning rate update strategy includes step size adjustment, exponential decay, adaptive adjustment and periodic adjustment strategies.

6. According to a physical information guided deep learning geoelectric model electromagnetic forward simulation method according to claim 1, it is characterized in that: In S4, the automatic differentiation is implemented based on the built-in differentiation engine torch.autograd in PyTorch, and the network output is obtained by chain derivation method. right The partial derivative of ; The minimization of the loss function is achieved by cumulative gradient descent during the back propagation process.

7. The method of electromagnetic forward modeling of a deep learning geoelectric model guided by physical information according to claim 1, characterized in that: In S5, the step S4 obtains Then, the magnetic field response of the point in the geoelectric model is predicted based on the relationship between the electric field and the magnetic field. The expression is: ； In the formula, is the magnetic field response of the geoelectric model under TE polarization mode; The neural network is a deep neural network that adds physical constraints to the loss function; the spatial coordinates of any point are normalized coordinates.

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