CN117390961A - Method for solving electromagnetic response of ground model by using physical information neural network - Google Patents

Abstract

The invention provides a method for solving electromagnetic response of an earth model by using a physical information neural network, which comprises the following steps: firstly, acquiring space coordinates of known data points, and extracting part of sampling points from a calculation region of a ground model to be solved to acquire the space coordinates; then, calculating and outputting electromagnetic field values corresponding to the data points and the space coordinates of the sampling points by using a neural network; then, taking the residual error of the output obtained after the data point coordinates are input into the neural network and the real electromagnetic response as the data error of the loss function, and taking the electromagnetic control equation and the boundary condition which are met by all points as physical constraint to add the loss function; finally, the space distribution condition of the electromagnetic response of the ground model can be obtained through an automatic differentiation technology and error counter propagation minimization loss function; so far, the physical information neural network can be utilized to solve electromagnetic response at any position in the ground model.

Description

A method for solving electromagnetic response of geoelectric model using physical information neural network

Technical field

The present invention relates to the fields of geophysics and artificial intelligence, and in particular, to a method for solving the electromagnetic response of a geoelectric model using a physical information neural network.

Background technique

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

Due to the large calculation space and high model complexity, in order to balance calculation accuracy and calculation efficiency, the usual traditional approach is to give a geoelectric model, divide the model space area into many grids, and use numerical simulations such as finite difference and finite element to method to calculate the electromagnetic fields at these grid nodes, and then approximate the electromagnetic response of the entire model space through interpolation fitting. However, although the gridding interpolation technology simplifies the solution of the calculation space to a certain extent, it essentially sacrifices a certain calculation accuracy in exchange for higher calculation efficiency. In addition, although gridding interpolation technology has very good technology migration, in the era of big data, simple and repeated large-scale matrix solving is slightly weak and inefficient when dealing with massive model calculations.

In recent years, in the era of big data and with the advancement of artificial intelligence technology, artificial neural networks have provided many intelligent solutions for human social life, and also provided new ideas and methods for geophysics. Neural networks realize the mapping between network input and output through the weight overlap of neurons, and have very powerful nonlinear expression capabilities. However, there are relatively mature physical mechanisms and mathematical logic behind solving geophysical problems. In particular, the electromagnetic response must strictly satisfy Maxwell's electromagnetic theory. However, traditional neural networks are overly dependent on data and lack the support of physical laws, which seriously restricts their use. Generalization usage and its predictive accuracy. Therefore, only by adding physical constraints to the data features extracted by the neural network can we maximize the advantages of the neural network and achieve intelligent solutions to the electromagnetic response of the geoelectric model.

Contents of the invention

In order to solve the above technical problems or at least partially solve the above technical problems, the present invention provides a method for solving the electromagnetic response of a geoelectric model using a physical information neural network.

In order to achieve the above objects, the technical solutions of the present invention are as follows:

A method of using physical information neural network to solve the electromagnetic response of a geoelectric model, including the following steps:

Step 1. Construct an electromagnetic response training data set;

Step 2: Construct a fully connected deep neural network;

Among them, the loss function loss of the fully connected deep neural network is as follows:

Among them, loss_u is the data error loss_u, loss_f is the physical constraint error; α is the balance coefficient;

Step 3: Input the electromagnetic response training data set into the fully connected deep neural network to minimize the loss function to obtain the trained fully connected deep neural network;

Step 4: Input the true electromagnetic response and corresponding spatial coordinates of the sampling points in the calculation area of the geoelectric model to be solved into the trained fully connected deep neural network, and predict the electromagnetic response of the predicted point.

In a further improvement, the electromagnetic response training data set includes the spatial coordinates of known data points and their corresponding real electromagnetic responses.

As a further improvement, the spatial coordinates of the known data points are surface observation points and Min-Max is normalized to the [0,1] interval.

Further improvement, in step two,

,

According to the constraints of Helmholtz equation, we get:

,

in, is the number of data points,/> is the number of sampling points,/> is the network output of the fully connected deep neural network for the nth training data,/> is the true value of the nth training data;/> Is the unit imaginary number/> ,/> is the angular frequency,/> is the frequency,/> is the magnetic permeability,/> is the conductivity,/> is the Laplacian operator.

As a further improvement, the fully connected deep neural network includes an input layer, an intermediate hidden layer and an output layer; the intermediate hidden layer includes 10 fully connected layers, each layer including 32 neurons; the size of the input layer is 1, and the size of the output layer is 1. Size is 4.

As a further improvement, the activation function of the fully connected deep neural network is a nonlinear activation function, and the nonlinear activation function includes Sine and Tanh.

As a further improvement, in the third step, the fully connected deep neural network calculates and outputs the real and imaginary parts of the electromagnetic field separately, and the constraints of the Helmholtz equation are disassembled into the real part of the electric field. , imaginary part of electric field/> , real part of magnetic field/> , the imaginary part of the magnetic field Four parts:

in, is the partial derivative symbol,/> is the normalized spatial coordinate,/> for/> The real part of ,/> for/> The imaginary part of ;/> for/> The real part of ,/> for/> The imaginary part of ;/> is the electric field value after normalization of the field value,/> is the magnetic field value after normalization of the field value.

Further improvement, in the third step, the partial derivative of the output of the fully connected deep neural network with respect to the input is obtained through automatic differentiation, that is, the partial differential term in the Helmholtz equation, using the optimizer and learning rate, and using the error inverse Iteratively updates the parameters of the fully connected deep neural network through forward propagation to minimize the loss function.

As a further improvement, the partial differential term is the second-order partial derivative term of the electric/magnetic field in the Helmholtz equation; the optimizer includes gradient descent type, adaptive type and quasi-Newton type optimizers; the learning rate Update strategies include step size adjustment, exponential decay, adaptive adjustment and periodic adjustment strategies.

The method of the present invention uses 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, the Helmholtz equation and the Helmholtz equation are added to the traditional neural network. Physical constraints such as boundary value conditions realize the electromagnetic response solution of the geoelectric model driven by both data and physical models, providing an intelligent solution for the calculation of electromagnetic response in electromagnetic exploration.

Description of the drawings

In order to explain the present invention or the technical solutions in the prior art more clearly, the drawings needed to be used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings in the following description are of the present invention. For some embodiments of the invention, those of ordinary skill in the art can also obtain other drawings based on these drawings without exerting creative efforts.

Figure 1 is a schematic diagram of the method of the present invention.

Figure 2 is a flow chart of the method of the present invention.

Figure 3 is a fully connected neural network provided by an embodiment of the present invention.

Figure 4 is a geoelectric model provided by an embodiment of the present invention.

Figure 5 is a schematic diagram of the error between the electric field response neural network prediction result and the true value provided by the embodiment of the present invention.

Detailed ways

The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts fall within the scope of protection of the present invention.

Figure 1 is a schematic diagram of a method for solving the electromagnetic response of a geoelectric model using a physical information neural network 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 should include two polarization modes: TE/TM. The embodiment of the present invention takes one-dimensional magnetotelluric TE polarization as an example. The geoelectric model is a one-dimensional layered resistivity model. All codes are written based on the PyTorch deep learning framework. As shown in Figure 2, the method specifically includes the following steps:

Step 1. Prepare data, obtain the spatial coordinates of known data points and their corresponding real electromagnetic responses, and extract a certain number of sampling points from the calculation area of the geoelectric model to be solved to obtain their corresponding spatial coordinates. The predicted points are calculated Any point within the area;

The data points mentioned in step 1 are usually surface observation points, that is, the spatial coordinate z=0, and their corresponding electromagnetic responses are usually normalized by the surface field value. The above boundary conditions or initial value conditions =1 or/> =1 is given. In the example of the present invention, the initial value condition of TE polarization is/> =1; the sampling points are a certain number of spatial random sampling points. In the embodiment of the present invention, the Latin hypercube sampling method is used to extract 2000 random points in the interval [0, 10000] in the z direction and normalize them. One to [0,1]; the prediction points are 10,000 random points within the [0,1] range. After the coordinate z is normalized to [0,1] by Min-Max:

Step 2. Build a network, using a fully connected deep neural network. An activation function is added to the hidden layer. The input is the normalized data point and sampling point coordinates, and the output is the normalized surface electric (magnetic) field value. The real and imaginary parts of the electric and magnetic fields;

In the embodiment of the present invention, the fully connected neural network in step 2 is shown in Figure 3. The input layer size is 1, the output layer size is 4, the middle hidden layer contains 10 fully connected layers, and each layer contains 32 neurons. , the activation function adopts Sine activation function. enter , the output field value is normalized to:

The outputs are real part/> , imaginary part/> and/> real part/> , imaginary part/> .

Step 3: Add constraints. The residual difference between the output obtained after the data point coordinates are input into the neural network and its true electromagnetic response is used as the data error. The Helmholtz equation and boundary conditions satisfied by the electromagnetic field of the geoelectric model are added to the loss function as physical information constraints. ;

In the embodiment of the present invention, the residual loss_u between the network output u' and the real value u described in step 3 is calculated using the RMSE calculation method:

.

According to the Helmholtz equation:

,

Among them, u is the electromagnetic field value, is the angular frequency,/> is the magnetic permeability,/> is the conductivity, let

,

When the network output u' strictly satisfies the Helmholtz equation, the error in this item is zero. Otherwise, the error in 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 separate the real and imaginary parts of the electromagnetic field for calculation and output, the constraints of the Helmholtz equation also need to be decomposed into the real part of the electric field. , imaginary part of electric field/> , real part of magnetic field/> , imaginary part of the magnetic field/> Four parts:

In addition, in the embodiment of the present invention, the upper and lower boundary conditions of the geoelectric model are simply given by the Dirichlet boundary, that is, the upper boundary condition is , lower boundary condition/> . To facilitate loss calculation, this boundary condition can be used as a data point for loss function calculation.

Step 4, train the network, use automatic differentiation to find the partial derivative of the network output to the input, that is, the partial differential term in the Helmholtz equation, use an appropriate optimizer and learning rate, and use error backpropagation to iteratively update the network parameters. Minimize the loss function;

In the embodiment of the present invention, the automatic differentiation described in step 4 is implemented based on the built-in differentiation engine torch.autograd in PyTorch. The network output can be obtained through the chain derivation method. ,/> ,/> and/> pair input/> The second-order partial derivative of

In the embodiment of the present invention, the optimizers are AdamW and LBFGS, and the training cycle of AdamW is set to 10,000 rounds, the learning rate is 0.001, the maximum number of iterations of each optimization step of LBFGS is 10,000, and the learning rate is 1, the update history size is 100, the termination tolerance of the first-order optimal is 10 ^-9 , the termination tolerance of parameter changes is 10 ^-9 , and the linear search algorithm adopts the strong_wolfe condition.

In the embodiment of the present invention, the loss function loss is the weighted sum of the data error loss_u and the physical constraint error loss_f , that is:

Among them, α is the balance coefficient, which is used to balance the weight of loss_u and loss_f . In the embodiment of the present invention, it is set . Minimization of loss is achieved by cumulative gradient descent during error backpropagation.

Step 5, predict the results. When the training loss no longer decreases, the loss function tends to converge, the neural network training is completed, and the neural network model parameters are saved. At this time, the spatial coordinates of any point in the neural network are input to the geoelectric model, and the geoelectric model can be obtained. electromagnetic response of the point.

A set of actual results from the complete implementation of the method of the present invention are shown below. The geoelectric model corresponding to this result is shown in Figure 4. The model background resistivity is , the model depth is 10000m, where the resistivity settings are set according to different depths from 3000m to 5000m/> , can be divided into uniform half-space model, high-resistance model and low-resistance model. The points on the right represent 10,000 prediction points of uniform sampling, and the sampling interval is 1m. For different detection frequencies f1 =0.001Hz, f2 =0.1Hz, f3 =1.0Hz, f4 =10Hz, after a series of steps in the above embodiment, the electric field response prediction results of the neural networks corresponding to the three models (/> Real part PINN-Real,/> Imaginary part PINN-Imag) and real value (finite difference result/> Real part GT-Real,/> The comparison results of the imaginary part (GT-Imag) are shown in Figure 5. The prediction accuracy of the comparison results is very consistent with the real value, which verifies the effectiveness of the method of the present invention.

Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention, but not to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: The technical solutions described in the foregoing embodiments can still be modified, or some or all of the technical features can be equivalently replaced; and these modifications or substitutions do not deviate from the essence of the corresponding technical solutions from the technical solutions of the embodiments of the present invention. scope.

Claims (9)

1. A method for solving the electromagnetic response of a geoelectric model using a physical information neural network, which is characterized by including the following steps: Step 1. Construct an electromagnetic response training data set; Step 2: Construct a fully connected deep neural network; Among them, the loss function loss of the fully connected deep neural network is as follows: ; in, is the data error loss_u, /> is the physical constraint error;/> is the balance coefficient; Step 3: Input the electromagnetic response training data set into the fully connected deep neural network to minimize the loss function to obtain the trained fully connected deep neural network; Step 4: Input the true electromagnetic response and corresponding spatial coordinates of the sampling points in the calculation area of the geoelectric model to be solved into the trained fully connected deep neural network, and predict the electromagnetic response of the predicted point. 2. The method of using physical information neural network to solve the electromagnetic response of a geoelectric model as claimed in claim 1, characterized in that the electromagnetic response training data set includes the spatial coordinates of known data points and their corresponding real electromagnetic responses. 3. The method for solving electromagnetic response of geoelectric model by using physical information neural network as claimed in claim 2, characterized in that the spatial coordinates of the known data points are surface observation points and Min-Max is normalized to [0 ,1] interval. 4. The method of using physical information neural network to solve the electromagnetic response of the geoelectric model as claimed in claim 1, characterized in that, in the second step, , According to the constraints of Helmholtz equation, we get: , in, is the number of data points,/> is the number of sampling points,/> is the network output of the fully connected deep neural network for the nth training data,/> is the true value of the nth training data;/> Is the unit imaginary number/> ,/> is the angular frequency,/> is the frequency,/> is the magnetic permeability,/> is the conductivity,/> is the Laplacian operator. 5. The method of using physical information neural network to solve the electromagnetic response of geoelectric model as claimed in claim 1, characterized in that the fully connected deep neural network includes an input layer, an intermediate hidden layer and an output layer; the intermediate hidden layer includes 10 The layer is a fully connected layer, each layer contains 32 neurons; the input layer has a size of 1 and the output layer has a size of 4. 6. The method of using a physical information neural network to solve the electromagnetic response of a geoelectric model as claimed in claim 1, characterized in that the activation function of the fully connected deep neural network is a nonlinear activation function, and the nonlinear activation function includes Sine, Tanh. 7. The method of using a physical information neural network to solve the electromagnetic response of a geoelectric model as claimed in claim 4, characterized in that in the third step, the fully connected deep neural network separates the real and imaginary parts of the electromagnetic field to calculate and output. The constraints of the Mholtz equation are decomposed into the real part of the electric field , imaginary part of electric field/> , real part of magnetic field/> , imaginary part of the magnetic field/> Four parts: , in, is the partial derivative symbol,/> is the normalized spatial coordinate,/> for/> The real part of ,/> for/> The imaginary part of; for/> The real part of ,/> for/> The imaginary part of ;/> is the electric field value after normalization of the field value,/> is the magnetic field value after normalization of the field value. 8. The method of using a physical information neural network to solve the electromagnetic response of a geoelectric model as claimed in claim 1, characterized in that in the third step, the partial derivative of the output of the fully connected deep neural network with respect to the input is obtained through automatic differentiation. , that is, the partial differential term in the Helmholtz equation, uses an optimizer and learning rate, and uses error backpropagation to iteratively update the fully connected deep neural network parameters to minimize the loss function. 9. The method for solving the electromagnetic response of a geoelectric model using a physical information neural network as claimed in claim 8, wherein the partial differential term is the second-order partial derivative term of the electric/magnetic field in the Helmholtz equation; The optimizer includes gradient descent type, adaptive type and quasi-Newton type optimizer; the update strategy of the learning rate includes step size adjustment, exponential decay, adaptive adjustment and periodic adjustment strategy.