CN117390961A - Method for solving electromagnetic response of ground model by using physical information neural network - Google Patents
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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
Technical Field
The invention relates to the field of geophysics and artificial intelligence, in particular to a method for solving electromagnetic response of an earth model by using a physical information neural network.
Background
Electromagnetic prospecting has wide application value in prospecting applications of groundwater resources, mineral resources, geothermal resources, oil and gas resources, and deep structures. The electromagnetic method exploration utilizes the large electrical difference between the underground target abnormal body and the surrounding medium to define the resistivity abnormal region, thereby realizing the description of the space position and occurrence form of the underground target abnormal body. The physical mapping relation between the resistivity distribution of the ground model and the electromagnetic response is that the electromagnetic response of any point in the ground model meets Maxwell's equation.
Because the calculation space is large and the model complexity is high, in order to balance the calculation precision and the calculation efficiency, the common traditional method is to divide the model space area into a plurality of grids by giving an electric model, calculate the electromagnetic fields at the grid nodes by means of numerical simulation methods such as finite difference, finite element and the like, and then obtain the electromagnetic response of the whole model space by interpolation fitting approximation. However, mesh-subdivision interpolation techniques, while simplifying the solution to the computation space to some extent, essentially sacrifice some computational accuracy in exchange for higher computational efficiency. In addition, although the mesh subdivision interpolation technology has very good technical mobility, in the big data age, the simple repeated large matrix solution is slightly debilitated and has low efficiency when dealing with mass model calculation.
In recent years, with the progress of artificial intelligence technology in the age background of big data, the artificial neural network provides a plurality of intelligent solutions for human society life, and also provides new ideas and methods for geophysics. The neural network realizes the mapping between the network input and the network output through the weight superposition of the neurons, and has very strong nonlinear expression capability. However, for solving the geophysical problem, there are relatively mature physical mechanisms and mathematical logics behind, and especially the electromagnetic response needs to strictly meet the Max Wei Dianci theory, while the traditional neural network is excessively dependent on data, lacks support of physical laws, and severely restricts generalization use and prediction accuracy of the neural network. Therefore, only if the data characteristics extracted by the neural network are added with physical constraints, the advantages of the neural network can be furthest exerted, and the intelligent solution of the electromagnetic response of the ground model is realized.
Disclosure of Invention
In order to solve the technical problems or at least partially solve the technical problems, the invention provides a method for solving electromagnetic response of an earth model by utilizing a physical information neural network.
In order to achieve the above purpose, the technical scheme of the invention is as follows:
a method for solving an electromagnetic response of a ground model by using a physical information neural network, comprising the following steps:
step one, constructing an electromagnetic response training data set;
step two, constructing a full-connection deep neural network;
wherein the loss function of the fully connected deep neural networklossThe following is shown:
wherein loss_u is a data error loss_u, and loss_f is a physical constraint error; alpha is a balance coefficient;
step three, inputting an electromagnetic response training data set into a full-connection deep neural network to minimize a loss function loss so as to obtain a trained full-connection deep neural network;
inputting the real electromagnetic response of the sampling points of the calculation region of the ground model to be solved and the corresponding space coordinates into a trained full-connection depth neural network, and predicting to obtain the electromagnetic response of the predicted point.
Further refinements, the electromagnetic response training dataset includes spatial coordinates of known data points and their corresponding true electromagnetic responses.
Further refinement, 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 the second step,
，
from the constraints of the helmholtz equation, we get:
，
wherein,number of data points>For the number of sampling points, +.>For the network output of the full-connection deep neural network to the nth training data, +.>Is the true value of the nth training data; />Is the unit imaginary number->，/>For angular frequency +.>For frequency +.>For permeability (I)>For conductivity, & gt>Is a laplace operator.
Further improvements, the fully connected deep neural network comprises an input layer, an intermediate hidden layer and an output layer; the middle hidden layer contains 10 fully connected layers, each layer containing 32 neurons; the size of the input layer is 1 and the size of the output layer is 4.
Further improved, the activation function of the fully-connected deep neural network is a nonlinear activation function, and the nonlinear activation function comprises Sine and Tanh.
In the third step, the full-connection depth neural network calculates and outputs the real part and the imaginary part of the electromagnetic field separately, and the constraint of the Helmholtz equation is disassembled into the real part of the electric fieldImaginary part of electric field->Real part of magnetic field->Imaginary part of magnetic fieldFour parts:
wherein,is a partial guide symbol>For the normalized spatial coordinates, +.>Is->Real part of->Is->Is the imaginary part of (2); />Is->Real part of->Is->Is the imaginary part of (2); />Electric field value normalized to field value, < ->The magnetic field value is normalized by the field value.
In the third step, partial derivatives of the output to the input of the fully connected deep neural network, namely partial differential terms in a helmholtz equation, are obtained through automatic differentiation, an optimizer and a learning rate are adopted, and error back propagation iteration is used for updating parameters of the fully connected deep neural network, so that a loss function is minimized.
Further improvement, the partial differential term is a second-order partial differential term of an electric/magnetic field in a Helmholtz equation; the optimizers include gradient descent classes, adaptive classes, and quasi-newton class optimizers; the updating strategies of the learning rate comprise step length adjustment, exponential decay, self-adaptive adjustment and periodical adjustment strategies.
According to the method, the electromagnetic response of the ground model is solved by using the physical information neural network, so that the precision loss caused by the mesh subdivision interpolation technology in the traditional numerical method is avoided, meanwhile, physical constraints such as a Helmholtz equation and side value conditions are added on the basis of the traditional neural network, the electromagnetic response solving of the ground model under the dual driving of data and the physical model is realized, and an intelligent scheme is provided for the calculation of the electromagnetic response in the electromagnetic exploration.
Drawings
In order to more clearly illustrate the invention or the technical solutions of the prior art, the following description will briefly explain the drawings used in the embodiments or the description of the prior art, and it is obvious that the drawings in the following description are some embodiments of the invention, and other drawings can be obtained according to the drawings without inventive effort for a person skilled in the art.
FIG. 1 is a schematic representation of the process of the present invention.
Fig. 2 is a flow chart of the method of the present invention.
Fig. 3 is a schematic diagram of a fully-connected neural network according to an embodiment of the present invention.
Fig. 4 is a schematic diagram of a ground model according to an embodiment of the present invention.
Fig. 5 is a schematic diagram of an error between a predicted result and a true value of an electric field response neural network according to an embodiment of the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made apparent and fully in view of the accompanying drawings, in which some, but not all embodiments of the invention are shown. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Fig. 1 is a schematic diagram of a method for solving an electromagnetic response of a geoelectrical model using a physical information neural network according to an embodiment of the present invention. In the embodiment of the invention, the electromagnetic response of the earth model comprises the earth electromagnetic (MT) of a natural field source and the Controllable Source Electromagnetic (CSEM) of an artificial field source, and the TE/TM two polarization modes are contained. As shown in fig. 2, the method specifically includes the following steps:
step 1, preparing data, obtaining the space coordinates of known data points and the corresponding real electromagnetic response thereof, extracting a certain number of sampling points from a calculation area of a ground model to be solved to obtain the corresponding space coordinates, wherein the predicted point is any point in the calculation area;
the data points described in step 1 are typically surface observation points, i.e., the spatial coordinates z=0, whose corresponding electromagnetic response is typically normalized by the surface field values, the above boundary conditions or initial conditions=1 or->The initial condition of TE polarization in the present example is +.1>=1; the sampling points are a certain number of space random sampling points, and in the embodiment of the invention, a Latin hypercube sampling method is adopted to form intervals [0, 10000 ] in the z direction]2000 random points are extracted and normalized to [0,1]]Within the range; the predicted point is [0,1]]10000 random points in the range. The coordinate z is normalized to [0,1] by Min-Max]And (2) after that:
step 2, constructing a network, adopting a fully-connected deep neural network, adding an activation function into a hidden layer, inputting the data point and the sampling point coordinates after normalization, and outputting real parts and imaginary parts of an electric field and a magnetic field after normalization of an earth surface electric (magnetic) field value;
in the embodiment of the present invention, as shown in fig. 3, in the fully-connected neural network in step 2, the input layer has a size of 1, the output layer has a size of 4, the middle hidden layer includes 10 fully-connected layers, each layer includes 32 neurons, and the activation function adopts a Sine activation function. Input deviceTransport and deliverThe field value is normalized as follows:
the outputs are respectivelyReal part of->Imaginary part->And +.>Real part of->Imaginary part->。
Step 3, adding constraints, namely taking the residual error of the output obtained after the data point coordinates are input into the neural network and the real electromagnetic response of the data point coordinates as a data error, and taking the Helmholtz equation and the boundary condition met by the electromagnetic field of the electric model as physical information constraints to add a loss function;
in the embodiment of the present invention, the network output in step 3u’And true valueuResidual of (2)loss_uThe method adopts the RMSE calculation mode to calculate:
.
according to the helmholtz equation:
，
wherein,ufor the value of the electromagnetic field,for angular frequency +.>For permeability (I)>For conductivity, let
，
When the network outputsu’When the Helmholtz equation is strictly satisfied, the error is zero, otherwise, the calculated error is the error brought by the fact that the network output does not strictly satisfy the Helmholtz equation. Notably, in the embodiment of the present invention, since the network needs to separate the real part and the imaginary part of the electromagnetic field into the calculated outputs, the constraint of the helmholtz equation also needs to be disassembled into the real part of the electric fieldImaginary part of electric field->Real part of magnetic field->Imaginary part of magnetic field->Four parts:
in addition, in the embodiment of the invention, the upper and lower boundary conditions of the ground model are simply given by Dirichlet boundary, namely, the upper boundary condition isLower boundary condition->. To facilitate the calculation of the loss, this boundary condition may be used as a data point for the loss function calculation.
Step 4, training the network, namely obtaining a partial derivative of the output of the network to the input, namely a partial derivative term in a Helmholtz equation by utilizing automatic differentiation, updating network parameters by utilizing error back propagation iteration by adopting a proper optimizer and learning rate, so that a loss function is minimized;
in the embodiment of the invention, the automatic differentiation in the step 4 is realized based on a built-in differential engine torch.autograd in PyTorch, and the network output can be obtained by a chain derivation method、/>、/>And +.>Input->Second partial derivative of (2)
In the embodiment of the invention, the optimizers are two optimizers, namely AdamW and LBFGS, and the training period of AdamW is set to 10000 rounds, the learning rate is 0.001, the maximum iteration number of each optimization step of LBFGS is 10000, the learning rate is 1, the update history is 100, and the first-order optimal termination tolerance is 10 -9 Termination tolerance of parameter variation is 10 -9 The linear search algorithm uses the strong_wolfe condition.
In an embodiment of the present invention, the loss functionlossFor data errorsloss_uError with physical constraintloss_fIs a weighted sum of (1), namely:
wherein,αis a balance coefficient for balancingloss_uAndloss_fthe weight of the (b) is set in the embodiment of the invention 。 lossIs achieved by a cumulative gradient drop during error back propagation.
And 5, predicting a result, namely when the training loss is not reduced any more, converging a loss function, finishing the training of the neural network, storing the parameters of the neural network model, and inputting the space coordinates of any point into the neural network at the moment to obtain the electromagnetic response of the point in the ground model.
A set of actual results of a complete implementation of the method of the present invention is shown below. As shown in FIG. 4, the model background resistivity of the model corresponding to the result isThe model depth is 10000m, wherein +.A. is set according to different resistivity in the depth range of 3000m to 5000m>The method can be divided into a uniform half-space model, a high-resistance model and a low-resistance model, wherein the right point represents 10000 prediction points which are uniformly sampled, and the sampling interval is 1m. For different detection frequenciesf1=0.001Hz、f2=0.1Hz、f3=1.0Hz、f4=10 Hz, and through a series of steps in the above embodiments, the electric field response prediction results (++) of the neural network corresponding to the three models>Real part PINN-Real, < ->Imaginary part PINN-image) and the real value (finite difference result +.>Real part GT-Real, ">Imaginary part GT-image) such asAs shown in FIG. 5, the prediction accuracy of the comparison result has very good consistency with the true value, and the effectiveness of the method of the invention is verified.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solution of the present invention, and not for limiting the same; although the invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical scheme described in the foregoing embodiments can be modified or some or all of the technical features thereof can be replaced by equivalents; such modifications and substitutions do not depart from the spirit of the invention.
Claims (9)
1. A method for solving an electromagnetic response of a ground model by using a physical information neural network, comprising the steps of:
step one, constructing an electromagnetic response training data set;
step two, constructing a full-connection deep neural network;
wherein the loss function of the fully connected deep neural networklossThe following is shown:
；
wherein,for data errorsloss_u，/>Is a physical constraint error; />Is a balance coefficient;
step three, inputting the electromagnetic response training data set into the fully-connected deep neural network to the loss functionlossMinimizing to obtain a trained full-connection deep neural network;
inputting the real electromagnetic response of the sampling points of the calculation region of the ground model to be solved and the corresponding space coordinates into a trained full-connection depth neural network, and predicting to obtain the electromagnetic response of the predicted point.
2. The method for solving an electromagnetic response of a geoelectrical model using a physical information neural network of claim 1, wherein the electromagnetic response training dataset includes spatial coordinates of known data points and their corresponding real electromagnetic responses.
3. The method for solving an electromagnetic response of a geoelectrical model using a physical information neural network of claim 2, wherein the spatial coordinates of the known data points are surface observation points and Min-Max is normalized to the [0,1] interval.
4. The method for solving electromagnetic response of electric model by physical information neural network according to claim 1, wherein in the second step,
，
from the constraints of the helmholtz equation, we get:
，
wherein,number of data points>For the number of sampling points, +.>For the network output of the full-connection deep neural network to the nth training data, +.>Is the true value of the nth training data; />Is the unit imaginary number->，/>For angular frequency +.>For frequency +.>For permeability (I)>For conductivity, & gt>Is a laplace operator.
5. The method for solving an electromagnetic response of a geoelectrical model with a physical information neural network of claim 1, wherein the fully connected deep neural network comprises an input layer, an intermediate hidden layer, and an output layer; the middle hidden layer contains 10 fully connected layers, each layer containing 32 neurons; the size of the input layer is 1 and the size of the output layer is 4.
6. The method for solving electromagnetic response of ground model by physical information neural network according to claim 1, wherein the activation function of the fully connected deep neural network is a nonlinear activation function, and the nonlinear activation function comprises Sine and Tanh.
7. Solving for the geoelectrical model with physical information neural network electromagnetic as recited in claim 4The response method is characterized in that in the third step, the full-connection depth neural network divides the real part and the imaginary part of the electromagnetic field to calculate and output, and the constraint of the Helmholtz equation is disassembled into the real part of the electric fieldImaginary part of electric field->Real part of magnetic field->Imaginary part of magnetic field->Four parts:
，
wherein,is a partial guide symbol>For the normalized spatial coordinates, +.>Is->Real part of->Is->Is the imaginary part of (2);is->Real part of->Is->Is the imaginary part of (2); />Electric field value normalized to field value, < ->The magnetic field value is normalized by the field value.
8. The method for solving the electromagnetic response of the ground model by using the physical information neural network according to claim 1, wherein in the third step, the partial derivative of the output to the input of the fully connected depth neural network is obtained through automatic differentiation, namely, the partial derivative term in the helmholtz equation is adopted, an optimizer and a learning rate are adopted, and the parameters of the fully connected depth neural network are updated by using error back propagation iteration, so that the loss function is minimized.
9. The method for solving an electromagnetic response of a geoelectrical model using a physical information neural network according to claim 8, wherein the partial differential term is a second order partial differential term of an electric/magnetic field in a helmholtz equation; the optimizers include gradient descent classes, adaptive classes, and quasi-newton class optimizers; the updating strategies of the learning rate comprise step length adjustment, exponential decay, self-adaptive adjustment and periodical adjustment strategies.
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