CN114966840A - Three-dimensional seismic wave field model building method based on general solution neural network - Google Patents
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Abstract
A model building method of a three-dimensional seismic wave field based on a general solution neural network aims to solve the problem that a three-dimensional wave equation cannot be solved by a traditional physical information neural network. The three-dimensional seismic wave field modeling method comprises the following steps: firstly, establishing a three-dimensional seismic wave equation in an isotropic medium; secondly, deducing a general solution of an equation; thirdly, determining the number of solution domains and data points; fourthly, establishing a full-connection layer neural network containing 4 sub-networks to respectively represent f and g 1 ，g 2 ，g 3 Each sub-network comprises 10 hidden layers; fifthly, designing a loss function; sixthly, taking the loss function as a targetAnd (4) training the network by using the function, and training by using an Adam self-adaptive optimization function algorithm. The invention deduces the general solution form of the three-dimensional wave equation, establishes the neural network by taking the amount of the general solution of the equation as input, and provides the three-dimensional seismic wave field modeling based on the general solution neural network, so that the network output automatically meets the three-dimensional wave equation, and the dependence of the network on residual points is greatly reduced.
Description
Technical Field
The invention belongs to the field of seismology, and particularly relates to a three-dimensional seismic wave field modeling method based on a general solution neural network.
Background
China is located between the Pacific earthquake zone and the Eurasian earthquake zone, and the earthquake structure has strong activity and wide earthquake distribution. In recent years, earthquakes occur in many places in the world, which causes serious casualties and economic property loss, and brings about serious social influence. The development of urban earthquake-resistant toughness research is a necessary means for reducing post-disaster loss and accelerating post-disaster recovery, the urban regional building earthquake damage simulation research is a basic task of urban earthquake-resistant toughness research, and the earthquake dynamic field simulation is a key problem of urban regional building earthquake damage simulation. The seismic motion field is closely related to the propagation of seismic waves in a medium, and a three-dimensional seismic wave field is usually obtained by solving a three-dimensional wave equation, so that the solving of the wave equation is an important step in seismic motion simulation.
In recent years, with the development of computer technology, neural networks are more and more widely applied to the fields of geophysical engineering, seismic engineering and the like, however, most of the 'black box' models obtained by the neural networks lack the limitation of physical laws and are difficult to explain. Although a scholars proposes a physical information neural network added with physical constraints, the network has the defects of difficult training control, difficult explanation of network structure and the like, and the three-dimensional wave equation cannot be solved.
Disclosure of Invention
The invention aims to solve the problem that the traditional physical information neural network cannot solve a three-dimensional wave equation, and provides a three-dimensional seismic wave field modeling method based on a general solution neural network.
The invention discloses a three-dimensional seismic wave field modeling method based on a general solution neural network, which is realized according to the following steps:
step one, establishing a control equation, wherein a three-dimensional seismic wave equation in an isotropic medium is as follows:
wherein V P Represents the wave velocity, V, of the P wave S Represents the S wave velocity u x ,u y ,u z Respectively representing the displacement of the particles in the xyz direction;
step two, derivation of general equation solution:
according to Stokes decomposition, u x ,u y And u z The displacement three components are respectively expressed as:
at this time, the spherical P wave potential function is expressed as
The S-wave potential function is expressed by psi x ＝g 1 (t-r/V S )/r，ψ y ＝g 2 (t-r/V S )/r，ψ z ＝g 3 (t-r/V S ) R, wherein r represents the distance from the seismic source to the survey point,
f，g 1 ，g 2 ，g 3 for any continuously conductable function, # x 、ψ y 、ψ z Respectively representing x, y and z components of the S wave potential function, and t represents time; when the input of the neural network is t-r/V P And t-r/V S When the network is in the open solution form, the network automatically meets the open solution form;
step three, determining the number of solution domains and data points:
a small number of data points are adopted for training assistance, so that the network convergence speed is increased;
step four, establishing a deep neural network:
establishing a full-connection layer neural network containing 4 sub-networks to respectively characterize f and g 1 ，g 2 ，g 3 The input of a sub-network is t-r/V P When the subnetwork output is
The inputs of the other three sub-networks are t-r/V S The output of the three sub-networks is psi x ＝g 1 (t-r/V S )/r，ψ y ＝g 2 (t-r/V S )/r，ψ z ＝g 3 (t-r/V S ) Each sub network comprises 10 hidden layers, and an activation function is y ═ sigmoid (x) ×, so as to obtain a deep neural network model;
step five, designing a loss function:
using a loss function to represent whether training is converged;
step six, network training:
and (5) training the network by taking the loss function in the step five as a target function, and training by adopting an Adam self-adaptive optimization function algorithm, thereby completing the modeling of the three-dimensional seismic wave field.
According to the method, firstly, a general solution form of the three-dimensional wave equation is deduced, a neural network is established by taking the quantity of the general solution as input, and a three-dimensional seismic wave field modeling based on the general solution neural network is provided, so that the network output automatically meets the three-dimensional wave equation, and the dependence of the network on residual points is greatly reduced; meanwhile, a small number of known data points are added in the training process, on the premise of keeping high solving precision, the training convergence speed is accelerated, compared with the traditional physical information neural network, the model completes the solving problem of the three-dimensional wave equation, the dependence on data is less, and the interpretability of the three-dimensional wave equation solving network is improved.
The three-dimensional seismic wave field modeling method provided by the invention can be applied to the fields of seismology, seismic engineering and the like.
Drawings
FIG. 1 is an overall framework flow diagram of an embodiment three-dimensional seismic wavefield modeling based on a generalized solution neural network;
FIG. 2 is a diagram of a general solution neural network for solving a three-dimensional wave equation in an embodiment;
FIG. 3 is a loss reduction diagram of the training process in the example;
FIG. 4 is a graph comparing the results of the solution and the results of the solution at (-3km,3km, -3km) in the examples, wherein 1 represents U x Solution result, 2 represents U y Solution, 3 for U z Solution, 4 for U x Analytical solution, 5 represents U y Analytical solution, 6 represents U z Resolving a solution;
FIG. 5 is a graph comparing the solution result at (-3km,3km,0km) position with the analytic solution result in the example, in which 1 represents U x Solution result, 2 represents U y Solution, 3 for U z Solution, 4 for U x Analytical solution, 5 represents U y Analytical solution, 6 represents U z Resolving a solution;
FIG. 6 is a graph comparing the solution results at (-3km, -3km, -3km) and the analytic solution results in the example, in which 1 represents U x Solution result, 2 represents U y Solution, 3 for U z Solution, 4 for U x Analytical solution, 5 represents U y Analytical solution, 6 represents U z Resolving a solution;
FIG. 7 is a graph comparing the results of the solution and the results of the solution in the examples (0km,3km, -3km), wherein 1 represents U x Solution result, 2 represents U y Solution, 3 for U z Solution, 4 for U x Analytical solution, 5 represents U y Analytical solution, 6 represents U z And (6) resolving the solution.
Detailed Description
The first embodiment is as follows: the three-dimensional seismic wave field modeling method based on the general solution neural network is implemented according to the following steps:
step one, establishing a control equation, wherein a three-dimensional seismic wave equation in an isotropic medium is as follows:
wherein V P Represents the wave velocity, V, of the P wave S Represents the S wave velocity u x ,u y ,u z Respectively representing the displacement of the particles in the xyz direction;
step two, derivation of general equation solution:
according to Stokes decomposition, u x ,u y And u z The displacement three components are respectively expressed as:
at this time, the spherical surface P wave potentialThe function is expressed as
The S-wave potential function is expressed by psi x ＝g 1 (t-r/V S )/r，ψ y ＝g 2 (t-r/V S )/r，ψ z ＝g 3 (t-r/V S ) R, wherein r represents the distance from the seismic source to the survey point,
f，g 1 ，g 2 ，g 3 for any continuously conductable function, # x 、ψ y 、ψ z Respectively representing x, y and z components of the S wave potential function, and t represents time; when the input of the neural network is t-r/V P And t-r/V S When the network is in the open solution form, the network automatically meets the open solution form;
step three, determining the number of solution domains and data points:
a small number of data points are adopted for training assistance, so that the network convergence speed is increased;
step four, establishing a deep neural network:
establishing a full-connection layer neural network containing 4 sub-networks to respectively represent f and g 1 ，g 2 ，g 3 The input of a sub-network is t-r/V P When the sub-network outputs as
The inputs of the other three sub-networks are t-r/V S The outputs of the three sub-networks are psi x ＝g 1 (t-r/V S )/r，ψ y ＝g 2 (t-r/V S )/r，ψ z ＝g 3 (t-r/V S ) Each sub network comprises 10 hidden layers, and an activation function is y ═ sigmoid (x) ×, so as to obtain a deep neural network model;
step five, designing a loss function:
using a loss function to represent whether training is converged;
step six, network training:
and (5) training the network by taking the loss function in the step five as a target function, and training by adopting an Adam self-adaptive optimization function algorithm, thereby completing the modeling of the three-dimensional seismic wave field.
The second embodiment is as follows: the difference between this embodiment and the first embodiment is that in step three, the solution domain of x, y, z is [ -4.5, 4.5km ], and the solution domain of t is [0, 6s ].
The third concrete implementation mode: the difference between the second embodiment and the second embodiment is that the number of data points in the third step is 400-800.
The fourth concrete implementation mode: the difference between this embodiment and one of the first to third embodiments is that each hidden layer in step four includes 256 neurons.
The fifth concrete implementation mode: the difference between this embodiment and the first to the fourth embodiment is that the formula of the loss function in the fifth step is as follows:
Loss＝|u-u real |；
wherein u is real Representing the true displacement value, and u is the predicted displacement value.
The sixth specific implementation mode: the difference between this embodiment and one of the first to fifth embodiments is that the Adam adaptive optimization function algorithm described in step six is as follows:
(1) calculating a first moment estimate and a second moment estimate of the gradient by the following formula:
m t ＝β 1 ·m t-1 +(1-β 1 )·g t ，ν t ＝β 2 ·ν t-1 +(1-β 2 )·g t 2 ；
in the formula, g t Is a gradient in which m t Is the mean value of the gradient at time t, v t Is the non-central variance value, m, at time t of the gradient t-1 Is the mean value at time t-1 of the gradient, V t-1 The exponential decay rate beta of the moment estimate, which is the non-central variance value at time t-1 of the gradient 1 And beta 2 Within the interval [0, 1);
(2) correcting the first order moment estimate and the second order moment estimate by calculating the formula:
(3) the final formula for parameter update is:
in the formula, theta t For updated parameters, η is the learning rate and ε is a small constant for numerical stability.
The seventh embodiment: this embodiment is different from the sixth embodiment in that β 1 Take 0.9, beta 2 Take 0.999.
The specific implementation mode is eight: this embodiment is different from the sixth embodiment in that e is 10 -8 。
The specific implementation method nine: the difference between this embodiment and the first to eighth embodiments is that in step six, a multi-learning-rate training method (strategy) is adopted, and the training rounds are 20,20 and 20 respectively.
The specific implementation mode is ten: the difference between this embodiment and the ninth embodiment is that the three-round training corresponds to learning rates of 0.001,0.0001,0.00001, and batch sizes of 44.
Example (b): the three-dimensional seismic wave field modeling method based on the general solution neural network is implemented according to the following steps:
step one, establishing a control equation, wherein a three-dimensional seismic wave equation in an isotropic medium is as follows:
wherein V P Represents the wave velocity, V, of the P wave S Represents the S wave velocity u x ,u y ,u z Respectively representing the displacement of the particles in the x, y and z directions;
step two, derivation of general equation solution:
according to Stokes decomposition, u x ,u y And u z The displacement three components are respectively expressed as:
at this time, the spherical P wave potential function is expressed as
The S-wave potential function is expressed by psi x ＝g 1 (t-r/V S )/r，ψ y ＝g 2 (t-r/V S )/r，ψ z ＝g 3 (t-r/V S ) R, wherein r represents the distance from the seismic source to the survey point,
f，g 1 ，g 2 ，g 3 for any continuously conductable function, # x 、ψ y 、ψ z Respectively representing x, y and z components of the S wave potential function, and t represents time; when the input of the neural network is t-r/V P And t-r/V S When the network is in the open solution form, the network automatically meets the open solution form;
step three, determining the number of solution domains and data points:
a small number of data points are adopted for training in an auxiliary mode, the network convergence speed is accelerated, the solution domains of x, y and z are set to be [ -4.5 and 4.5km ], the solution domain of t is set to be [0 and 6s ], and the number of the data points is 704;
step four, establishing a deep neural network:
establishing a full-connection layer neural network containing 4 sub-networks to respectively represent f and g 1 ，g 2 ，g 3 The input of a sub-network is t-r/V P When the sub-network outputs as
The inputs of the other three sub-networks are t-r/V S The outputs of the three sub-networks are psi x ＝g 1 (t-r/V S )/r，ψ y ＝g 2 (t-r/V S )/r，ψ z ＝g 3 (t-r/V S ) Each subnetwork comprises 10 hidden layers, each layer comprises 256 neurons, and an activation function is y ═ sigmoid (x) ×, so that a deep neural network model is obtained;
step five, designing a loss function:
whether to train convergence is characterized by using a loss function, wherein the formula of the loss function is as follows:
Loss＝|u-u real |；
wherein u is real Representing the true displacement value, and u is the predicted displacement value;
step six, network training:
and (5) training the network by taking the loss function in the step five as a target function, training by adopting an Adam adaptive optimization function algorithm, and training by adopting a multi-learning-rate training strategy, wherein the training rounds are respectively 20,20 and 20, the corresponding learning rates are 0.001,0.0001 and 0.00001, and the batch sizes are all 44, so that the modeling of the three-dimensional seismic wave field is completed.
The neural network training data are derived from uniform sampling in a fixed interval, displacement components at any receiving point can be obtained after three-dimensional seismic wave field modeling is completed, seismic motion simulation is achieved, fig. 3 shows that network errors tend to be stable when training is finished, and fig. 4-7 show that the solving result of the deep neural network is well matched with analytic solutions.
Claims (10)
1. The method for establishing the three-dimensional seismic wave field model based on the general solution neural network is characterized by being realized according to the following steps:
step one, establishing a control equation, wherein a three-dimensional seismic wave equation in an isotropic medium is as follows:
wherein V P Represents the wave velocity, V, of the P wave S Represents the S wave velocity u x ,u y ,u z Respectively representing the displacement of the particles in the xyz direction;
step two, derivation of general equation solution:
according to Stokes decomposition, u x ,u y And u z The displacement three components are respectively expressed as:
at this time, the spherical P wave potential function is expressed as
The S-wave potential function is expressed by psi x ＝g 1 (t-r/V S )/r，ψ y ＝g 2 (t-r/V S )/r，ψ z ＝g 3 (t-r/V S ) R, wherein r represents the distance from the seismic source to the survey point,
f，g 1 ，g 2 ，g 3 for any continuously conductable function, # x 、ψ y 、ψ z Respectively representing x, y and z components of the S wave potential function, and t represents time; when the input of the neural network is t-r/V P And t-r/V S When the network is in the open solution form, the network automatically meets the open solution form;
step three, determining the number of solution domains and data points:
a small number of data points are adopted for training assistance, so that the network convergence speed is increased;
step four, establishing a deep neural network:
establishing a full-connection layer neural network containing 4 sub-networks to respectively represent f and g 1 ，g 2 ，g 3 The input of a sub-network is t-r/V P When the sub-network outputs as
The inputs of the other three sub-networks are t-r/V S The outputs of the three sub-networks are psi x ＝g 1 (t-r/V S )/r，ψ y ＝g 2 (t-r/V S )/r，ψ z ＝g 3 (t-r/V S ) Each sub network comprises 10 hidden layers, and an activation function is y ═ sigmoid (x) ×, so as to obtain a deep neural network model;
step five, designing a loss function:
using a loss function to represent whether training is converged;
step six, network training:
and (5) training the network by taking the loss function in the step five as a target function, and training by adopting an Adam self-adaptive optimization function algorithm, thereby completing the modeling of the three-dimensional seismic wave field.
2. The method of claim 1, wherein x, y, z is set to have a solution domain of [ -4.5, 4.5km ], and t is set to have a solution domain of [0, 6s ] in step three.
3. The method for modeling a three-dimensional seismic wavefield based on the solution neural network as claimed in claim 1, wherein the number of data points in the third step is 400-800.
4. The method of claim 1, wherein each hidden layer in step four comprises 256 neurons.
5. The method of modeling a three-dimensional seismic wavefield in accordance with claim 1, wherein the loss function in step five is formulated as follows:
Loss＝|u-u real |；
wherein u is real Representing the true displacement value, and u is the predicted displacement value.
6. The method for modeling a three-dimensional seismic wavefield based on a solution-passing neural network as claimed in claim 1, wherein the Adam adaptive optimization function algorithm in step six is as follows:
(1) calculating a first moment estimate and a second moment estimate of the gradient by the following formula:
m t ＝β 1 ·m t-1 +(1-β 1 )·g t ，ν t ＝β 2 ·ν t-1 +(1-β 2 )·g t 2 ；
in the formula, g t Is a gradient in which m t Is the mean value of the gradient at time t, v t Is the non-central variance value, m, at time t of the gradient t-1 Is a gradientAverage value of time t-1, V t-1 The exponential decay rate beta of the moment estimate, which is the non-central variance value at time t-1 of the gradient 1 And beta 2 Within the interval [0, 1);
(2) correcting the first order moment estimate and the second order moment estimate by calculating the formula:
(3) the final formula for parameter update is:
in the formula, theta t For updated parameters, η is the learning rate and ε is a small constant for numerical stability.
7. The method of claim 6, wherein β is a function of the three-dimensional seismic wavefield of the solution neural network 1 Take 0.9, beta 2 Take 0.999.
8. The method of claim 6 in which ε is taken to be 10 -8 。
9. The method for building the three-dimensional seismic wave field model based on the solution neural network as claimed in claim 1, wherein a multi-learning-rate training method is adopted in the sixth step, and the training rounds are respectively 20,20 and 20.
10. The method of claim 9 in which the three rounds of training correspond to a learning rate of 0.001,0.0001,0.00001 and a batch size of 44.
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