Local MQRBF-FD sound wave propagation simulation method based on Adam-BP neural network

Abstract

The invention discloses a local MQRBF-FD sound wave propagation simulation method based on an Adam-BP neural network, and the method comprises the following steps: 1, optimizing the MQRBF shape parameters of a sine function combination based on IRW, and generating a data set containing frequency components and optimal parameters; step 2, constructing an Adam-BP neural network model; 3, carrying out Fourier expansion on the solution of the acoustic wave equation to form a sine function combination, carrying out adaptive optimization on shape parameters based on the constructed Adam-BP neural network model, and executing local MQRBF interpolation at key nodes by using the optimized shape parameters to complete spatial discretization; and 4, discretizing time by adopting a finite difference method, forming a sparse linear equation set by an obtained discrete equation system, and efficiently solving the sparse linear equation set by utilizing a conjugate gradient method and ILU decomposition pretreatment to realize high-precision simulation of sound wave propagation in a complex medium. According to the method, the shape parameters can be automatically optimized according to the local characteristics of the medium, so that the accuracy and the stability of sound wave propagation simulation in a complex and heterogeneous medium are remarkably improved.

Description

Technical Field

[0001] The present invention relates to the technical field of acoustic wave propagation simulation, and particularly to a local MQRBF-FD acoustic wave propagation simulation method based on an Adam-BP neural network. Background Art

[0002] Traditional numerical methods face significant challenges in dealing with acoustic wave propagation problems under complex boundary conditions and heterogeneous media. Although the finite difference method (FD) is known for its simple algorithm and high computational efficiency, it is prone to accuracy decay and numerical oscillations in irregular grids or heterogeneous regions with drastic changes in wave velocity. Especially in reflection interfaces or high-gradient media regions, it is necessary to rely on global grid refinement to maintain accuracy, resulting in an exponential increase in computational cost.

[0003] The finite difference method based on radial basis functions (MQRBF-FD) provides a new idea for acoustic wave propagation simulation in complex media by combining the meshless characteristics of radial basis functions with the high efficiency of the finite difference method. However, the core bottleneck of this method lies in the sensitivity of the shape parameter: the shape parameter directly determines the smoothness and spatial distribution characteristics of the radial basis function. Improper parameter selection will lead to an imbalance between interpolation accuracy and numerical stability - too high a parameter is prone to interpolation oscillation, and too low a parameter will result in overfitting, making it difficult to accurately characterize the local wave field characteristics of heterogeneous media. In highly inhomogeneous media, the shape parameter needs to be dynamically adapted to local wave velocity changes. However, traditional optimization methods mostly rely on global search strategies, which cannot effectively adapt to the local heterogeneity of the media and face problems such as slow convergence speed and insufficient stability in large-scale simulations.

[0004] In the prior art, there is still a lack of an efficient solution for the dynamic optimization of the shape parameter of MQRBF in complex boundaries and heterogeneous media. Global optimization algorithms are difficult to balance local characteristic adaptation and computational efficiency, resulting in limited simulation accuracy; static parameter selection strategies cannot cope with the changes in media characteristics during the dynamic evolution of the wave field, and are prone to cumulative errors. In addition, traditional methods have significant deficiencies in capturing complex wave field behaviors such as interface reflection and wavefront diffraction, which limits their practical applications in fields such as deep exploration and medical acoustics.

[0005] In summary, how to construct an MQRBF-FD acoustic wave propagation simulation method that can adaptively optimize the shape parameter, balance high accuracy and efficient calculation, has become a key technical challenge to break through the bottleneck of heterogeneous media modeling. Summary of the Invention

[0006] To overcome the above technical problems, the purpose of the present invention is to provide a local MQRBF-FD (Multi-Quadratic Radial Basis Function-Finite Difference Method) acoustic wave propagation simulation method based on an Adam-BP (Adaptive Momentum Optimization-Backpropagation Neural Network) neural network. This method, through an improved random walk optimization algorithm, combined with an Adam-BP neural network model, can automatically optimize the shape parameters according to the local characteristics of the medium, thereby significantly improving the accuracy and stability of acoustic wave propagation simulation in complex and heterogeneous media, and solving the problems of numerical instability and accuracy decay of the traditional finite difference method in complex media.

[0007] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0008] The local MQRBF-FD acoustic wave propagation simulation method based on an Adam-BP neural network includes the following steps:

[0009] Step 1: Use an improved random walk algorithm to optimize the shape parameters of MQRBF in the Fourier expansion of the solutions of a large number of acoustic wave equations, generating a data set containing acoustic wave frequency components and their corresponding optimal parameters;

[0010] Step 2: Based on the data set, train a BP neural network and use an Adam optimizer for parameter optimization to construct an Adam-BP neural network model for precisely adjusting the shape parameters of the acoustic wave equation; this model can dynamically predict the optimal shape parameters, break through the efficiency bottleneck of the traditional trial-and-error method, and achieve precise matching of parameters with local medium characteristics.

[0011] Step 3: Fourier-expand the solution of the acoustic wave equation into a combination of sine functions, adaptively optimize the shape parameters based on the Adam-BP neural network model constructed in Step 2, and perform local MQRBF interpolation at key nodes using the optimized shape parameters to complete spatial discretization;

[0012] Step 4: On the basis of completing spatial discretization in Step 3, use the finite difference (FD) method to discretize time, and the resulting discrete equation system forms a sparse linear equation system. By combining the implicit midpoint method and an adaptive time step strategy, the computational efficiency and numerical stability are optimized. Finally, use the conjugate gradient method (CG) and ILU decomposition preprocessing to efficiently solve this sparse linear equation system, thereby achieving high-precision simulation of acoustic wave propagation in complex media.

[0013] The specific content of Step 1 is as follows:

[0014] The acoustic wave propagation satisfies the following two-dimensional acoustic wave equation:

[0015]

[0016] Among them, \(u(x,y,t)\) represents the sound pressure or displacement field, represents the spatial propagation of the wave, the density \(\rho(x,y)\) and the bulk modulus \(K(x,y)\), and the wave speed \(v(x,y)\) is the spatially varying wave speed, which is determined by the physical properties of the medium:

[0017]

[0018] Equation (2) is the physical basis of the wave speed in Equation (1), which converts the medium properties into the input parameters required for numerical calculation and provides key data for subsequent spatial discretization.

[0019] To simulate the acoustic wave propagation in complex media, efficient numerical methods are needed to handle the challenges brought by irregular grids and heterogeneous material properties.

[0020] The MQRBF-FD (Multiquadric Radial Basis Function - Finite Difference) method constructs the spatial derivative through MQRBF and combines the finite difference to discretize the time derivative, forming a numerical framework with both high accuracy and high efficiency. The general form of MQRBF is:

[0021]

[0022] Among them, \(r = \|x - c\|\) represents the distance between the evaluation point and the center, \(c\) is the shape parameter, and the optimization goal of the shape parameter \(c\) is to minimize the maximum error \(MaxError(c)\), which is defined as:

[0023] \(MaxError(c)=\max\) x \(\vert f(x)-s(x,c)\vert\ (4)\)

[0024] Equation (4) drives the optimization of the shape parameter. Among them, \(s(x,c)\) is the sine function interpolation basis function, \(f(x)\) is the target function, and \(c\) opt is the optimal shape parameter, and its optimization process is expressed as:

[0025]

[0026] Fourier-expand the solution of the acoustic wave equation (Equation (1)) into a combination of sine functions, use the improved random walk (IRW) algorithm to optimize the MQRBF shape parameters of a large number of sine function combinations according to Equation (5), and establish a dataset "LCSF-MQSP900K" containing a large number of sine function combinations and the corresponding optimal shape parameters of MQRBF based on the optimization results, covering the mapping relationship between different frequency combinations and the optimal parameter \(c\) opt to provide data support for the training of the Adam-BP model.

[0027] The specific content of the second step is as follows:

[0028] Using the data set established in Step 1, train the Adam-BP neural network model. The expression of the Adam-BP neural network model is:

[0029] △w ij = ηδ j x i , △b j = ηδ j , δ j = f′(s j )(y j - t j )(6)

[0030] Among them, w ij represents the weight from the i-th input neuron to the j-th hidden neuron, b j is the bias of the j-th neuron, δ j is the error term, f′(s j ) is the derivative of the activation function, y j and t j are the network output and the target output respectively, and η is the learning rate; on this basis, the Adam optimizer introduces the momentum and adaptive learning rate strategies, and the core formula is as follows:

[0031]

[0032] Among them, g t is the gradient, θ t is the current model parameter, α is the learning rate, β1 and β2 are the momentum coefficients, m t and v t are the exponential moving averages of the gradient and the squared gradient respectively, and are the exponential moving averages after bias correction, and ∈ is the numerical stability term. Through the above training process, the Adam-BP neural network model can efficiently predict the optimal shape parameter c opt corresponding to the sine function combination, so as to provide a dynamically optimized parameter basis for the subsequent local MQRBF interpolation.

[0033] The specific content of Step 3 is as follows:

[0034] On the basis of Step 2, Step 3 completes the spatial discretization of the wave field through local MQRBF interpolation. First, expand the solution of the acoustic wave equation into the form of a sine function combination by Fourier, and use the optimized model in Step 2 to select the optimal MQRBF shape parameter. Subsequently, based on the MQRBF kernel function defined in formula (3), perform local interpolation. In each subdomain, the MQRBF interpolation realizes the high-precision calculation of the spatial derivative through the neighborhood nodes:

[0035]

[0036] Among them, k represents the number of neighborhood nodes, and (x i , y i ) are the coordinates of the neighborhood nodes, and w i (t) is the time-dependent interpolation weight. Through formula (9), each subdomain independently completes the interpolation calculation, significantly improving the parallel efficiency, and finally completes the spatial discretization of the wave field, providing an accurate spatial discretization result for subsequent time integration and wave field evolution.

[0037] Specifically, step four is as follows:

[0038] After completing the spatial discretization in step three, the time is discretized using the finite difference method (FD). The second-order time derivative of the wave field is discretized using the second-order central difference scheme:

[0039]

[0040] Among them, respectively represent the wave field values at the current, previous, and next steps, and △t is the time step. To ensure the numerical stability of long-time simulations, the implicit midpoint method is combined to optimize the time integration.

[0041] Furthermore, the specific expression is: the wave field velocity component is calculated at the midpoint of the time step by ; the wave field displacement component is updated at the next time step by ; the wave field velocity component is calculated at the next time step by . Among them, u n represents the wave field displacement component at the current time step, v n is the wave field velocity component at the current time step, △t is the time step, is the spatial Laplacian operator. At the same time, an adaptive time step strategy is adopted to dynamically adjust the time step according to the local error:

[0042]

[0043] Among them, ∈ actual is the current error, and ∈ tol is the preset tolerance.

[0044] Through the finite difference method, the second-order time derivative of the acoustic wave equation is transformed into a difference format, and together with the previous spatial discretization, a sparse linear equation system is constructed:

[0045] Ax n+1 = b n (12)

[0046] The matrix A in Equation (12) is jointly constructed by the spatial derivative matrix in Equation (9) and the time discretization term in Equation (10).

[0047] The preconditioned conjugate gradient method (PCG) combined with incomplete LU decomposition (ILU) is used for parallel solution:

[0048] Preconditioning acceleration: ILU decomposition improves the matrix condition number;

[0049] Boundary synchronization: Based on the local wave field values obtained during the time integration process, the continuity of the wave field between subdomains is ensured by directly sharing the wave field values of the boundary nodes of adjacent subdomains.

[0050] Efficient iteration: PCG minimizes the residual ||Ax n+1 -b n ||, and each subdomain is solved independently.

[0051] Through the efficient solution of the sparse linear equations, the solution of the current time step of the global wave field is successfully updated, and the propagation behavior of sound waves in complex media is fully simulated, finally achieving high-precision and high-efficiency elastic wave propagation simulation.

[0052] Spatial discretization provides a high-precision interpolation basis for time discretization. The finite difference method constructs a sparse linear equation system through discretization in the time direction. The combination of the conjugate gradient method and the implicit midpoint method further guarantees the solution efficiency and energy conservation, while the adaptive time step strategy realizes resource optimization by precisely controlling the time resolution. Through this closely coordinated framework, Step 4 successfully realizes the efficient simulation of sound wave propagation in complex media, meeting the dual requirements of accuracy and efficiency.

[0053] Advantages of the present invention:

[0054] Through the above steps, the final output of the Adam-BP neural network is the optimal shape parameter required by the MQRBF-FD method when solving the acoustic wave equation. MQRBF uses the optimal shape parameter for local interpolation to complete spatial discretization. Spatial discretization provides a high-precision interpolation basis for time discretization. The finite difference method constructs a sparse linear equation system through discretization in the time direction. The combination of the conjugate gradient method and the implicit midpoint method further guarantees the solution efficiency and energy conservation, while the adaptive time step strategy realizes resource optimization by precisely controlling the time resolution. Through this closely coordinated framework, the efficient simulation of sound wave propagation in complex media is successfully realized, meeting the dual requirements of accuracy and efficiency.

[0055] The present invention significantly improves the computational efficiency and accuracy of acoustic wave propagation simulation by combining local interpolation, shape parameter adaptive optimization and efficient numerical solution. Local interpolation uses MQRBF, which does not rely on regular grids and can provide high-precision spatial approximation under irregular grids and complex geometric structures; shape parameter adaptive optimization dynamically adjusts the shape parameters of MQRBF through the Adam-BP neural network model, achieving a precise balance between capturing complex wave characteristics and maintaining numerical stability; the efficient numerical solution combines the implicit midpoint method and the conjugate gradient method, which not only ensures the energy conservation of wave field evolution, but also greatly reduces the computational cost of solving large sparse linear equations. The synergistic effect of these methods enables the framework of the present invention to achieve efficient and accurate wave field simulation under complex boundary conditions, heterogeneous media and irregular grids.

[0056] In traditional methods, the selection of shape parameters often relies on experience or trial and error, which makes it difficult to adapt to changes in local medium properties. The IRW algorithm can effectively minimize the maximum error by generating random vectors and iteratively optimizing shape parameters. It solves the problems of numerical instability and reduced accuracy caused by improper shape parameter selection in traditional methods by gradually updating shape parameters and converging to the optimal solution.

[0057] The adaptive optimization model constructed by the neural network realizes the automatic selection of shape parameters without manual intervention, and can dynamically adjust the optimal shape parameters according to the characteristics of the local medium. The optimization model combines deep learning and adaptive momentum optimization methods to ensure the high accuracy and numerical stability of the model in complex media. The shape parameters selected by the model are used for local MQRBF interpolation, which avoids the computational burden brought by global grid encryption, so that the present invention can efficiently capture wave field changes, especially in areas with drastic wave field changes or uneven material properties, significantly improving the computational efficiency.

[0058] This invention breaks through the bottleneck of traditional methods in shape parameter optimization in complex media, and provides a new solution for sound wave propagation simulation through intelligent adaptive optimization model and efficient numerical solution technology. Its high precision and high stability make it show significant technical advantages in the fields of acoustics, seismic exploration and medical imaging. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 It is a flowchart of the local MQRBF-FD sound wave propagation simulation method based on Adam-BP neural network.

[0060] Figure 2 It is a flowchart of optimizing shape parameters using improved random walk (IRW) optimization algorithm.

[0061] Figure 3 is a non-uniform grid node distribution diagram of an embodiment.

[0062] Figure 4 It is the comparison of the L2 errors of displacement and velocity under different methods of the embodiments in the logarithmic scale.

[0063] Figure 5 It is the comparison of wavefront propagation under different methods of the embodiments. Specific embodiments

[0064] The present invention will be further described in detail below with reference to the accompanying drawings.

[0065] The local MQRBF-FD acoustic wave propagation simulation method based on the Adam-BP neural network of the present invention realizes a high-precision acoustic wave simulation scheme based on an adaptive optimization shape parameter model and local interpolation technology.

[0066] The following will introduce the present invention in detail step by step. The specific operation process is as Figure 1 shown:

[0067] Step 1: Generate a data set using an improved random walk optimization algorithm;

[0068] The solution of the two-dimensional acoustic wave equation (Equation (1)) can be expressed as a combination of frequency components through Fourier expansion:

[0069]

[0070] For different combinations of frequency components, this method optimizes the shape parameters through an improved random walk (IRW) algorithm. The optimization objective is to minimize MaxError in Equation (4) and find the optimal shape parameter c described in Equation (5) opt , and the process is as Figure 2 shown.

[0071] The steps of the IRW algorithm for optimizing shape parameters are as follows:

[0072] Initialization: Set the initial shape parameter c0, the number of iterations i and k, define the step size precision θ and the error control tolerance ∈.

[0073] Generate random vectors: At each iteration, generate N n-dimensional random vectors u ik , and these vectors are within the interval (-q, q), and are normalized to update the shape parameter c.

[0074] Error comparison: Calculate the maximum error MaxError(c i ) corresponding to each shape parameter c i ), and compare it with the error of the previous round. If the current error MaxError(c i ) is less than the error of the previous round MaxError(c i-1) If so, update the shape parameter and continue the iteration; if the current error is larger, check whether the step size exceeds the threshold θ: if it exceeds the threshold, halve the step size λ and start over, if it does not exceed the threshold, end the iteration.

[0075] Termination condition: When the maximum error MaxError(c) is less than the set tolerance ∈ or reaches the maximum number of iterations M, stop the optimization and output the optimal shape parameter c opt .

[0076] During the optimization process, it is explored that there is an inverse relationship between the frequency ω of the sine function and the optimal shape parameter c opt This relationship indicates that the higher the frequency, the smaller the required shape parameter, thus providing an effective reference for optimization. The specific expression is:

[0077]

[0078] Using the Pandas library of Python 3.10, the IRW algorithm was used to select the optimal shape parameters for the linear combination of 900,000 sine functions. A dataset named "LCSF-MQSP900K" was established. Each instance in this dataset encapsulates two different elements: ω of the sine function combination 1, ω 2, ......, ω n and its corresponding c opt .

[0079] Step 2: Establish an adaptive optimization model for shape parameters based on Adam-BP;

[0080] In the present invention, the BP algorithm and the Adam optimizer are combined, and through the adaptive ability of deep learning and efficient gradient processing, rapid convergence of the large-scale complex dataset established in Step 1 is achieved.

[0081] In the present invention, the network structure of the Adam-BP neural network model includes an input layer, two hidden layers, and an output layer. Each hidden layer contains 100 neurons, and the activation function is ReLU. The learning rate of the model is set to α = 0.003, the bias is uniformly initialized to 1, the batch size is 128, and the number of training epochs is 50. During the optimization process, the Adam-BP model achieves accurate prediction of shape parameters by minimizing the mean square error (MSE) loss function. The comparison between its prediction results and the target values shows that the mean square error of the model is 0.163512, and the prediction accuracy is as high as 97.98%. The further convergence trend of the error function indicates that the Adam-BP model can converge rapidly after 50 epochs of training, and the optimization process is stable.

[0082] Step 3: Perform spatial discretization based on local MQRBF under the optimal shape parameter;

[0083] In this step, the solution of the acoustic wave equation is first Fourier-expanded. Based on the shape parameter adaptive optimization model in Step 2, the optimal shape parameter is selected, and spatial discretization is achieved through local MQRBF under the optimal shape parameter. For spatial derivatives, the first and second derivatives are calculated based on the derivative form of the MQ radial basis function:

[0084]

[0085] To further improve the interpolation accuracy in complex regions, especially in the boundary regions where the wave field changes violently, the combination of polynomial basis functions and radial basis functions is introduced, and its extended matrix is expressed as:

[0086]

[0087] where Φ is the radial basis function matrix, and its elements are calculated by formula (3). λ is the regularization parameter used to improve numerical stability. P is the polynomial basis function term, which is combined with MQRBF to further suppress numerical oscillations during the interpolation process and improve the accuracy in complex regions. Through the synergistic effect of the above formulas, Step 3 achieves high-precision spatial discretization of the wave field, providing a reliable numerical basis for subsequent time discretization and wave field evolution.

[0088] Step 4: Time-direction discretization and efficient sparse linear equation system solution;

[0089] After completing the spatial discretization in Step 3, to simulate the dynamic evolution of acoustic wave propagation, the finite difference method is used to discretize the second-order time derivative term in the acoustic wave equation in the time direction (see formula (10)). By combining formula (10) with the spatial derivative discretization result (formula (9)) in Step 2, a spatio-temporal coupled discrete equation is constructed. To improve the computational efficiency and take into account numerical stability, the present invention introduces an error-driven adaptive time step strategy. The specific process is as follows:

[0090] Local error estimation: Calculate the local error ε of the current time step in each subdomain actual ,

[0091] Dynamic step size adjustment: According to the ratio of the error to the preset tolerance ∈ tol , use formula (11) to update the next time step size. When the wave field changes violently (ε actual < ε tol ): Reduce the step size to improve the accuracy; when the wave field is stable (ε actual > ε tol ): Increase the step size to optimize the efficiency.

[0092] After the above-mentioned time and space discretization, the wave field evolution is transformed into the problem of solving a sparse linear equation system, and its mathematical form is formula (12). In order to efficiently solve the equation system, the preconditioned conjugate gradient method (PCG) combined with the incomplete LU decomposition (ILU) is used as the core solver. The preconditioned matrix M -1 Constructed via ILU decomposition:

[0093] M=L·U(18)

[0094] Among them, L and U are the lower triangular and upper triangular matrices that retain the sparsity of the original matrix, significantly improving the matrix condition number and reducing the number of iterations. Through the above process, the solution of the sparse linear equations and the wave field update are completed efficiently in the entire domain, ultimately achieving high-precision simulation of sound wave propagation.

[0095] Through the above steps, the present invention successfully constructs a local MQRBF-FD acoustic wave propagation simulation method based on the Adam-BP neural network, which can accurately analyze the multi-scale dynamic behavior of acoustic wave propagation in complex media. Its core technical breakthrough is: the adaptive dynamic optimization of MQRBF shape parameters is realized through the Adam-BP model, which significantly improves the calculation accuracy and numerical stability, and solves the core bottleneck of low parameter adaptation efficiency and serious error accumulation in heterogeneous media in traditional methods.

[0096] In summary, the local MQRBF-FD acoustic wave propagation simulation method based on Adam-BP neural network proposed in this paper constructs an acoustic wave propagation simulation framework with both efficiency and robustness through the innovative integration of deep learning and efficient numerical technology. This framework breaks through the limitations of traditional global optimization strategies by real-time adaptation of local wave field characteristics based on intelligent models; local interpolation of MQRBF with optimal shape parameters accurately captures complex wave field behaviors such as interface reflection and diffraction; implicit midpoint method and CG-ILU collaborative solution ensure the balance between energy conservation and computational efficiency.

[0097] Step 1 is used as the training data source for the deep learning model in step 2. It is used as the training data basis for the deep learning model. The frequency components are efficiently extracted through the improved random walk (IRW) algorithm to establish the "LCSF-MQSP900K" data set, and the correlation between shape parameters and wave field characteristics is accurately mapped. Step 2 is to train the Adam-BP neural network based on the step 1 data set, build a dynamic optimization model for shape parameters, and realize intelligent adaptation of medium characteristics. Step 3 is based on the optimized parameters in step 2, completes high-precision spatial discretization through local MQRBF interpolation, and combines polynomial basis functions to suppress numerical oscillations. Step 4 is based on the spatial discretization of step 3, uses the finite difference method to complete time discretization, combines the adaptive step size strategy with the CG-ILU sparse solution, and realizes the efficient evolution of the global wave field.

[0098] Example:

[0099] The local MQRBF-FD acoustic wave propagation simulation method based on the Adam-BP neural network of the present invention has been verified in the P-wave propagation example in heterogeneous media.

[0100] For the P-wave propagation problem in two-dimensional heterogeneous media, the size of the computational domain is 60×60 km, and the wave speed distribution includes the wave speed c1 = 2500 m / s of the upper medium and the wave speed c2 = 4500 m / s of the lower medium. The two regions are separated by a sine interface, and the interface expression is:

[0101]

[0102] The wave speed changes drastically at the interface, resulting in complex wave field behaviors, including reflected waves and transmitted waves. These phenomena pose the main challenges in wave simulation. To accurately capture the propagation of the wave field in complex media, a finer grid (Δx = Δy = 0.5 km) is used in the interface region where the wave speed changes drastically, while a coarser grid (Δx = Δy = 2 km) is adopted in the region far from the interface to optimize the calculation efficiency, as Figure 3 shown.

[0103] In the solution processing of the wave equation, the solution is first expressed as a combination of frequency components through Fourier expansion. After extracting the key frequency components, the Adam-BP model is used to complete the high-precision prediction of the shape parameters. Subsequently, in wave field interpolation and derivative calculation, the local MQRBF technique is adopted to accurately capture the wave field changes in complex media. Combining the finite difference method for time discretization and numerical solution method, the numerical simulation of the wave field is completed.

[0104] The simulation results further verify the accuracy and stability of the present method. For example, in the L2 error comparison (as Figure 4 shown), the error of the present method in complex wave field simulation is significantly lower than that of the FD, SEM, and FV methods. The wavefront propagation behaviors under different methods demonstrate the accuracy and stability of the present method. Compared with other numerical methods, such as Figure 5 shown. The wavefront of our method maintains higher smoothness and stability of energy transfer.

[0105] The method proposed by the present invention, through the Adam-BP neural network model driven by deep learning, combines the adaptive optimization of MQRBF shape parameters and efficient numerical solution methods, and successfully applies to the simulation problem of complex P-wave propagation in heterogeneous media. This method demonstrates excellent high precision, high stability, and high computational performance in complex wave field simulation, breaks through the limitations of traditional methods in heterogeneous media and irregular grids, and provides an efficient and reliable numerical simulation tool for acoustic imaging, seismic exploration, and related fields.

Claims

1. A local MQRBF-FD acoustic wave propagation simulation method based on Adam-BP neural network, characterized in that: The following steps are involved: Step 1: Using the improved random walk algorithm, the shape parameters of MQRBF in the Fourier expansion of the acoustic wave equation are optimized to generate a data set containing the acoustic wave frequency components and their corresponding optimal parameters; Step 2: Based on the data set, a BP neural network is trained, and an Adam optimizer is used to optimize parameters, so as to construct an Adam-BP neural network model for accurately adjusting the shape parameters of the acoustic wave equation; Step 3: Expand the solution of the acoustic wave equation into a combination of sine functions through Fourier transformation, adaptively optimize the shape parameters based on the Adam-BP neural network model constructed in step 2, and use the optimized shape parameters to perform local MQRBF interpolation at key nodes to complete spatial discretization; Step 4: Based on the spatial discretization completed in step 3, the finite difference method is used to discretize the time. The obtained discrete equation system forms a sparse linear equation system. The conjugate gradient method and ILU decomposition preprocessing are used to efficiently solve the sparse linear equation system to realize the simulation of sound wave propagation in complex media.

2. The local MQRBF-FD acoustic wave propagation simulation method based on Adam-BP neural network according to claim 1 is characterized in that: The step 1 is specifically as follows: The propagation of sound waves satisfies the following two-dimensional sound wave equation: Among them, u(x,y,t) represents the sound pressure or displacement field, represents the spatial propagation of the wave, the density ρ(x,y) and the bulk modulus K(x,y), and the wave speed v(x,y) is the spatially varying wave speed: The MQRBF-FD method constructs the spatial derivative through MQRBF and discretizes the time derivative by combining finite differences to form a numerical framework. The general form of MQRBF is: Among them, r = ‖xc‖ represents the distance between the evaluation point and the center, c is the shape parameter, and the optimization goal of the shape parameter c is to minimize the maximum error MaxError(c), which is defined as: MaxError(c)=max x |f(x)-s(x,c)| (4) Formula (4) drives shape parameter optimization, where s(x,c) is the sine function interpolation basis function, f(x) is the objective function, and c opt is the optimal shape parameter, and its optimization process is expressed as: The solution of the acoustic wave equation is Fourier expanded into a combination of sine functions. The MQRBF shape parameters of the sine function combination are optimized using the improved random walk algorithm according to formula (5). Based on the optimization results, a data set "LCSF-MQSP900K" containing the sine function combination and the corresponding MQRBF optimal shape parameters is established, covering different frequency combinations and optimal parameters c opt The mapping relationship.

3. The local MQRBF-FD acoustic wave propagation simulation method based on Adam-BP neural network according to claim 1 is characterized in that: The step 2 is specifically as follows: Use the data set established in step 1 to train the Adam-BP neural network model. The expression of the Adam-BP neural network model is: △w ij =hd j x i ,△b j =hd j ,d j =f′(s j )(y j -t j )(6) Among them, w ij represents the weight from the i-th input neuron to the j-th hidden neuron, b j is the bias of the jth neuron, δ j is the error term, f′(s j ) is the derivative of the activation function, y j and t j are network output and target output respectively, η is the learning rate; Adam optimizer introduces momentum and adaptive learning rate strategy, the formula is as follows: Among them, g t is the gradient, θ t is the current model parameter, α is the learning rate, β1 and β2 are momentum coefficients, and m t and v t are the exponential moving averages of the gradient and the square of the gradient, respectively. and is the bias-corrected exponential moving average, and ∈ is a numerical stability term.

4. The local MQRBF-FD acoustic wave propagation simulation method based on Adam-BP neural network according to claim 3 is characterized in that: The step three is specifically as follows: Based on step 2, step 3 completes the spatial discretization of the wave field through local MQRBF interpolation. First, the solution of the acoustic wave equation is Fourier expanded into a combination of sine functions, and the optimal MQRBF shape parameters are selected using the optimization model of step 2. Then, local interpolation is performed based on the MQRBF kernel function defined in formula (3). In each subdomain, MQRBF interpolation realizes high-precision calculation of spatial derivatives through neighborhood nodes: Among them, k represents the number of neighboring nodes, (x i ,y i ) are the coordinates of the neighborhood nodes, w i (t) is the time-dependent interpolation weight.

5. The local MQRBF-FD acoustic wave propagation simulation method based on Adam-BP neural network according to claim 1 is characterized in that: The step 4 is specifically as follows: After completing the spatial discretization in step 3, the time discretization uses the finite difference method to obtain the time second-order derivative of the wave field. Discretize using the second-order central difference format: in, They represent the wave field values ​​of the current, previous step and next step respectively, and △t is the time step.

6. The local MQRBF-FD acoustic wave propagation simulation method based on Adam-BP neural network according to claim 5 is characterized in that: The specific expression is: The wave field velocity component at the midpoint of the time step is calculate; The wavefield displacement component is given by renew; The wave field velocity component is given by calculate; Among them, u n represents the displacement component of the wave field at the current time step, v n is the wave field velocity component of the current time step, △t is the time step, is the spatial Laplace operator; at the same time, an adaptive time step strategy is adopted to dynamically adjust the time step according to the local error: Among them, ∈ actual is the current error, ∈ tol To preset tolerance; Through the finite difference method, the time second-order derivative of the acoustic wave equation is converted into a difference format, and together with the previous spatial discretization, a sparse linear equation system is constructed: Ax n+1 =b n (12) The matrix A of formula (12) is constructed jointly by the spatial derivative matrix of formula (9) and the time discrete terms of formula (10).

7. The local MQRBF-FD acoustic wave propagation simulation method based on Adam-BP neural network according to claim 6 is characterized in that: The preconditioned conjugate gradient method is combined with incomplete LU decomposition to solve in parallel: Preconditioning acceleration: ILU decomposition improves matrix condition number; Boundary synchronization: Based on the local wave field values ​​obtained during the time integration process, the wave field values ​​of the adjacent sub-domain boundary nodes are directly shared to ensure the continuity of the wave field between sub-domains; Efficient iteration: PCG minimizes residual || Ax n+1 -b n ||, each subdomain is solved independently; Through the efficient solution of sparse linear equations, the propagation behavior of sound waves in complex media is simulated and elastic wave propagation simulation is realized.

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