Fourier feature embedded PINN based method for underwater two-dimensional acoustic field calculation

Abstract

The application discloses an underwater two-dimensional sound field calculation method based on a Fourier feature embedded PINN, relates to the technical field of sound field calculation, and comprises the following steps: encoding into a partial differential equation physical loss function to construct PINNs; determining corresponding sound source conditions, sea surface boundary conditions and seabed boundary conditions, and encoding into a boundary condition physical loss function; constructing a data-assisted loss function according to observation data of a calculation region; modifying the frequency corresponding to the Fourier base by initially selecting different sigma parameters; pre-training the PINNs with different sigma parameters selected initially at a set step size, calculating a point-by-point average error according to the sound pressure output by the PINNs after the training is completed, and finding an optimal sigma parameter; and performing two-stage training according to the optimal sigma parameter, and calculating a sound propagation loss prediction result from the real part and the imaginary part of the sound pressure output by the PINNs. The application can enable the network to learn high-frequency components, thereby predicting multi-scale marine sound propagation phenomena.

Description

Technical Field

[0001] This application relates to the field of sound field calculation technology, and in particular to a method for calculating underwater two-dimensional sound fields based on Fourier feature embedding PINN. Background Technology

[0002] Sound waves are currently the only known information carrier capable of long-distance propagation in water. Rapid and high-precision calculation of sound fields is crucial for the design and use of sonar equipment, as well as for marine environmental perception and parameter acquisition. However, existing traditional data-driven neural network methods have significant limitations: data-driven models require extensive acoustic data for training, but high-fidelity underwater acoustic data is often difficult to obtain in practice due to the cost of offshore operations; data-driven models also have insufficient generalization ability outside the data acquisition area. Therefore, their practical application is limited. Furthermore, high-frequency sound sources and phenomena such as reflection and refraction caused by sound waves interacting with the sea surface and seabed give underwater acoustic propagation multi-scale characteristics, often exhibiting both high-frequency and low-frequency features simultaneously.

[0003] PINNs, based on the universal fitting theorem, encode physical constraints with data, seamlessly integrating physical principles with observational data, and solving the PDE by minimizing the loss function. This provides a powerful alternative to traditional methods. Compared to purely data-driven ocean acoustic field calculation models, PINNs reduce the fitting space of nonlinear relationships, significantly lowering the demand for acoustic measurement data. It shifts from acoustic data-driven to acoustic data-assisted, enabling network training under small sample conditions and exhibiting stronger generalization capabilities. However, the inherent spectral bias of the fully connected layer neural networks used in PINNs poses a challenge to solving these types of problems. In both theoretical and practical problems, PINNs often struggle to solve for high-frequency components. Summary of the Invention

[0004] In view of this, embodiments of this application provide an underwater two-dimensional sound field calculation method based on Fourier feature embedding PINN to predict multi-scale ocean sound fields.

[0005] One aspect of this application provides a method for calculating underwater two-dimensional acoustic fields based on Fourier feature embedding PINN, the method comprising the following steps:

[0006] The wavenumber values ​​and computational domain of the Helmholtz equation are determined based on the marine environmental parameters of the computational region, and encoded into a partial differential equation physical loss function, thereby constructing PINNs; where the real and imaginary parts of the sound pressure are respectively used as output terms of PINNs.

[0007] Based on the sound source information and marine environmental parameters, the corresponding sound source conditions, sea surface boundary conditions, and seabed boundary conditions are determined and encoded into boundary condition physical loss functions;

[0008] A data-aided loss function is constructed based on the observation data of the computational region;

[0009] After normalizing the input coordinates, a Fourier feature layer is constructed using the Fourier feature mapping function. Based on the wavenumber value and combined with the neural tangent kernel of PINNs, different σ parameters are initially selected to modify the frequency corresponding to the Fourier basis, thereby controlling the convergence rate of PINNs for different frequency components.

[0010] Pre-training is performed on PINNs with different initially selected σ parameters with a set step size. After training, the point-by-point average error is calculated based on the sound pressure output of the PINNs to find the optimal σ parameter.

[0011] Based on the optimal σ parameter, a two-stage training process is performed using ADAM and L-BFGS. The sound propagation loss prediction result is calculated from the real and imaginary parts of the sound pressure output by PINNs.

[0012] In some embodiments, before determining the wavenumber values ​​and computational domain of the Helmholtz equations based on the marine environmental parameters of the computational region, the method further includes the following steps:

[0013] Transform the wave equation of ocean acoustics into the frequency domain Helmholtz equation in cylindrical coordinates:

[0014] ;

[0015] in, For sound pressure, Horizontal distance For depth, The wavenumber is used to construct the PINNs, which include an input layer, a Fourier feature layer, a hidden layer, and an output layer. The input layer represents the horizontal distance and depth, and the output layer represents the real and imaginary parts of the sound pressure.

[0016] In some embodiments, the sound source conditions are:

[0017] ;

[0018] ;

[0019] in, For control parameters, The horizontal distance from the sound source. For the depth of the sound source;

[0020] The sea surface boundary conditions are as follows:

[0021] ;

[0022] Sound pressure level; It is always equal to;

[0023] The seabed boundary conditions are as follows:

[0024] ;

[0025] ;

[0026] Where a and b represent the upper and lower surfaces of the boundary, respectively; This refers to the sound pressure level on the upper surface of the seabed boundary. This represents the acoustic pressure below the seabed boundary. The density of seawater, Density at the seabed; This indicates taking the derivative with respect to depth z;

[0027] The sea surface boundary conditions are implemented through network changes, when At that time, the network output satisfies a sound pressure level of 0:

[0028] ;

[0029] ;

[0030] in, For the network output, For depth, i.e., the input parameter of the PDE; The loss function for encoding boundary conditions. For the number of boundary points selected, The network predicts the acoustic pressure on the upper surface of the seabed boundary. This represents the acoustic pressure at the lower surface of the seabed boundary.

[0031] In some embodiments, constructing a data-assisted loss function based on the observation data of the computational region includes the following steps:

[0032] The data-aided loss function is constructed by randomly selecting the observed data as follows:

[0033] ;

[0034] in, The auxiliary loss function for the data, The number of data points selected. For the real part of the sound pressure predicted by the network, The real part of the sound pressure in the observed data. The imaginary part of the sound pressure predicted by the network. This represents the imaginary part of the sound pressure in the observed data.

[0035] Therefore, the total loss function is: ;

[0036] For the loss weights of the PDE constraint, The loss weights are determined by the boundary conditions. The loss weights are constraints imposed on the observed data.

[0037] In some embodiments, the Fourier feature layer is:

[0038] ;

[0039] in, Input coordinates, For coefficients, [ It belongs to the Gaussian distribution. Parameters;

[0040] For input coordinates The tangential nucleus of the nerve is determined by the following formula:

[0041] ;

[0042] in,[ It belongs to the Gaussian distribution. By modifying The value is used to regulate the tangential nucleus of the nerve.

[0043] In some embodiments, the pre-training of PINNs with different initially selected σ parameters with a set step size, and the calculation of the point-by-point average error based on the sound pressure output by the PINNs after training, to find the optimal σ parameter, includes the following steps:

[0044] Repeat the following steps to dynamically find the optimal solution. parameter:

[0045] Different initial selections based on the wavenumber values Multiple PINNs of parameters are trained with a set step size. After training, the point-by-point average error is calculated based on the sound pressure output of the network, and the target parameters are selected.

[0046] In some embodiments, the two-stage training using ADAM and L-BFGS based on the optimal σ parameter includes the following steps:

[0047] According to the optimal The parameters are adjusted so that the PDE and BC points are reselected every 100 training steps. Combined with the residual adaptive refinement method, ADAM and L-BFGS are used for two-stage training.

[0048] Another aspect of this application embodiment provides an underwater two-dimensional acoustic field calculation device based on Fourier feature embedding PINN, the device comprising:

[0049] The network construction unit is used to determine the wave number value and computational domain of the Helmholtz equation based on the marine environmental parameters of the computational region, and encodes them into a partial differential equation physical loss function, thereby constructing PINNs; wherein, the real part and imaginary part of the sound pressure are respectively used as the output terms of PINNs;

[0050] The boundary coding unit is used to determine the corresponding sound source conditions, sea surface boundary conditions, and seabed boundary conditions based on sound source information and marine environmental parameters, and encode them into a boundary condition physical loss function.

[0051] The data assistance unit is used to construct a data assistance loss function based on the observation data of the computational region;

[0052] The parameter control unit is used to normalize the input coordinates and then construct a Fourier feature layer using the Fourier feature mapping function. Based on the wavenumber value and combined with the neural tangent kernel of PINNs, different σ parameters are initially selected to modify the frequency corresponding to the Fourier basis, thereby controlling the convergence rate of PINNs for different frequency components.

[0053] The pre-training unit is used to pre-train PINNs with different initially selected σ parameters with a set step size. After training, the point-by-point average error is calculated based on the sound pressure output of the PINNs to find the optimal σ parameter.

[0054] The sound field calculation unit is used to perform two-stage training using ADAM and L-BFGS based on the optimal σ parameter, and calculates the sound propagation loss prediction result from the real and imaginary parts of the sound pressure output by PINNs.

[0055] Another aspect of this application embodiment provides an electronic device, including a processor and a memory;

[0056] The memory is used to store programs;

[0057] The processor executes the program to implement any of the methods described above.

[0058] Another aspect of this application provides a computer-readable storage medium storing a program that is executed by a processor to implement the method described in any of the above embodiments.

[0059] This application includes at least the following beneficial effects:

[0060] This application determines the wavenumber values ​​and computational domain of the Helmholtz equation based on the marine environmental parameters of the computational region, and encodes them as a partial differential equation physical loss function, thereby constructing PINNs. The real and imaginary parts of the sound pressure are used as output terms of the PINNs. Based on the sound source information and marine environmental parameters, the corresponding sound source conditions, sea surface boundary conditions, and seabed boundary conditions are determined and encoded as boundary condition physical loss functions. A data-aided loss function is constructed based on the observation data of the computational region. After normalizing the input coordinates, a Fourier feature layer is constructed using a Fourier feature mapping function. Based on wavenumber values ​​and the neural tangent kernel of PINNs, different σ parameters are initially selected to modify the frequencies corresponding to the Fourier basis, thereby controlling the convergence rate of PINNs for different frequency components. PINNs with different initially selected σ parameters are pre-trained with a set step size. After training, the point-by-point average error is calculated based on the sound pressure output of PINNs to find the optimal σ parameter. Based on the optimal σ parameter, a two-stage training process is performed using ADAM and L-BFGS, and the sound propagation loss prediction result is obtained by calculating the real and imaginary parts of the sound pressure output of PINNs. This application provides an algorithm to mitigate spectral bias, enabling the network to learn high-frequency components and thus predict multi-scale ocean sound propagation phenomena. Attached Figure Description

[0061] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0062] Figure 1 A flowchart illustrating the underwater two-dimensional acoustic field calculation method based on Fourier feature embedding PINN provided in this application embodiment;

[0063] Figure 2 An example flowchart of an underwater two-dimensional acoustic field calculation method based on Fourier feature embedding PINN provided for embodiments of this application;

[0064] Figure 3 A schematic diagram of a Fourier feature embedding PINNs structure for marine acoustics provided in an embodiment of this application;

[0065] Figure 4 Example diagram of the sound pressure field (real part) prediction results of the KRAKEN model and Fourier feature embedding PINNs provided in the embodiments of this application;

[0066] Figure 5 The structural block diagram of the underwater two-dimensional acoustic field calculation device based on Fourier feature embedding PINN provided in the embodiments of this application is shown. Detailed Implementation

[0067] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0068] Before providing a detailed description of the embodiments of this application, some related technologies involved in the embodiments of this application will be described first, as follows:

[0069] PINN: Physical Information Neural Network.

[0070] NTK: Neural tangential nucleus.

[0071] Adam: Adaptive Moment Estimation, is currently one of the most commonly used optimization algorithms in deep learning.

[0072] L-BFGS: Finite Memory BFGS Algorithm, a second-order optimization algorithm belonging to the family of quasi-Newton methods.

[0073] KRAKEN: A computational model for marine acoustics.

[0074] Addressing the challenges of fitting high-frequency components and encoding acoustic boundary conditions for sound sources, sea surface, and seabed caused by the multi-scale characteristics of underwater acoustic propagation and the spectral bias of PINNs, this invention proposes a method for calculating underwater two-dimensional sound fields based on Fourier feature embedding within PINNs. This method solves the encoding problems of sound source, sea surface, and seabed boundary conditions, enabling super-resolution reconstruction of underwater sound fields by combining physical constraints with limited observational data. Compared to the baseline PINN method, this approach reduces training convergence time, effectively mitigates the spectral bias problem, and achieves more accurate results.

[0075] Reference Figure 1 This application provides a method for calculating underwater two-dimensional acoustic fields based on Fourier feature embedding PINN, specifically including the following steps S100~S150:

[0076] S100: Determine the wavenumber values ​​and computational domain of the Helmholtz equation based on the marine environmental parameters of the computational region, and encode them into a partial differential equation physical loss function, thereby constructing PINNs; where the real and imaginary parts of the sound pressure are respectively used as output terms of PINNs;

[0077] S110: Determine the corresponding sound source conditions, sea surface boundary conditions, and seabed boundary conditions based on sound source information and marine environmental parameters, and encode them as boundary condition physical loss functions;

[0078] S120: Construct a data-aided loss function based on the observation data of the computational region;

[0079] S130: After normalizing the input coordinates, a Fourier feature layer is constructed using the Fourier feature mapping function. Based on the wavenumber value and combined with the neural tangent kernel of PINNs, different σ parameters are initially selected to modify the frequency corresponding to the Fourier basis, thereby controlling the convergence rate of PINNs for different frequency components.

[0080] S140: Perform pre-training on PINNs with different initially selected σ parameters with a set step size. After training, calculate the point-by-point average error based on the sound pressure output by the PINNs and find the optimal σ parameter.

[0081] S150: Based on the optimal σ parameter, a two-stage training is performed using ADAM and L-BFGS. The sound propagation loss prediction result is calculated from the real and imaginary parts of the sound pressure output by PINNs.

[0082] The following section will provide a detailed introduction and explanation of the solutions in the embodiments of this application, using specific application examples.

[0083] Reference Figure 2 This embodiment provides a method for calculating the underwater two-dimensional acoustic field based on Fourier feature embedding PINN, including the following steps:

[0084] Step 1: Transform the wave equation, the core of ocean acoustics, into the frequency domain Helmholtz equation in cylindrical coordinates, and determine the wavenumber in the Helmholtz equation based on the ocean environmental parameters of the calculation area. The values ​​of and the computational domain of the equation are encoded into a partial differential equation physical loss function to construct PINNs. Since the sound pressure value is a complex number, the real and imaginary parts of the sound pressure are processed separately and used as the two output terms of the network. Figure 3 A schematic diagram of a PINNs structure for embedding Fourier features for marine acoustics.

[0085] Step 2: Based on the sound source information and marine environmental parameters, determine the corresponding sound source conditions, sea surface boundary conditions, and seabed boundary conditions, and encode them as boundary condition physical loss functions. For the sound source, a Gaussian sound source is used instead of the discontinuous Dirac function. The sea surface boundary condition is a pressure release boundary condition (Dirichlet boundary condition), and the seabed boundary condition is a fluid seabed boundary condition (Neumann boundary condition). Since the sea surface boundary condition is a first-type Dirichlet boundary condition, it is enforced through a more effective method of changing the network output. The second-type Neumann boundary condition for the seabed is enforced by encoding it as a loss function.

[0086] Step 3: To address the multi-scale characteristics of marine acoustic problems and the discontinuities introduced by point sources, a data-assisted loss function is constructed using a small amount of observation data to improve the fitting effect of PINNs on marine sound propagation problems.

[0087] Step 4: After normalizing the input coordinates, use Fourier feature mapping. The function constructs the Fourier feature layer based on the wavenumber obtained in step one. Based on the tangential nuclei of PINNs, different... The parameters are modified to correspond to the frequencies of the Fourier basis, thereby controlling the convergence rate of PINNs for different frequency components.

[0088] Step 5: Different initial choices The parameters PINNs are pre-trained with a set step size. After training, the point-by-point average error is calculated from the sound pressure output by the network to find the optimal value. parameter.

[0089] Step 6: Based on the optimal... The parameters are trained in two stages using ADAM+L-BFGS to achieve higher accuracy. The sound propagation loss prediction result is calculated from the real and imaginary parts of the sound pressure output by the network.

[0090] Furthermore, the frequency domain Helmholtz equation in cylindrical coordinates in Step 1 is as follows:

[0091] ;

[0092] in For sound pressure, Horizontal distance For depth, The wavenumber is used. The constructed PINNs consist of an input layer, a Fourier feature layer, a hidden layer, and an output layer. The input layer represents the horizontal distance and depth, and the output layer represents the real and imaginary parts of the sound pressure level.

[0093] Furthermore, the specific conditions for the sound source in Step 2 are as follows:

[0094] ;

[0095] ;

[0096] in For control parameters, The horizontal distance from the sound source. The depth of the sound source.

[0097] The sea surface boundary conditions are:

[0098] ;

[0099] This refers to sound pressure level. It is always equal to.

[0100] The seabed boundary conditions are:

[0101] ;

[0102] ;

[0103] Where a and b represent the upper and lower surfaces of the boundary, respectively. This refers to the sound pressure level on the upper surface of the seabed boundary. This represents the acoustic pressure below the seabed boundary. The density of seawater, This represents the density of the seabed. This indicates taking the derivative with respect to depth z.

[0104] Among them, the sea surface boundary conditions are enforced through network changes, when At that time, the network output will automatically satisfy the condition that the sound pressure level is 0.

[0105] ;

[0106] ;

[0107] in, The output of the neural network, This refers to the depth (the input parameter of the PDE, not the depth of the neural network). The loss function for encoding boundary conditions. For the number of boundary points selected, The network predicts the acoustic pressure on the upper surface of the seabed boundary. This represents the acoustic pressure at the lower surface of the seabed boundary. The density of seawater, This represents the density of the seabed. This indicates taking the derivative with respect to depth z.

[0108] Furthermore, in Step 3, a certain number of observation data are randomly selected as auxiliary data:

[0109] ;

[0110] The loss function that encodes the observed data. The number of data points selected. For the real part of the sound pressure predicted by the network, The real part of the sound pressure in the observed data. The imaginary part of the sound pressure predicted by the network. This represents the imaginary part of the sound pressure in the observed data.

[0111] Therefore, the total loss function is: .

[0112] For the loss weights of the PDE constraint, The loss weights are determined by the boundary conditions. The loss weights are constraints imposed on the observed data.

[0113] Furthermore, the Fourier feature layer in Step 4 is as follows:

[0114] ;

[0115] Input coordinates, For coefficients, [ It belongs to the Gaussian distribution. The parameters.

[0116] For input coordinates The tangential nucleus of the nerve is determined by the following formula:

[0117] ;

[0118] in,[ It belongs to the Gaussian distribution. It can be modified The value of NTK is adjusted.

[0119] Further, Step 5: Based on the initial selection of different wavenumber k values... Several PINNs of parameters are trained with set step sizes. After training, the point-by-point average error is calculated from the sound pressure output by the network, and the most suitable parameters are selected. This process is then repeated to dynamically find the optimal parameters. parameter;

[0120] Further, Step 6: Based on the optimal... The parameters are adjusted, and the PDE and BC points are reselected every 100 training steps. Combined with the residual adaptive refinement method, ADAM+L-BFGS is used for two-stage training to achieve higher accuracy for PINNs. After training, the sound propagation loss prediction result is calculated from the real and imaginary parts of the sound pressure output by the network.

[0121] More specifically, this embodiment provides the following detailed implementation methods:

[0122] Considering an ideal shallow sea environment with constant horizontal pressure, a sea depth of 50m, a calculation distance of 50m, a sound source depth of 20m, and a sound source frequency of 50Hz, assuming a constant seawater sound speed of 1500m / s and a seabed sound speed of 1800m / s, and a seawater density of 1000g / s... The density of the seabed is 1800g / .

[0123] Step 1: Calculate the wave number. The constructed neural network has 8 hidden layers, with 50 neurons in each layer. The activation function chosen is tanh. Points were selected during network training. , , The learning rate uses the cosine annealing algorithm.

[0124] Based on the above, PDE can be defined as follows:

[0125] .

[0126] Step 2: Based on the sound source information and marine environmental parameters, determine the corresponding sound source conditions, sea surface boundary conditions, and seabed boundary conditions, and encode them as boundary condition physical loss functions. The sound source uses a Gaussian sound source instead of the discontinuous Dirac function. The sea surface boundary condition is a pressure release boundary condition (Dirichlet boundary condition), and the seabed boundary condition is a fluid seabed boundary condition (Neumann boundary condition). Since the sea surface boundary condition is a first-type Dirichlet boundary condition, it is enforced through a more effective method of changing the network output. The second-type Neumann boundary condition for the seabed is enforced by encoding it as a loss function. Based on the conditions, the sound source and boundary conditions can be obtained as follows:

[0127] Sound source:

[0128] ;

[0129] Sea surface:

[0130] ;

[0131] Forced execution through network output changes:

[0132] ;

[0133] seabed:

[0134] ;

[0135] .

[0136] Step 3: Addressing the multi-scale characteristics of marine acoustics and the discontinuities introduced by point sources, a data-assisted loss function is constructed using limited observational data to improve the fitting performance of PINNs on marine sound propagation. A KRAKEN model is used to simulate this environment, generating a low-resolution sound pressure field. One hundred data points are randomly selected from this field for network training.

[0137] Step 4: After normalizing the input coordinates, use Fourier feature mapping. The function constructs a Fourier mapping layer based on the wavenumber obtained in step one. Based on the tangential nuclei of PINNs, different... The frequency corresponding to the Fourier basis is modified by adjusting the parameters, thereby controlling the convergence rate of PINNs for different frequency components; based on the wavenumber, a preliminary selection is made. The parameters are 0.001, 0.01, and 0.1.

[0138] Step 5: Different initial choices The parameters PINNs are pre-trained with a set step size. After training, the point-by-point average error is calculated from the sound pressure output by the network to find the optimal value. Parameters; finally, the optimal result is obtained. The parameter is 0.01.

[0139] Step 6: Based on the optimal... The parameters were trained using ADAM for 1000 epochs, followed by a two-stage training process using L-BFGS until convergence to achieve higher accuracy. The sound propagation loss prediction results were calculated from the real and imaginary parts of the sound pressure output by the network, and a graphical comparison is shown below. Figure 4 As shown, Figure 4 The prediction results of the sound pressure field (real part) of the KRAKEN model and Fourier features embedded in PINNs.

[0140] To evaluate the prediction results of PINNs, the results were compared with those of the KRAKEN model, and point-by-point errors were calculated. The results show that the sound field obtained by Fourier feature embedding PINNs is very close to that of the KRAKEN model, and super-resolution is achieved. Compared with the baseline PINNs method, the method in this embodiment accelerates convergence during training, effectively alleviates the spectral bias problem, and achieves more accurate results.

[0141] The beneficial effects of this embodiment include:

[0142] This embodiment solves the encoding problem of sound source, sea surface, and seabed boundary conditions. Compared with the PINNs benchmark method, Fourier feature embedding accelerates the convergence of the training process and effectively alleviates the spectral bias problem, achieving more accurate high-resolution results.

[0143] Reference Figure 5 This application provides an underwater two-dimensional acoustic field calculation device based on Fourier feature embedding PINN, comprising:

[0144] The network construction unit is used to determine the wave number value and computational domain of the Helmholtz equation based on the marine environmental parameters of the computational region, and encodes them into a partial differential equation physical loss function, thereby constructing PINNs; wherein, the real part and imaginary part of the sound pressure are respectively used as the output terms of PINNs;

[0145] The boundary coding unit is used to determine the corresponding sound source conditions, sea surface boundary conditions, and seabed boundary conditions based on sound source information and marine environmental parameters, and encode them into a boundary condition physical loss function.

[0146] The data assistance unit is used to construct a data assistance loss function based on the observation data of the computational region;

[0147] The parameter control unit is used to normalize the input coordinates and then construct a Fourier feature layer using the Fourier feature mapping function. Based on the wavenumber value and combined with the neural tangent kernel of PINNs, different σ parameters are initially selected to modify the frequency corresponding to the Fourier basis, thereby controlling the convergence rate of PINNs for different frequency components.

[0148] The pre-training unit is used to pre-train PINNs with different initially selected σ parameters with a set step size. After training, the point-by-point average error is calculated based on the sound pressure output of the PINNs to find the optimal σ parameter.

[0149] The sound field calculation unit is used to perform two-stage training using ADAM and L-BFGS based on the optimal σ parameter, and calculates the sound propagation loss prediction result from the real and imaginary parts of the sound pressure output by PINNs.

[0150] It is understood that the content of the above method embodiments is applicable to the present device embodiments. The specific functions implemented by the present device embodiments are the same as those of the above method embodiments, and the beneficial effects achieved are also the same as those achieved by the above method embodiments.

[0151] In some alternative embodiments, the functions / operations mentioned in the block diagrams may not occur in the order shown in the operation diagrams. For example, depending on the functions / operations involved, two consecutively shown blocks may actually be executed substantially simultaneously, or the blocks may sometimes be executed in reverse order. Furthermore, the embodiments presented and described in the flowcharts of this application are provided by way of example to provide a more comprehensive understanding of the technology. The disclosed methods are not limited to the operations and logic flows presented herein. Alternative embodiments are contemplated in which the order of various operations is changed and sub-operations described as part of a larger operation are executed independently.

[0152] Furthermore, although this application is described in the context of functional modules, it should be understood that, unless otherwise stated, one or more of the described functions and / or features may be integrated into a single physical device and / or software module, or one or more functions and / or features may be implemented in a separate physical device or software module. It is also understood that a detailed discussion of the actual implementation of each module is unnecessary for understanding this application. Rather, given the properties, functions, and internal relationships of the various functional modules in the apparatus disclosed herein, the actual implementation of the module will be understood within the scope of conventional technology for an engineer. Therefore, those skilled in the art can implement the application set forth in the claims using ordinary techniques without excessive experimentation. It is also understood that the specific concepts disclosed are merely illustrative and not intended to limit the scope of this application, which is determined by the full scope of the appended claims and their equivalents.

[0153] If the aforementioned functions are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, or a portion of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this application. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0154] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus, or device (such as a computer-based system, a processor-including system, or other system that can fetch and execute instructions from, an instruction execution system, apparatus, or device). For the purposes of this specification, "computer-readable medium" can be any means that can contain, store, communicate, propagate, or transmit programs for use by, or in conjunction with, an instruction execution system, apparatus, or device.

[0155] More specific examples of computer-readable media (a non-exhaustive list) include: electrical connections (electronic devices) having one or more wires, portable computer disk drives (magnetic devices), random access memory (RAM), read-only memory (ROM), erasable and editable read-only memory (EPROM or flash memory), fiber optic devices, and portable optical disc read-only memory (CDROM). Furthermore, computer-readable media can even be paper or other suitable media on which the program can be printed, because the program can be obtained electronically, for example, by optically scanning the paper or other medium, followed by editing, interpreting, or otherwise processing as necessary, and then stored in computer memory.

[0156] It should be understood that various parts of this application can be implemented using hardware, software, firmware, or a combination thereof. In the above embodiments, multiple steps or methods can be implemented using software or firmware stored in memory and executed by a suitable instruction execution system. For example, if implemented in hardware, as in another embodiment, it can be implemented using any one or a combination of the following techniques known in the art: discrete logic circuits having logic gates for implementing logical functions on data signals, application-specific integrated circuits (ASICs) having suitable combinational logic gates, programmable gate arrays (PGAs), field-programmable gate arrays (FPGAs), etc.

[0157] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this application. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0158] Although embodiments of this application have been shown and described, those skilled in the art will understand that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of this application, the scope of which is defined by the claims and their equivalents.

[0159] The above is a detailed description of the preferred embodiments of this application, but this application is not limited to the embodiments described. Those skilled in the art can make various equivalent modifications or substitutions without departing from the spirit of this application, and these equivalent modifications or substitutions are all included within the scope defined by the claims of this application.

Claims

1. A method for calculating underwater two-dimensional acoustic fields based on Fourier feature embedding PINN, characterized in that, The method includes the following steps: The wavenumber values ​​and computational domain of the Helmholtz equation are determined based on the marine environmental parameters of the computational region, and encoded into a partial differential equation physical loss function, thereby constructing PINNs; where the real and imaginary parts of the sound pressure are respectively used as output terms of PINNs. Based on the sound source information and marine environmental parameters, the corresponding sound source conditions, sea surface boundary conditions, and seabed boundary conditions are determined and encoded into boundary condition physical loss functions; A data-aided loss function is constructed based on the observation data of the computational region; After normalizing the input coordinates, a Fourier feature layer is constructed using the Fourier feature mapping function. Based on the wavenumber value and combined with the neural tangent kernel of PINNs, different σ parameters are initially selected to modify the frequency corresponding to the Fourier basis, thereby controlling the convergence rate of PINNs for different frequency components. Pre-training is performed on PINNs with different initially selected σ parameters with a set step size. After training, the point-by-point average error is calculated based on the sound pressure output of the PINNs to find the optimal σ parameter. Based on the optimal σ parameter, a two-stage training is performed using ADAM and L-BFGS, and the sound propagation loss prediction result is calculated from the real and imaginary parts of the sound pressure output by PINNs. The sound source conditions are as follows: ； ； in, For control parameters, The horizontal distance from the sound source. For the depth of the sound source; Horizontal distance For depth, Wave number; The sea surface boundary conditions are as follows: ； Sound pressure level; It is always equal to; The seabed boundary conditions are as follows: ； ； Where a and b represent the upper and lower surfaces of the boundary, respectively; This refers to the sound pressure level on the upper surface of the seabed boundary. This represents the acoustic pressure below the seabed boundary. The density of seawater, The density of the seabed; This indicates taking the derivative with respect to depth z; The sea surface boundary conditions are implemented through network changes, when At that time, the network output satisfies a sound pressure level of 0: ； ； in, For the network output, For depth, i.e., the input parameter of the PDE; The loss function for encoding boundary conditions. For the number of boundary points selected, The network predicts the acoustic pressure on the upper surface of the seabed boundary. This represents the acoustic pressure at the lower surface of the seabed boundary.

2. The underwater two-dimensional acoustic field calculation method based on Fourier feature embedding PINN according to claim 1, characterized in that, Before determining the wavenumber values ​​and computational domain of the Helmholtz equations based on the marine environmental parameters of the computational region, the method further includes the following steps: Transform the wave equation of ocean acoustics into the frequency domain Helmholtz equation in cylindrical coordinates: ； in, For sound pressure, Horizontal distance For depth, The wavenumber is used to construct the PINNs, which include an input layer, a Fourier feature layer, a hidden layer, and an output layer. The input layer represents the horizontal distance and depth, and the output layer represents the real and imaginary parts of the sound pressure.

3. The underwater two-dimensional acoustic field calculation method based on Fourier feature embedding PINN according to claim 1, characterized in that, The step of constructing a data-assisted loss function based on the observation data of the computational region includes the following steps: The data-aided loss function is constructed by randomly selecting the observed data as follows: ； in, The data is used as an auxiliary loss function. The number of data points selected. The real part of the sound pressure level predicted by the network. The real part of the sound pressure in the observed data. The imaginary part of the sound pressure predicted by the network. The imaginary part of the sound pressure in the observed data; Therefore, the total loss function is: ; For the loss weights of the PDE constraint, The loss weights are determined by the boundary conditions. The loss weights are constraints imposed on the observed data.

4. The underwater two-dimensional acoustic field calculation method based on Fourier feature embedding PINN according to claim 1, characterized in that, The Fourier feature layer is: ； in, Input coordinates, For coefficients, [ It belongs to the Gaussian distribution. Parameters; For input coordinates The tangential nucleus of the nerve is determined by the following formula: ； in,[ It belongs to the Gaussian distribution. By modifying The value is used to regulate the neural tangent nucleus.

5. The underwater two-dimensional acoustic field calculation method based on Fourier feature embedding PINN according to claim 1, characterized in that, The process of pre-training PINNs with different initially selected σ parameters with a set step size, and then calculating the point-by-point average error based on the sound pressure output by the PINNs after training to find the optimal σ parameter, includes the following steps: Repeat the following steps to dynamically find the optimal solution. parameter: Different initial selections based on the wavenumber values Multiple PINNs of parameters are trained with a set step size. After training, the point-by-point average error is calculated based on the sound pressure output of the network, and the target parameters are selected.

6. The underwater two-dimensional acoustic field calculation method based on Fourier feature embedding PINN according to claim 1, characterized in that, The two-stage training using ADAM and L-BFGS based on the optimal σ parameter includes the following steps: According to the optimal The parameters are adjusted so that the PDE and BC points are reselected every 100 training steps. Combined with the residual adaptive refinement method, ADAM and L-BFGS are used for two-stage training.

7. An underwater two-dimensional acoustic field calculation device based on Fourier feature embedding PINN, characterized in that, The device is used to implement the underwater two-dimensional acoustic field calculation method based on Fourier feature embedding PINN as described in claim 1, and the device includes: The network construction unit is used to determine the wave number value and computational domain of the Helmholtz equation based on the marine environmental parameters of the computational region, and encodes them into a partial differential equation physical loss function, thereby constructing PINNs; wherein, the real part and imaginary part of the sound pressure are respectively used as the output terms of PINNs; The boundary coding unit is used to determine the corresponding sound source conditions, sea surface boundary conditions, and seabed boundary conditions based on sound source information and marine environmental parameters, and encode them into a boundary condition physical loss function. The data assistance unit is used to construct a data assistance loss function based on the observation data of the computational region; The parameter control unit is used to normalize the input coordinates and then construct a Fourier feature layer using the Fourier feature mapping function. Based on the wavenumber value and combined with the neural tangent kernel of PINNs, different σ parameters are initially selected to modify the frequency corresponding to the Fourier basis, thereby controlling the convergence rate of PINNs for different frequency components. The pre-training unit is used to pre-train PINNs with different initially selected σ parameters with a set step size. After training, the point-by-point average error is calculated based on the sound pressure output of the PINNs to find the optimal σ parameter. The sound field calculation unit is used to perform two-stage training using ADAM and L-BFGS based on the optimal σ parameter, and calculates the sound propagation loss prediction result from the real and imaginary parts of the sound pressure output by PINNs.

8. An electronic device, characterized in that, The electronic device includes a processor and a memory; The memory is used to store programs; The processor executes the program to implement the method as described in any one of claims 1 to 6.

9. A computer-readable storage medium, characterized in that, The storage medium stores a program that is executed by a processor to implement the method as described in any one of claims 1 to 6.

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