Earth-ionospheric waveguide three-dimensional electromagnetic field prediction method based on u-net architecture and spherical coordinate system physical law and application

Abstract

The application provides an earth-ionospheric waveguide three-dimensional electromagnetic field prediction method based on a U-Net architecture and a spherical coordinate system physical law and application, and comprises the following steps: constructing a physical information neural network of the U-Net architecture; obtaining multiple groups of ionospheric parameters, generating a three-dimensional electromagnetic field as a label for each group of ionospheric parameters based on simulation simulation to obtain a training sample set; iteratively training the physical information neural network with the training sample set, and in the iterative training process, taking the spherical coordinate system physical law, the ground impedance boundary condition and the ionospheric impedance boundary condition as constraints; stopping iteration and saving the optimal parameters of the physical information neural network when the loss function meets the preset condition to complete training. The scheme takes the spherical coordinate system physical law, the ground impedance boundary condition and the ionospheric impedance boundary condition as constraints to ensure the prediction accuracy of the physical information neural network at the ground boundary and the ionospheric boundary.

Description

Technical Field

[0001] This application relates to the field of computational electromagnetics, and in particular to a method and application for predicting the three-dimensional electromagnetic field of the Earth-ionospheric waveguide based on the U-Net architecture and the physical laws of the spherical coordinate system. Background Technology

[0002] Very low frequency (VLF, 3-30kHz) electromagnetic waves can propagate over long distances with low attenuation within a spherical resonant cavity structure formed by the Earth's surface and the upper ionosphere. This propagation channel is called the Earth-ionospheric waveguide (EIWG). The propagation characteristics of VLF electromagnetic waves within the EIWG are strongly correlated with multiple physical field parameters, including the electron density distribution in the D region of the ionosphere, ground electromagnetic parameters, and the geomagnetic field distribution. Accurate prediction of its three-dimensional electric field distribution is a core foundation for major engineering and scientific research fields such as global long-wave communication system design, ionospheric remote sensing and space weather monitoring, global lightning location, and monitoring of the electromagnetic effects of nuclear explosions. It has irreplaceable engineering application value and scientific research significance.

[0003] Currently, the mainstream approach for modeling VLF electromagnetic wave propagation within Earth-ionospheric waveguides is the traditional numerical solution method based on Maxwell's equations. Widely used methods in the industry include the Long Wave Capability (LWPC) model, the full-wave solution model, and waveguide mode equation solvers. These methods achieve field distribution calculations through discretized solutions to Maxwell's equations, possessing high computational accuracy and meeting the needs of propagation characteristic analysis under conventional steady-state scenarios. However, due to the inherent principles of numerical solutions, this type of method faces insurmountable technical bottlenecks: First, its computational efficiency is extremely low, with a single-path simulation taking several seconds to minutes, and simulations of the full-field distribution in three-dimensional space on a global scale taking hours, completely failing to meet the millisecond-level response requirements of scenarios such as real-time space weather monitoring, large-scale parameter scanning, and real-time evaluation of communication links; Second, its modeling capabilities are limited, with traditional solutions mostly only able to output one-dimensional field strength amplitudes along the propagation path, making it difficult to achieve multi-component, amplitude-phase integrated prediction of the electric field in the full three-dimensional space within the waveguide, and unable to adapt to the needs of three-dimensional electromagnetic field distribution analysis in complex scenarios; Third, it has poor adaptability to complex ionospheric environments, requiring repeated model reconstruction and iterative calculations for extreme dynamic scenarios such as ionospheric inhomogeneous disturbances and geomagnetic storms, severely limiting its engineering practicality.

[0004] In recent years, with the development of deep learning technology, purely data-driven neural network methods have been attempted to be introduced into the field of electromagnetic propagation modeling. These methods achieve field distribution prediction by fitting the mapping relationship of a large amount of simulation data, which can compress the inference time to the millisecond level, solving the pain point of low efficiency of traditional numerical methods. However, purely data-driven models have inherent defects that cannot be overcome: their predictive ability depends entirely on the coverage of the training data, and they do not have physical extrapolation ability. For complex ionospheric disturbances and extreme space weather scenarios not covered by the training data, they are prone to non-physical interpretations that violate Maxwell's basic laws of physics. The prediction results lack physical consistency and cannot be directly applied to engineering and scientific research scenarios with extremely high reliability requirements, making it difficult to become a reliable solution for electromagnetic modeling of Earth-ionospheric waveguides.

[0005] The emergence of Physical Information Neural Networks (PINNs) offers a new technical approach to balancing efficiency and physical reliability in electromagnetic modeling. This method embeds physical control equations into the loss function of the neural network, forcing the network output to conform to fundamental physical laws during training. This significantly improves the model's generalization ability and physical consistency while maintaining inference efficiency. However, current PINN research in the electromagnetic field largely focuses on local electromagnetic problems in high-frequency, small-scale, Cartesian coordinate systems, and has not yet developed a mature solution for the special system of Earth-ionospheric waveguides. Specifically, existing PINN schemes cannot adapt to the global spherical geometry of Earth-ionospheric waveguides, making it difficult to achieve complete embedding and constraint of Maxwell's equations in spherical coordinates; they cannot handle the high-dimensional global coupling mapping from the two-dimensional spatial distribution of ionospheric parameters to the three-dimensional electromagnetic field distribution, making end-to-end full-field prediction difficult; and they fail to achieve deep coupling between ionospheric anisotropic impedance boundary conditions and network training, making it impossible to accurately characterize the anisotropic reflection characteristics of ionospheric magnetized plasma and guarantee prediction accuracy and physical consistency under complex boundary conditions.

[0006] In summary, existing technologies cannot simultaneously meet the core requirements of high accuracy, high efficiency, strong generalization, and physical consistency for the prediction of three-dimensional electric field distribution in Earth-ionospheric waveguides. This severely restricts the application of VLF electromagnetic wave propagation-related technologies in real-time, large-scale, and complex dynamic scenarios. There is an urgent need to develop a neural network training method that adapts to the complex physical characteristics of Earth-ionospheric waveguides in order to solve many of the pain points of existing technologies. Summary of the Invention

[0007] This application provides a physical information neural network training method and application for predicting the three-dimensional electromagnetic field distribution of the Earth-ionospheric waveguide. This scheme uses the physical laws of the spherical coordinate system, the ground impedance boundary conditions, and the ionospheric impedance boundary conditions as constraints to ensure the prediction accuracy of the physical information neural network at the ground boundary and the ionospheric boundary.

[0008] In a first aspect, embodiments of this application provide a method for predicting the three-dimensional electromagnetic field of an Earth-ionospheric waveguide based on a U-Net architecture and the physical laws of spherical coordinates, the method comprising:

[0009] S100, Constructing a physical information neural network based on the U-Net architecture; S200: Obtain multiple sets of ionospheric parameters, and generate a three-dimensional electromagnetic field for each set of ionospheric parameters based on simulation to obtain a training sample set; S300. The physical information neural network is iteratively trained using the training sample set until the iteration conditions are met. During the iterative training process, spherical coordinate system physical laws, ground impedance boundary conditions, and ionospheric impedance boundary conditions are constructed as constraints. After each iteration, the loss function is calculated. The spherical coordinate system physical laws are used to constrain the physical laws of mutual coupling between the radial, polar, and azimuth electric field components and magnetic field components in the Earth-ionospheric waveguide free space region. The ground impedance boundary conditions are used to constrain the tangential components of the electric field and magnetic field at the polar and azimuth angles of the ground boundary to satisfy a preset ground surface impedance relationship. The ionospheric impedance boundary conditions constrain the reflection relationship between the tangential components of the electric field and magnetic field at the lower boundary of the ionosphere based on a complex impedance matrix. The complex impedance matrix is ​​calculated based on ionospheric parameters. S400. The trained physical information neural network uses a two-dimensional distribution map of ionospheric parameters, which includes the reflection height and sharpness of each effective grid point in the target area, as input to predict the three-dimensional electromagnetic field of the target area. The three-dimensional electromagnetic field includes the amplitude and phase of the electric field component and the magnetic field component of each effective grid point in the target area.

[0010] Secondly, embodiments of this application provide an electronic device, including a memory and a processor. The memory stores a computer program, and the processor is configured to run the computer program to execute a three-dimensional electromagnetic field prediction method for Earth-ionospheric waveguides based on the U-Net architecture and the physical laws of spherical coordinates.

[0011] Thirdly, embodiments of this application provide a readable storage medium storing a computer program. When the computer program is executed by a processor, it implements a three-dimensional electromagnetic field prediction method for Earth-ionospheric waveguides based on the U-Net architecture and the physical laws of spherical coordinates.

[0012] The main contributions and innovations of this invention are as follows: This application's embodiments embed Maxwell's equations expanded in spherical coordinates as physical constraints into the network iterative training process, perfectly adapting to the global spherical geometry of the Earth-ionospheric waveguide. This forces the electric and magnetic field components output by the network to satisfy the physical laws of electromagnetic coupling in free space, fundamentally ensuring the physical consistency of the prediction results. This application's embodiments also introduce ground impedance boundary conditions as constraints for network training, accurately characterizing the reflection and absorption characteristics of VLF waves by the ground's lossy medium. This forces the tangential components of the electric and magnetic fields at the ground boundary to satisfy a preset impedance relationship, ensuring the model's prediction accuracy at the ground boundary. Furthermore, this application's embodiments introduce ionospheric impedance boundary conditions based on the 2×2 complex impedance matrix of anisotropic magnetized plasma as constraints for network training. This accurately characterizes the anisotropic reflection characteristics of the lower ionospheric boundary under the influence of the geomagnetic field, achieving deep coupling of ionospheric physical parameters with neural network training through boundary conditions. This accurately adapts to complex scenarios such as ionospheric inhomogeneous disturbances and geomagnetic storms, significantly improving the model's generalization ability and prediction accuracy in complex ionospheric environments.

[0013] Details of one or more embodiments of this application are set forth in the following drawings and description to make other features, objects and advantages of this application more readily apparent. Attached Figure Description

[0014] The accompanying drawings, which are included to provide a further understanding of this application and form part of this application, illustrate exemplary embodiments and are used to explain this application, but do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a schematic diagram of a three-dimensional electromagnetic field prediction method for Earth-ionospheric waveguides based on U-Net architecture and the physical laws of spherical coordinates, according to an embodiment of this application. Figure 2 This is an architecture diagram of a physical information neural network according to an embodiment of this application; Figure 3 This is a schematic diagram of the hardware structure of an electronic device according to an embodiment of this application. Detailed Implementation

[0015] Exemplary embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings. When the following description relates to the drawings, unless otherwise indicated, the same numerals in different drawings denote the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with one or more embodiments of this specification. Rather, they are merely examples of apparatuses and methods consistent with some aspects of one or more embodiments of this specification as detailed in the appended claims.

[0016] It should be noted that the steps of the corresponding methods are not necessarily performed in the order shown and described in this specification in other embodiments. In some other embodiments, the methods may include more or fewer steps than described in this specification. Furthermore, a single step described in this specification may be broken down into multiple steps in other embodiments; and multiple steps described in this specification may be combined into a single step in other embodiments.

[0017] Example 1 This application provides a three-dimensional electromagnetic field prediction method for Earth-ionospheric waveguides based on the U-Net architecture and the physical laws of the spherical coordinate system. The method uses the physical laws of the spherical coordinate system, ground impedance boundary conditions, and ionospheric impedance boundary conditions as constraints to ensure the prediction accuracy of the physical information neural network at the ground and ionospheric boundaries. Specifically, refer to... Figure 1 The method includes: S100, Constructing a physical information neural network based on the U-Net architecture; S200: Obtain multiple sets of ionospheric parameters, and generate a three-dimensional electromagnetic field for each set of ionospheric parameters based on simulation to obtain a training sample set; S300. The physical information neural network is iteratively trained using the training sample set until the iteration conditions are met. During the iterative training process, spherical coordinate system physical laws, ground impedance boundary conditions, and ionospheric impedance boundary conditions are constructed as constraints. After each iteration, the loss function is calculated. The spherical coordinate system physical laws are used to constrain the physical laws of mutual coupling between the radial, polar, and azimuth electric field components and magnetic field components in the Earth-ionospheric waveguide free space region. The ground impedance boundary conditions are used to constrain the tangential components of the electric field and magnetic field at the polar and azimuth angles of the ground boundary to satisfy a preset ground surface impedance relationship. The ionospheric impedance boundary conditions constrain the reflection relationship between the tangential components of the electric field and magnetic field at the lower boundary of the ionosphere based on a complex impedance matrix. The complex impedance matrix is ​​calculated based on ionospheric parameters. S400. The trained physical information neural network uses a two-dimensional distribution map of ionospheric parameters, which includes the reflection height and sharpness of each effective grid point in the target area, as input to predict the three-dimensional electromagnetic field of the target area. The three-dimensional electromagnetic field includes the amplitude and phase of the electric field component and the magnetic field component of each effective grid point in the target area.

[0018] In step S100, the physical information neural network is constructed using an end-to-end model based on the U-Net encoder-decoder architecture to efficiently process two-dimensional inputs such as ionospheric parameters and generate high-dimensional outputs. The architecture of the physical information neural network is as follows: Figure 2 As shown.

[0019] Furthermore, the encoder of the physical information neural network is a 3-level downsampling structure, with each level containing residual convolutional blocks for progressively extracting multi-scale features.

[0020] For example, the first-level downsampling in the physical information neural network encoder uses a residual convolutional block with a stride of 1 to maintain the resolution of the original input and improve the initial features; the second-level downsampling consists of two residual convolutional blocks connected together, with the first residual convolutional block having a vertical stride of 1 and a horizontal stride of 2, and the second residual convolutional block having a stride of 1; the third-level downsampling has the same structure as the second-level downsampling.

[0021] For example, the bottleneck layer of the physical information neural network is a dilated convolutional layer with an inflation rate of 2, used to expand the receptive field.

[0022] For example, the decoder of the physical information neural network includes a 3-level upsampling structure symmetrical to the encoder. The encoder features are spliced ​​together by skip connections, and the spliced ​​features are fused together by two residual convolutional blocks with a stride of 1 to obtain the final decoding result.

[0023] For example, the output layer of the physical information neural network uses 1×1 convolution to generate the final prediction result, and the final prediction result is post-processed to obtain the corresponding three-dimensional electromagnetic field.

[0024] Specifically, each residual block in the physical information neural network contains two 3×3 convolutions, batch normalization, a GELU activation function, and a residual connection structure.

[0025] Specifically, the constructed physical information neural network uses ionospheric parameters as output, and outputs a three-dimensional electromagnetic field corresponding to the ionospheric parameters.

[0026] In S200, a long-wave propagation capability model (LWPC) or a parallel full-wave simulator developed based on the Julia language is used to generate multiple sets of ionospheric parameters. Each set of ionospheric parameters includes a two-dimensional distribution map composed of the reflection height and sharpness of each effective grid point in any region. A pre-trained simulation model is used to simulate the ionospheric parameters to generate a three-dimensional electromagnetic field.

[0027] For example, the dimension of the ionospheric parameter is Where B is the batch size, representing the number of samples trained each time; 2 is the number of channels, corresponding to the reflection height and sharpness; 60 is the number of azimuth grids, and 200 is the number of distance grids. The number of azimuth grids and the number of distance grids are used to define the two-dimensional distribution space of ionospheric parameters.

[0028] Furthermore, the three-dimensional electromagnetic field includes the amplitude and phase of the electric field component at each effective grid point within the corresponding region, as well as the amplitude and phase of the magnetic field component at each effective grid point. The electric field components include radial electric field components, polar electric field components, and azimuth electric field components, and the magnetic field components include radial magnetic field components, polar magnetic field components, and azimuth magnetic field components.

[0029] In other words, each electric field component and magnetic field component of the three-dimensional electromagnetic field output contains amplitude and phase information. For example, the dimension of the output three-dimensional electromagnetic field is... Where B is the number of samples processed at one time, 60 is the 60 grid points in the azimuth direction, 15 is the 15 grid points in the polar direction, and 400 is the 400 grid points in the radial direction.

[0030] Furthermore, to improve the training efficiency of the physical information neural network, a perturbation augmentation method is used to augment the training sample set.

[0031] In some embodiments, a multi-scale perturbation method is used to superimpose random perturbations of different spatial scales onto a two-dimensional distribution map of ionospheric parameters to simulate ionospheric inhomogeneities existing in nature. For example, kilometer-scale perturbations are superimposed to simulate ionospheric irregularities, hundred-kilometer-scale mesoscale perturbations are superimposed to simulate the effects of gravity waves and tidal waves, and thousand-kilometer-scale perturbations are superimposed to simulate global effects such as solar activity and geomagnetic storms.

[0032] In other embodiments, a non-Gaussian distributed Blob perturbation mode is used in the two-dimensional distribution map of ionospheric parameters to simulate common local enhancement or attenuation regions in the ionosphere, such as patchy perturbations in the polar ionosphere, irregular structures in the mid-latitude ionosphere, and bubble-like perturbations in the equatorial ionosphere.

[0033] Specifically, by adding perturbations to the training sample set to make the training data cover a variety of VLF propagation scenarios from normal to extreme, the neural network's ability to generalize to complex and non-ideal ionospheric conditions is significantly enhanced.

[0034] In step S300, in constructing the physical laws of the spherical coordinate system, the Maxwell equations satisfied by the VLF time harmonic field in the spherical coordinate system are obtained. The Maxwell equations are expanded into a set of coupled differential equations in the spherical coordinate system. The physical constraints of the electric field component on the electromagnetic component, the physical constraints of the electromagnetic component on the electric field component, the physical constraints of the electric field component on the electromagnetic component, and the physical constraints of the electromagnetic component on the electric field component are obtained from the set of coupled differential equations as the physical laws of the spherical coordinate system.

[0035] Specifically, in spherical coordinates, the VLF time harmonic field The Maxwell equations that are satisfied are:

[0036] Where E represents all electric field components and H represents all electromagnetic components. The permeability of free space, The vacuum permittivity, For curl operator, Angular frequency, It is the imaginary unit.

[0037] Next, Maxwell's equations are expanded into a set of coupled differential equations in spherical coordinates, that is, E is expanded into radial electric field components. Polar electric field components and azimuth electric field components Expand H into radial magnetic field components Polar magnetic field components and azimuth magnetic field components The following equation is obtained:

[0038] in, For the radial electric field component, The polar angle electric field component, For the azimuth electric field component, For radial magnetic field components, The polar angle magnetic field component, This refers to the azimuth magnetic field component. The radial distance of the current spatial point. The polar angle of the current spatial point. The azimuth of the current spatial point. The permeability of free space, The vacuum permittivity, The imaginary unit, Let be the angular frequency, and , The frequency of the VLF signal.

[0039] In other words, by using the physical laws of the spherical coordinate system as constraints, the electric field components and electromagnetic components obtained by the physical information neural network are coupled and mutually constrained to ensure that the output of the physical information neural network conforms to the physical laws.

[0040] In step S300, the ground boundary is a lossy medium, neither an ideal conductor nor an ideal lossless medium. When the VLF signal is incident on the ground, reflection and absorption will occur. To ensure the accuracy of the physical information neural network's prediction at the ground boundary, a ground impedance boundary condition is introduced as a constraint. In constructing the ground impedance boundary condition, the tangential components of the electric field and magnetic field at the ground boundary are obtained. The tangential components of the electric field include polar angle electric field components and azimuth angle electric field components, and the tangential components of the magnetic field include polar angle magnetic field components and azimuth angle magnetic field components. The preset impedance relationship formula is expressed as:

[0041] in, The polar angle electric field component, For the azimuth electric field component, This refers to the azimuth magnetic field component. The polar angle magnetic field component, The impedance is the surface impedance of the ground.

[0042] Specifically, the tangential component of the electric field is the electric field component parallel to the Earth's surface, and the tangential component of the magnetic field is the magnetic field component parallel to the Earth's surface. Since the transmitted wave of the VLF signal will attenuate rapidly in a lossy medium, the tangential components of the electric field and the tangential components of the magnetic field must satisfy the preset impedance relationship.

[0043] Specifically, the surface impedance depends on the ground conductivity and relative permittivity, expressed by the formula:

[0044] in, The imaginary unit, Angular frequency, The permeability of free space, For ground conductivity, The vacuum permittivity, is the relative permittivity.

[0045] In step S300, the ionosphere is magnetized plasma generated by solar radiation ionization. Under the influence of the Earth's magnetic field, it exhibits strong anisotropy. Therefore, the reflection characteristics of VLF signals with different polarization directions and propagation directions are completely different. To ensure the correct reflection of VLF signals in the ionosphere, ionospheric impedance boundary conditions are constructed as constraints. In constructing the ionospheric impedance boundary conditions, a 2×2 complex impedance matrix is ​​calculated based on the ionospheric parameters. The ionospheric impedance boundary conditions are defined based on the complex impedance matrix. The formula for the ionospheric impedance boundary conditions is expressed as follows:

[0046] in, It is a complex impedance matrix. , , as well as These are elements in the complex impedance matrix. The polar angle electric field component, For the azimuth electric field component, The polar angle magnetic field component, This represents the azimuth magnetic field component.

[0047] Specifically, the complex impedance matrix is ​​used to constrain the reflection relationship between the tangential components of the electric field and the magnetic field at the lower boundary of the ionosphere. The reflection relationship between the tangential components of the electric field and the magnetic field reflects the anisotropic reflection characteristics of the ionosphere on VLF waves under the influence of the geomagnetic field.

[0048] Specifically, the method for calculating the complex impedance matrix is ​​based on the transfer matrix method (TMM) combined with the integral form of the admittance matrix, and rigorously incorporates highly correlated electron density, collision frequency, and magnetization effects, derived from the standard full-wave modeling technique for magnetized plasma. The calculation process for the ionospheric complex impedance matrix can be implemented sequentially as follows: 1. Obtaining the frequency of electromagnetic waves lower boundary height of the ionosphere ionospheric parameters Geomagnetic parameters ; 2. The ionosphere at an altitude of 100 km is discretized as follows: Layers, each represented by a number from 1 to M; 3. Initialize the top-level (100 km) admittance matrix , as the initial boundary condition for hierarchical recursive calculation; 4. From top to bottom (i.e.) from Traverse down from level 1 to level 2) and perform the following calculations level by level: (1) Calculate the level 1 shell electron number density Collision frequency (based on (2) Calculate the first... Layer plasma angular frequency gyro angular frequency (based on (3) Construct the polarizability tensor With transmission matrix (4) By and Recursive Update ; 5. After completing the recursive calculations for all layers, invert the lower boundary admittance matrix: ; 6. Output the final ionospheric complex impedance matrix This completes the entire calculation process.

[0049] Specifically, the ionospheric impedance boundary condition enables deep coupling of physical parameters with neural network training through the boundary condition. In this scheme, the complex impedance matrix can be calculated in any way.

[0050] Specifically, the propagation of VLF electromagnetic waves in the Earth-ionospheric waveguide is essentially strongly coupled with the motion of ionospheric plasma and the changes in the geomagnetic field. This coupling is mainly manifested in the following ways: the direction of the geomagnetic field affects the anisotropy of the magnetized plasma in the ionosphere, changing the reflection and propagation characteristics of electromagnetic waves; and the ionospheric electron density distribution and the geomagnetic field jointly determine the equivalent impedance matrix of the ionosphere.

[0051] This scheme encodes this strong coupling relationship through input ionospheric parameters, rather than a simple geometric positional relationship. Specifically, the physical information neural network learns the complex mapping relationship between these physical parameters and the electromagnetic field distribution, thereby indirectly mastering the coupling mechanism of electromagnetic waves-plasma-geomagnetic field. Through the diversity of parameter changes in the training data, the network can learn the changes in propagation characteristics under different coupling strengths.

[0052] In step S400, the loss function includes data loss, governing equation loss, and boundary loss. The difference between the predicted three-dimensional electromagnetic field output by the physical information neural network and the corresponding label is used as the data loss. The predicted three-dimensional electromagnetic field output by the physical information neural network is substituted into the physical laws of the spherical coordinate system to obtain the governing equation loss. The predicted three-dimensional electromagnetic field output by the physical information neural network is substituted into the ground impedance boundary condition and the ionospheric impedance boundary condition to obtain the boundary loss.

[0053] Furthermore, in the calculation of data loss, the amplitude and phase in the predicted three-dimensional electromagnetic field are converted into complex values, expressed by the following formula:

[0054] in, The converted complex value. To predict the amplitude in a three-dimensional electromagnetic field, To predict the phase in a three-dimensional electromagnetic field, It is the imaginary unit.

[0055] Then, based on complex numerical calculations, the amplitude loss term and phase loss term are calculated, and the formula is expressed as:

[0056]

[0057] in, For amplitude loss, This represents the total number of valid grid cells within the corresponding area. For the set of indexes of the valid grid, For the index of the valid grid, Let be the complex value of the predicted magnitude at the i-th effective grid. The label amplitude complex value at the i-th valid grid; For phase loss term, Let be the predicted phase complex value at the i-th valid grid. Let be the complex value of the label phase at the i-th valid grid.

[0058] The data loss is then:

[0059] in, For data loss, For amplitude loss, For phase loss term, The weighting coefficients for the phase loss term are used in this scheme. It is 0.1.

[0060] Furthermore, in the calculation of the governing equation loss, the predicted three-dimensional electromagnetic field output by the physical information neural network is substituted into the physical laws of the spherical coordinate system. The spatial partial derivatives are discretized using the second-order central finite difference method, and the residuals of the six differential equations expanded from Maxwell's equations in the spherical coordinate system are calculated. The governing equation loss is constructed by the sum of the squares of the L2 norm of all equation residuals. Specifically, the radial electric field component of the predicted three-dimensional electric field on each effective grid in the corresponding region is calculated. Polar electric field components azimuth electric field component Radial magnetic field component Polar magnetic field components azimuth magnetic field components Substituting these equations into each equation of the Maxwell equations expanded in spherical coordinates, we obtain the residuals on both sides of each equation, and then calculate the control equation loss. The formula for the control equation loss is as follows:

[0061] in, To control the loss of the equation, , , , , , For the corresponding weighting coefficients, , , , , , Let be the residuals on both sides of each equation in the expansion of Maxwell's equations in spherical coordinates. This is the set of indices for all valid grid points inside the waveguide.

[0062] Specifically, the weight coefficients in the control equation loss are dynamically adjusted based on a Bayesian optimization strategy, automatically adjusted according to the convergence of each residual during the training process, thereby more strictly and directly constraining the electromagnetic field predicted by the network to meet basic physical laws.

[0063] Furthermore, in the calculation of boundary loss, the tangential components of the electric field and magnetic field of the predicted three-dimensional electric field on each effective grid in the corresponding region are substituted into the preset impedance relationship of the ground impedance boundary condition. The sum of the squares of the L2 norm of the two formulas of the preset impedance relationship yields the ground boundary loss, expressed as follows:

[0064] in, For ground boundary loss, The polar angle electric field component, For the azimuth electric field component, This refers to the azimuth magnetic field component. The polar angle magnetic field component, The impedance is the surface impedance of the ground.

[0065] The tangential components of the electric field and magnetic field of the predicted 3D electric field on each effective grid in the corresponding region are substituted into the ionospheric impedance boundary condition. The sum of squares of the L2 norm of the difference between the predicted 3D electric field and the ionospheric impedance boundary condition is used as the ionospheric boundary loss, as expressed by the formula:

[0066] in, For ionospheric boundary loss, , , as well as These are elements in the complex impedance matrix. The polar angle electric field component, For the azimuth electric field component, The polar angle magnetic field component, This represents the azimuth magnetic field component.

[0067] Therefore, the formula for boundary loss is expressed as:

[0068] in, For boundary loss, For ground boundary loss, This is the ionospheric boundary loss.

[0069] Example 2 This embodiment also provides an electronic device, see reference. Figure 3 It includes a memory 404 and a processor 402, wherein the memory 404 stores a computer program and the processor 402 is configured to run the computer program to perform the steps in any of the above method embodiments.

[0070] Specifically, the processor 402 may include a central processing unit (CPU), or an application-specific integrated circuit (ASIC), or one or more integrated circuits that can be configured to implement the embodiments of this application.

[0071] Memory 404 may include a mass storage device for data or instructions. For example, and not limitingly, memory 404 may include a hard disk drive (HDD), a floppy disk drive, a solid-state drive (SSD), flash memory, an optical disk drive, a magneto-optical disk drive, magnetic tape, or a Universal Serial Bus (USB) drive, or a combination of two or more of these. Where appropriate, memory 404 may include removable or non-removable (or fixed) media. Where appropriate, memory 404 may be internal or external to a data processing device. In a particular embodiment, memory 404 is non-volatile memory. In a particular embodiment, memory 404 includes read-only memory (ROM) and random access memory (RAM). Where appropriate, the ROM may be a mask-programmed ROM, a programmable read-only memory (PROM), an erasable read-only memory (EPROM), an electrically erasable read-only memory (EEPROM), an electrically alterable read-only memory (EAROM), or flash memory, or a combination of two or more of these. Where appropriate, the RAM can be Static Random-Access Memory (SRAM) or Dynamic Random-Access Memory (DRAM). DRAM can be Fast Page Mode Dynamic Random-Access Memory (FPMDRAM), Extended Data Out Dynamic Random-Access Memory (EDODRAM), Synchronous Dynamic Random-Access Memory (SDRAM), etc.

[0072] The memory 404 can be used to store or cache various data files that need to be processed and / or communicated, as well as possible computer program instructions executed by the processor 402.

[0073] The processor 402 reads and executes computer program instructions stored in the memory 404 to implement any of the physical information neural network training methods for predicting the three-dimensional electromagnetic field distribution of the Earth-ionospheric waveguide in the above embodiments.

[0074] Optionally, the electronic device may further include a transmission device 406 and an input / output device 408, wherein the transmission device 406 is connected to the processor 402, and the input / output device 408 is connected to the processor 402.

[0075] The transmission device 406 can be used to receive or send data via a network. Specific examples of the network described above may include wired or wireless networks provided by the communication provider of the electronic device. In one example, the transmission device includes a Network Interface Controller (NIC), which can connect to other network devices via a base station to communicate with the Internet. In another example, the transmission device 406 may be a Radio Frequency (RF) module used for wireless communication with the Internet.

[0076] The input / output device 408 is used to input or output information. In this embodiment, the input information may be ionospheric parameters, etc., and the output information may be a three-dimensional electromagnetic field, etc.

[0077] Optionally, in this embodiment, the processor 402 can be configured to perform the following steps via a computer program: S100, Constructing a physical information neural network based on the U-Net architecture; S200: Obtain multiple sets of ionospheric parameters, and generate a three-dimensional electromagnetic field for each set of ionospheric parameters based on simulation to obtain a training sample set; S300. The physical information neural network is iteratively trained using the training sample set until the iteration conditions are met. During the iterative training process, spherical coordinate system physical laws, ground impedance boundary conditions, and ionospheric impedance boundary conditions are constructed as constraints. After each iteration, the loss function is calculated. The spherical coordinate system physical laws are used to constrain the physical laws of mutual coupling between the radial, polar, and azimuth electric field components and magnetic field components in the Earth-ionospheric waveguide free space region. The ground impedance boundary conditions are used to constrain the tangential components of the electric field and magnetic field at the polar and azimuth angles of the ground boundary to satisfy a preset ground surface impedance relationship. The ionospheric impedance boundary conditions constrain the reflection relationship between the tangential components of the electric field and magnetic field at the lower boundary of the ionosphere based on a complex impedance matrix. The complex impedance matrix is ​​calculated based on ionospheric parameters. S400. The trained physical information neural network uses a two-dimensional distribution map of ionospheric parameters, which includes the reflection height and sharpness of each effective grid point in the target area, as input to predict the three-dimensional electromagnetic field of the target area. The three-dimensional electromagnetic field includes the amplitude and phase of the electric field component and the magnetic field component of each effective grid point in the target area.

[0078] It should be noted that the specific examples in this embodiment can refer to the examples described in the above embodiments and optional implementations, and will not be repeated here.

[0079] Generally, various embodiments can be implemented in hardware or dedicated circuitry, software, logic, or any combination thereof. Some aspects of the invention can be implemented in hardware, while others can be implemented by firmware or software executed by a controller, microprocessor, or other computing device, but the invention is not limited thereto. Although various aspects of the invention may be shown and described as block diagrams, flowcharts, or using some other graphical representation, it should be understood that, by way of non-limiting example, these blocks, apparatuses, systems, techniques, or methods described herein can be implemented in hardware, software, firmware, dedicated circuitry or logic, general-purpose hardware or controllers or other computing devices, or some combination thereof.

[0080] Embodiments of the present invention can be implemented by computer software, which may be executable by a data processor of a mobile device, such as a processor entity, or by hardware, or by a combination of software and hardware. Computer software or programs (also referred to as program products) including software routines, applets, and / or macros can be stored in any device-readable data storage medium, and they include program instructions for performing specific tasks. The computer program product may include one or more computer-executable components configured to perform the embodiments when the program is run. The one or more computer-executable components may be at least one piece of software code or a portion thereof. Additionally, it should be noted in this respect that, as Figure 3Any box in the logical flow can represent a program step, or interconnected logic circuits, boxes and functions, or a combination of program steps and logic circuits, boxes and functions. Software can be stored on physical media such as memory chips or blocks of storage implemented within a processor, magnetic media such as hard disks or floppy disks, and optical media such as DVDs and their data variants, CDs, etc. The physical medium is a non-transient medium.

[0081] Those skilled in the art should understand that the technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments have been described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0082] The above embodiments are merely illustrative of several implementation methods of this application, and their descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this application should be determined by the appended claims.

Claims

1. A method for predicting the three-dimensional electromagnetic field of an Earth-ionospheric waveguide based on the U-Net architecture and the physical laws of spherical coordinates, characterized in that, Includes the following steps: S100, Constructing a physical information neural network based on the U-Net architecture; S200: Obtain multiple sets of ionospheric parameters, and generate a three-dimensional electromagnetic field for each set of ionospheric parameters based on simulation to obtain a training sample set; S300. The physical information neural network is iteratively trained using the training sample set until the iteration conditions are met. During the iterative training process, spherical coordinate system physical laws, ground impedance boundary conditions, and ionospheric impedance boundary conditions are constructed as constraints. After each iteration, the loss function is calculated. The spherical coordinate system physical laws are used to constrain the physical laws of mutual coupling between the radial, polar, and azimuth electric field components and magnetic field components in the Earth-ionospheric waveguide free space region. The ground impedance boundary conditions are used to constrain the tangential components of the electric field and magnetic field at the polar and azimuth angles of the ground boundary to satisfy a preset ground surface impedance relationship. The ionospheric impedance boundary conditions constrain the reflection relationship between the tangential components of the electric field and magnetic field at the lower boundary of the ionosphere based on a complex impedance matrix. The complex impedance matrix is ​​calculated based on ionospheric parameters. S400. The trained physical information neural network uses a two-dimensional distribution map of ionospheric parameters, which includes the reflection height and sharpness of each effective grid point in the target area, as input to predict the three-dimensional electromagnetic field of the target area. The three-dimensional electromagnetic field includes the amplitude and phase of the electric field component and the magnetic field component of each effective grid point in the target area.

2. The method for predicting the three-dimensional electromagnetic field of the Earth-ionospheric waveguide based on the U-Net architecture and the physical laws of the spherical coordinate system according to claim 1, characterized in that, Each set of ionospheric parameters includes a two-dimensional distribution map consisting of the reflection height and sharpness of each effective grid point within any region.

3. The method for predicting the three-dimensional electromagnetic field of the Earth-ionospheric waveguide based on the U-Net architecture and the physical laws of the spherical coordinate system according to claim 1, characterized in that, The three-dimensional electromagnetic field includes the amplitude and phase of the electric field component at each effective grid point within the corresponding region, as well as the amplitude and phase of the magnetic field component at each effective grid point. The electric field components include radial electric field components, polar electric field components, and azimuth electric field components. The magnetic field components include radial magnetic field components, polar magnetic field components, and azimuth magnetic field components.

4. The method for predicting the three-dimensional electromagnetic field of the Earth-ionospheric waveguide based on the U-Net architecture and the physical laws of the spherical coordinate system according to claim 1, characterized in that, In constructing the physical laws of the spherical coordinate system, the Maxwell equations satisfied by the VLF time-harmonic field in the spherical coordinate system are obtained. The Maxwell equations are expanded into a set of coupled differential equations in the spherical coordinate system. The physical constraints of the electric field component on the electromagnetic component, the physical constraints of the electromagnetic component on the electric field component, the physical constraints of the electric field component on the electromagnetic component, and the physical constraints of the electromagnetic component on the electric field component are obtained from the set of coupled differential equations as the physical laws of the spherical coordinate system.

5. The method for predicting the three-dimensional electromagnetic field of the Earth-ionospheric waveguide based on the U-Net architecture and the physical laws of the spherical coordinate system according to claim 1, characterized in that, In constructing the ground impedance boundary conditions, the tangential components of the electric field and magnetic field at the ground boundary are obtained. The tangential components of the electric field include polar angle electric field components and azimuth angle electric field components, and the tangential components of the magnetic field include polar angle magnetic field components and azimuth angle magnetic field components. The preset impedance relationship is expressed by the following formula: in, The polar angle electric field component, For the azimuth electric field component, This refers to the azimuth magnetic field component. The polar angle magnetic field component, The impedance is the surface impedance of the ground.

6. The method for predicting the three-dimensional electromagnetic field of the Earth-ionospheric waveguide based on the U-Net architecture and the physical laws of the spherical coordinate system according to claim 1, characterized in that, In constructing ionospheric impedance boundary conditions, a 2×2 complex impedance matrix is ​​calculated based on ionospheric parameters. The ionospheric impedance boundary conditions are then defined based on this complex impedance matrix, and the formula for these boundary conditions is as follows: in, It is a complex impedance matrix. , , as well as These are elements in the complex impedance matrix. The polar angle electric field component, For the azimuth electric field component, The polar angle magnetic field component, This represents the azimuth magnetic field component.

7. The method for predicting the three-dimensional electromagnetic field of the Earth-ionospheric waveguide based on the U-Net architecture and the physical laws of the spherical coordinate system according to claim 1, characterized in that, The loss function includes data loss, governing equation loss, and boundary loss. The difference between the predicted three-dimensional electromagnetic field output by the physical information neural network and the corresponding label is used as the data loss. The predicted three-dimensional electromagnetic field output by the physical information neural network is substituted into the physical laws of the spherical coordinate system to obtain the governing equation loss. The predicted three-dimensional electromagnetic field output by the physical information neural network is substituted into the ground impedance boundary conditions and the ionospheric impedance boundary conditions to obtain the boundary loss.

8. The method for predicting the three-dimensional electromagnetic field of the Earth-ionospheric waveguide based on the U-Net architecture and the physical laws of the spherical coordinate system according to claim 7, characterized in that, The predicted three-dimensional electromagnetic field output by the physical information neural network is substituted into the physical laws of the spherical coordinate system. The spatial partial derivatives are discretized by the second-order central finite difference method. The residuals of the six differential equations of the Maxwell equations in the spherical coordinate system are calculated. The loss of the governing equation is obtained by constructing the sum of the squares of the L2 norm of the residuals of all equations.

9. An electronic device comprising a memory and a processor, characterized in that, The memory stores a computer program, and the processor is configured to run the computer program to execute the Earth-ionospheric waveguide three-dimensional electromagnetic field prediction method based on the U-Net architecture and the physical laws of the spherical coordinate system as described in any one of claims 1-8.

10. A readable storage medium, characterized in that, The readable storage medium stores a computer program, which, when executed by a processor, implements a three-dimensional electromagnetic field prediction method for Earth-ionospheric waveguides based on the U-Net architecture and the physical laws of spherical coordinates as described in any one of claims 1-8.

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