Ground-ionosphere waveguide propagation simulation method based on physical information neural network

Abstract

The invention discloses a ground-ionosphere waveguide propagation simulation method and device based on a physical information neural network. The method comprises the following steps: defining a planar ground-ionosphere waveguide propagation model; acquiring a data set; a physical information neural network is trained through the data set, the network is used for simulating a bit function, and the bit function is a Hertz vector function about # imgabs0 #; when the physical information neural network is trained, physical information constraints and prediction errors are combined to construct a loss function to train the physical information neural network; and simulating ground-ionosphere waveguide propagation by using the planar ground-ionosphere waveguide propagation model and the trained physical information neural network. According to the method, the propagation result of the electromagnetic wave can be quickly and accurately obtained, the calculation complexity is effectively reduced, and the simulation efficiency is improved.

Description

Technical Field

[0001] The present invention belongs to the field of communication technology, and in particular relates to a method for simulating earth-ionosphere waveguide propagation based on a physical information neural network, which is suitable for analyzing the propagation characteristics of very low frequency (VLF) electromagnetic waves in complex environments. Background Art

[0002] Very low frequency (VLF, 3-30kHz) electromagnetic waves, due to their extremely long wavelengths (10-100km), are repeatedly reflected between the Earth and the ionosphere, guiding their propagation forward. This propagation mode is known as waveguide propagation. VLF waves propagating in the Earth-ionosphere waveguide can provide reliable communication links over thousands of kilometers, possessing irreplaceable application value in strategic communications (such as deep-water submarine communications), ionospheric disturbance monitoring, lightning location, and global climate modeling.

[0003] However, modeling Earth-ionosphere waveguide propagation faces multiple challenges: the ionosphere, as a dynamic plasma medium, exhibits a continuous gradient of electron density with altitude and exhibits significant anisotropy due to the influence of the Earth's magnetic field. Traditional numerical simulation methods primarily employ the finite-difference time-domain (FDTD) method and the waveguide mode method, but these methods face the following bottlenecks in practical applications:

[0004] First, while the FDTD method can handle the propagation of radio waves in complex, inhomogeneous, and anisotropic media, its computational complexity is quite high, and the amount of calculation increases cubically with increasing frequency. Especially when simulating longer propagation distances, the FDTD method requires a large number of time steps, resulting in extremely long computational times, which is unacceptable for practical applications. The calculation process of the FDTD method involves spatial discretization and time iteration. However, as the calculation area increases or the frequency increases, the computational complexity and time increase significantly, resulting in the inefficiency of the FDTD method in practical applications, especially in large-scale simulations.

[0005] Waveguide mode theory has difficulty modeling the inhomogeneous ionosphere and dealing with lateral boundary mutations. For example, at the junction of the ocean (conductivity ~4S / m) and the land (conductivity ~0.01S / m), the traditional method uses the average impedance approximation, which leads to an underestimate of the mode conversion loss.

[0006] In summary, current analytical and numerical simulation methods for Earth-ionosphere waveguide propagation (such as waveguide mode theory and the finite-difference time-domain method) have limitations when dealing with inhomogeneous media. Traditional methods rely on grid discretization and require detailed modeling of the complex ionospheric structure. This results in computational complexity that increases exponentially with increasing dimensionality, leading to low computational efficiency and an inability to effectively simulate radio wave propagation in complex environments. Summary of the Invention

[0007] To address these challenges, this paper proposes a method for simulating Earth-ionosphere waveguide propagation based on physics-informed neural networks (PINNs). This method lies at the intersection of communications technology and computational electromagnetics. By embedding Maxwell's equations and boundary condition constraints into the neural network training process, this method constructs an implicit mapping between the waveguide environment and the electromagnetic field distribution, overcoming the limitations of traditional grid discretization.

[0008] The technical problem to be solved by the present invention is achieved through the following technical solutions:

[0009] A method for simulating Earth-ionosphere waveguide propagation based on a physical information neural network, the method comprising:

[0010] A planar Earth-ionosphere waveguide propagation model is defined; in the planar Earth-ionosphere waveguide propagation model, a cylindrical coordinate system is used to describe the propagation position of the VLF wave in the Earth-ionosphere waveguide;

[0011] Get a data set; the data samples in the data set are represented as Among them, ρ, z is the radial distance, azimuth, and altitude of a point between the Earth and the ionosphere in the cylindrical coordinate system; E ρ is the radial component of the electric field intensity in the cylindrical coordinate system, E z is the axial component of the electric field intensity in the cylindrical coordinate system, is the azimuthal component of the magnetic field intensity in the cylindrical coordinate system;

[0012] The data set is used to train a physical information neural network; the physical information neural network is used to simulate a bit function, and the bit function is about Hertz vector function; wherein, when training the physical information neural network, a loss function is constructed by combining physical information constraints and prediction errors to train the physical information neural network;

[0013] The planar Earth-ionosphere waveguide propagation model and the trained physical information neural network are used to simulate Earth-ionosphere waveguide propagation.

[0014] Optionally, acquiring the data set includes:

[0015] Construct a data space; the data points in the data space are represented as

[0016] The data set is obtained by performing random sampling and Latin hypercube sampling on the data space.

[0017] Optionally, the physical information constraints include: Helmholtz equation constraints, impedance boundary condition constraints and Laplace operator constraints;

[0018] The Helmholtz equation constraint is that the potential function simulated by the physical information neural network must satisfy the Helmholtz equation;

[0019] The impedance boundary condition constraint is an impedance boundary condition satisfied by the electromagnetic field at the ground;

[0020] The Laplace operator constraint is that the bit function simulated by the physical information neural network must satisfy the Laplace operator.

[0021] Optionally, the prediction error is a mean square error between a predicted electromagnetic field component and a real electromagnetic field component; the predicted electromagnetic field component is calculated based on a potential function simulated by the physical information neural network.

[0022] Optionally, the loss function is:

[0023] Loss = loss H +loss L +loss B +loss D ;

[0024] Among them, loss H is the loss with respect to the Helmholtz equation constraint, loss L is the loss of the Laplace operator constraint, loss B is the loss constrained by the impedance boundary condition, loss D is the prediction error, and Loss is the total loss.

[0025] Optionally, the loss regarding the Helmholtz equation constraint is calculated by the following formula:

[0026]

[0027] Where k is the sampling point number, K is the wave number, and m=N d +N b , N d is the number of sampling points in the data set, N b is the number of sampling points for boundary conditions; represents the bit function simulated by the physical information neural network trained using the k-th sampling point, is the Laplace operator, Loss H is the loss due to the Helmholtz equation constraint.

[0028] Optionally, the loss regarding the Laplace operator constraint is calculated by the following formula:

[0029]

[0030] Where k is the sampling point number, K is the wave number, and m=N d +N b , N d is the number of sampling points in the data set, N b is the number of sampling points for boundary conditions; Represents the bit function simulated by the physical information neural network trained using the k-th sampling point, Loss L is the loss constrained by the Laplace operator.

[0031] Optionally, the loss constrained by the impedance boundary condition is calculated by the following formula:

[0032] Loss B =Loss B1 +Loss B2 ,

[0033]

[0034] Among them, k is the sampling point number, N b is the number of sampling points for boundary conditions; represents the bit function simulated by the physical information neural network trained using the k-th sampling point, represents E calculated based on the Hertzian vector function predicted by the physical information neural network for the kth sampling point ρ , Represents the Hertz vector function calculated based on the physical information neural network prediction for the kth sampling point Δ g is the normalized surface impedance of the ground, Δ i is the surface impedance of the ionosphere, η0 is the wave impedance in the air, n is the number of sampling points in the region where ρ→0 and |z-z0|<δ, z0 is the height of the excitation source of the VLF electromagnetic wave, δ is a preset infinitesimal positive threshold, M is the magnetic dipole moment of the dipole, and the excitation source is a dipole.

[0035] Optionally, simulating Earth-ionosphere waveguide propagation using the planar Earth-ionosphere waveguide propagation model and the trained physical information neural network includes:

[0036]

[0037] in, represents the trained physical information neural network, θ represents the parameters of the physical information neural network; ω is the angular frequency, i represents the imaginary part, ε0 is the dielectric constant, and K represents the wave number.

[0038] The present invention also provides a device for simulating Earth-ionosphere waveguide propagation based on a physical information neural network, comprising: a planar Earth-ionosphere waveguide propagation model, a data set, a physical information neural network, and a simulation module;

[0039] In the planar Earth-ionosphere waveguide propagation model, a cylindrical coordinate system is used to describe the propagation position of the VLF wave in the Earth-ionosphere waveguide;

[0040] The data samples in the dataset are represented as Among them, ρ, z is the radial distance, azimuth, and altitude of a point between the Earth and the ionosphere in the cylindrical coordinate system; E ρ is the radial component of the electric field intensity in the cylindrical coordinate system, E z is the axial component of the electric field intensity in the cylindrical coordinate system, is the azimuthal component of the magnetic field intensity in the cylindrical coordinate system;

[0041] The data set is used to train the physical information neural network; the physical information neural network is used to simulate a bit function, the bit function is about Hertz vector function; wherein, when training the physical information neural network, a loss function is constructed by combining physical information constraints and prediction errors to train the physical information neural network;

[0042] The simulation module is used to simulate the Earth-ionosphere waveguide propagation by using the planar Earth-ionosphere waveguide propagation model and the trained physical information neural network.

[0043] The method for simulating Earth-ionosphere waveguide propagation based on physical information neural network provided by the present invention has the following advantages:

[0044] (1) Efficiency and accuracy: By introducing the physical information neural network (PINN) method based on the waveguide mode equation of electromagnetic waves in the Earth-ionosphere waveguide to approximate the potential function, and further solving the electromagnetic field components in combination with the physical model, the propagation results of electromagnetic waves can be obtained quickly and accurately. This process significantly improves the calculation speed and overcomes the time and resource limitations of traditional methods;

[0045] (2) Simplified computational complexity: This method draws on differential equation solving methods and uses PINN to calculate electromagnetic fields, thereby simplifying the complexity of traditional methods such as FDTD in Earth-ionosphere waveguides. Traditional methods require a large amount of computing resources and time steps, while the PINN method effectively reduces computational complexity and improves simulation efficiency by combining data-driven and physical constraints.

[0046] The present invention will be further described in detail below with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 1 is a flow chart of a method for simulating Earth-ionosphere waveguide propagation based on a physical information neural network provided by an embodiment of the present invention;

[0048] Figure 2 A technical roadmap for a method for simulating Earth-ionosphere waveguide propagation based on a physical information neural network provided by an embodiment of the present invention;

[0049] Figure 3 The input interface of the IRI2012 model is shown;

[0050] Figure 4 The input interface of the NRLMSISE-00 model is shown;

[0051] Figure 5 Shows the input interface of the IGRF model. DETAILED DESCRIPTION

[0052] The present invention will be further described in detail below with reference to specific examples, but the embodiments of the present invention are not limited thereto.

[0053] During the implementation of the present invention, the inventors discovered that the traditional FDTD method relies on grid discretization and requires detailed modeling of complex ionospheric structures, resulting in exponentially increasing computational complexity with increasing dimensionality, leading to low computational efficiency in large-scale simulations. To address this issue, an embodiment of the present invention provides a method for simulating Earth-ionosphere waveguide propagation based on a physical information neural network. By integrating a deep learning framework with physical models, it is used for Earth-ionosphere waveguide propagation model prediction. It can accurately and quickly solve for various components of the electromagnetic field in the Earth-ionosphere waveguide, thereby providing more accurate and reliable technical support for radio wave propagation research.

[0054] like Figure 1 As shown, the method for simulating Earth-ionosphere waveguide propagation based on a physical information neural network provided by an embodiment of the present invention includes the following steps:

[0055] S10. Define a planar Earth-ionosphere waveguide propagation model. In the planar Earth-ionosphere waveguide propagation model, a cylindrical coordinate system is used to describe the propagation position of the VLF wave in the Earth-ionosphere waveguide.

[0056] Specifically, when the propagation distance is relatively short, there is no need to consider factors such as the curvature of the earth. The ionosphere can be idealized as a reflection wall at a certain height, and the ground is another reflection wall. VLF (very low frequency) electromagnetic waves are guided to propagate in the space defined by these two reflection walls, and the ground can be approximately treated as a plane.

[0057] Assuming that there is a vertical electric dipole on the ground as the excitation source of VLF electromagnetic waves, TM waves are the only electromagnetic wave mode that can be excited by it in the Earth-ionosphere waveguide. Using a cylindrical coordinate system to simulate the propagation of VLF waves in the Earth-ionosphere waveguide, the field components can be expressed as:

[0058]

[0059] Where, ρ, z is the radial distance, azimuth, and altitude of a point between the Earth and the ionosphere in the cylindrical coordinate system; E ρ represents the radial component of the electric field strength in the cylindrical coordinate system, H ρ Expressed as the radial component of the magnetic field intensity in cylindrical coordinates, represents the azimuthal component of the magnetic field intensity in the cylindrical coordinate system, represents the azimuthal component of the electric field strength in the cylindrical coordinate system, e z It represents the axial component of the electric field strength in the cylindrical coordinate system, H z represents the axial component of the magnetic field intensity in the cylindrical coordinate system, i is the imaginary unit, ε0 is the dielectric constant, K represents the wave number, and Π represents the potential function, which is about Hertz vector function; ω = 2πf is the angular frequency, and f is the radiation frequency.

[0060] S20, obtain a data set; the data samples in the data set are represented as Among them, ρ, z is the radial distance, azimuth, and altitude of a point between the Earth and the ionosphere in the cylindrical coordinate system; E ρ is the radial component of the electric field intensity in the cylindrical coordinate system, E z is the axial component of the electric field intensity in the cylindrical coordinate system, is the azimuthal component of the magnetic field intensity in the cylindrical coordinate system.

[0061] Here, the purpose of acquiring the data set is to subsequently train a physical information neural network, thereby using the physical information neural network to simulate the potential function π, which is about The Hertzian vector function. In each data sample, As input data to the network, The true value information of the samples guides the training of the physical information neural network, so that the trained network can accurately simulate the potential function π. The specific structure of the physical information neural network and its training method will be explained in detail later.

[0062] Optionally, in one implementation, obtaining the data set in step S20 may include:

[0063] (1) Construct a data space; the data points in the data space are represented as

[0064] (2) The data set is obtained by random sampling and Latin hypercube sampling of the data space, such as Figure 2 As shown in .

[0065] Specifically, random sampling is performed on the data space. To ensure uniform data distribution, Latin Hypercube sampling is also used to sample the data space, thereby obtaining the dataset X. This effectively avoids the bias of traditional random sampling and is particularly suitable for high-dimensional problems, helping to improve training results and the generalization ability of the model.

[0066] S30, using the data set to train a physical information neural network; the physical information neural network is used to simulate a bit function, which is about Hertz vector function; when training the physical information neural network, the loss function is constructed by combining the physical information constraints and prediction errors to train the physical information neural network.

[0067] Specifically, based on the idea of PINN (physical information neural network) to solve partial differential equations, the potential function Π is approximated by a feedforward neural network, as shown in Figure 2 As shown in , the network includes an input layer, three hidden layers, each layer contains C (e.g., 128) neurons. The activation function uses Tanh, and the output layer is used to predict the electromagnetic field component.

[0068] Specifically, let X∈d m×3 Input into the first layer of feedforward neural network to obtain the latent vector H∈d m×c Then, the prediction result (Hertz vector) is obtained through a two-layer feedforward neural network, and the electromagnetic field prediction result (electric field E) is calculated through the relationship between the potential function π and the field quantity (Formula 1-3). p , magnetic field H p ). Where m represents the number of samples and d represents the dimension.

[0069] Furthermore, by defining a loss function, the network is trained by minimizing the loss, and combined with physical information constraints, accurate propagation results are obtained. Specifically, by defining the prediction error and introducing the physical information constraint into the loss function, the model (that is, the physical information neural network) always satisfies the physical laws during the iteration process and continuously reduces the prediction error, so as to iteratively train the model. In terms of the optimizer, the Adam optimizer is used, and the learning rate is set to 0.001. The Adam optimizer can adaptively adjust the learning rate of each parameter to improve the efficiency and stability of training. Thus, by minimizing the loss of the model to update and adjust the parameters θ of the physical information neural network, a model is obtained. As the optimal potential function Π simulation model, it is also the trained physical information neural network.

[0070] It can be understood that the prediction error is the error between the electromagnetic field components predicted by the physical information neural network and the actual electromagnetic field components. The predicted electromagnetic field components can be calculated using formulas (1)-(3) based on the potential function simulated by the physical information neural network.

[0071] For example, the prediction error may be the mean square error between the predicted electromagnetic field component and the actual electromagnetic field component, which is calculated as follows:

[0072]

[0073] Among them, k is the sampling point number, one sampling point is a data sample, N d is the number of sampling points contained in the data set; represents the electric field intensity calculated by the Hertzian vector function predicted by the physical information neural network for the k-th sampling point, represents the real electric field component at the kth sampling point, represents the magnetic field component calculated by the Hertzian vector function predicted by the physical information neural network for the k-th sampling point, Represents the true magnetic field component at the kth sampling point, Loss D It is the mean square error between the electromagnetic field components predicted by the physical information neural network and the actual electromagnetic field components, that is, the prediction error.

[0074] In the present invention, physical information constraints are introduced into the loss function so that the physical information neural network always satisfies physical laws during the iteration process. The physical information constraints include: Helmholtz equation constraints, impedance boundary condition constraints and Laplace operator constraints.

[0075] Among them, the Helmholtz equation constraint is: the potential function simulated by the physical information neural network must satisfy the Helmholtz equation.

[0076] Specifically, the potential function π must satisfy the Helmholtz equation:

[0077]

[0078] Where, Represents the Laplace operator, and the other parameters are explained above.

[0079] Accordingly, by defining the loss of the Helmholtz equation constraint, it is introduced into the loss function to form the Helmholtz equation constraint. Here, the loss of the Helmholtz equation constraint is calculated by the following formula:

[0080]

[0081] Where m = N d +N b , N d is the number of sampling points in the data set, N b is the number of sampling points for boundary conditions; Represents the bit function simulated by the physical information neural network trained using the k-th sampling point, Loss H is the loss due to the Helmholtz equation constraint.

[0082] In the present invention, the Laplace operator constraint is: the bit function simulated by the physical information neural network must satisfy the Laplace operator.

[0083] Specifically, the expression of the Lagrangian operator in the cylindrical coordinate system is:

[0084]

[0085] Accordingly, by defining the loss of the Laplace operator constraint, it is introduced into the loss function to form the Laplace operator constraint. Here, the loss of the Laplace operator constraint is calculated by the following formula:

[0086]

[0087] Among them, Loss L is the loss constrained by the Laplace operator.

[0088] In the present invention, the impedance boundary condition constraint is the impedance boundary condition satisfied by the electromagnetic field at the ground.

[0089] Specifically, on the ground, i.e. at z = 0, the ground normalized surface impedance is considered to be a constant, which is denoted as Δ g ; In the ionosphere, that is, at z = h, the ionosphere surface impedance is Δ i (The calculation method will be explained below), then the boundary conditions that should be satisfied at the waveguide boundary are:

[0090]

[0091] Where η0 is the wave impedance in air, and the meanings of the other parameters are as above.

[0092] Figure 2 middle,

[0093] Substituting formula (1)-(2) into the above two formulas, we can obtain:

[0094]

[0095] When the observation edge is infinitely close to the source, that is, ρ→0, z→z0, the influence of the waveguide wall can be ignored, and the formula is obtained:

[0096]

[0097] in, Ids is the electric moment of the dipole. The other parameters of the excitation source of the VLF electromagnetic wave are explained above. z0 is the height of the excitation source of the VLF electromagnetic wave, and ε0 is the dielectric constant.

[0098] Therefore, the impedance boundary condition satisfied by the electromagnetic field at the waveguide boundary is in the form of:

[0099]

[0100] in, represents E calculated by the Hertz vector function predicted by the physical information neural network for the kth sampling point ρ , Represents the Hertz vector function calculated based on the physical information neural network prediction for the kth sampling point Δ g is the normalized surface impedance of the ground, Δ i is the surface impedance of the ionosphere, η0 is the wave impedance in the air, n is the number of sampling points in the region where ρ→0 and |z-z0|<δ, and δ is a preset infinitesimal positive threshold.

[0101] It should be noted that only the sampling points on the boundary will consider Loss B1 .

[0102] When ρ→0, z→z0, we have:

[0103]

[0104] It should be noted that only the points sampled in the area of ρ→0, |z-z0|<δ need to calculate the Loss. B2 .

[0105] Therefore, the loss due to the impedance boundary condition can be calculated as follows:

[0106] Loss B =Loss B1 +Loss B2 (17).

[0107] Based on the above loss H 、loss L 、loss B 、loss D The loss function can be constructed as follows:

[0108] loss=loss H +loss L +loss B +loss D (18);

[0109] Among them, loss H is the loss of the Helmholtz equation constraint, loss L is the loss of the Laplace operator constraint, loss B is the loss constrained by the impedance boundary condition, loss D is the prediction error, and Loss is the total loss.

[0110] In addition, you can also give loss H 、loss L 、loss B 、loss D Assign weight factors so that another loss function is formed by calculating their weighted sum. For example, in order to avoid weight imbalance during training, loss D The weight of is set to 60%, and the total weight of the three physical constraints is set to 40%, so as to ensure that the model complies with the physical constraints while fitting the data as much as possible, thereby improving the accuracy and reliability of the model.

[0111] Therefore, the loss function is used to calculate the loss, and the model parameter θ is continuously adjusted according to the loss (the number of iterations can be set according to the actual situation, for example, it can be set to 200). When the loss converges and is less than the threshold ξ, or when the number of training iterations reaches the set value, the model training ends, and a physical information neural network that meets the physical information constraints can be obtained.

[0112] In addition, after training the physical information neural network, the performance of the trained physical information neural network on the test set can be evaluated using the following indicators, such as describing the accuracy of the prediction model based on relative error:

[0113]

[0114] Among them, M P Represents the predicted value of the field quantity, M T is the field quantity reference value from the data sample in the test set, and n is the number of data samples in the test set.

[0115] S40. Simulate Earth-ionosphere waveguide propagation using a planar Earth-ionosphere waveguide propagation model and a trained physical information neural network.

[0116] Specifically, use Substituting π, formulas (1)-(3) can be expressed as:

[0117]

[0118] Therefore, deep learning is integrated with physical models and data dual-driven methods, through physical information neural network Simulating ∏, the distribution of the electromagnetic field can be easily solved using ∏. When solving, equations (1)-(4), (6), (8), (10), and (11) need to be satisfied, thereby realizing the simulation of the Earth-ionosphere waveguide propagation.

[0119] The calculation method of the ionospheric surface impedance is explained in detail below.

[0120] First, the ionospheric parameters in the ground-ionosphere waveguide are calculated using the International Ionosphere Model (IRI), and the particle density of neutral gas and nitrogen, oxygen and oxygen atoms at each altitude; further, the electron density and collision frequency parameters of the ionosphere at any time and any position are obtained. The very low frequency / low frequency wave propagation calculation is completed. Among them, the global low ionosphere electron density parameters are provided by the International Ionosphere Model (IRI2012). By inputting local (international) time, geographic longitude and latitude, geomagnetic longitude and latitude, altitude and other parameters (such as Figure 3 ), and output the corresponding electron density.

[0121] The global lower ionospheric collision frequency is provided by the NRLMSISE-00 model. By inputting parameters such as year, month, day, local time, global time, geographic longitude, latitude, and altitude (such as Figure 4 ) to obtain the density of various molecular atoms, atmospheric temperature, etc. Then, by combining the relevant atomic and molecular densities, the corresponding collision frequency is obtained.

[0122] The global geomagnetic field model is provided by the International Electromagnetic Reference Field (IGRF). Figure 5 ) to obtain the corresponding magnetic inclination, magnetic field strength and other parameters.

[0123] Then, the ideal Earth-ionosphere waveguide model is further modified to obtain more accurate electromagnetic wave propagation prediction results. Specifically, in the present invention, the ionosphere is idealized as a horizontal layered model, and the specific steps for calculating the ionospheric surface impedance on this basis are as follows:

[0124] (1) The area below 65 km in the daytime and below 80 km in the nighttime ionosphere is set as vacuum. This is mainly because the height of the lower boundary of the ionosphere in the IRI model changes during the day and at night.

[0125] (2) The ionospheric altitude range during the day is set to 65km-150km, and the ionospheric altitude range at night is set to 80km-150km. The ionosphere is divided into one layer every 100m, and the changes in ionospheric parameters in each layer are considered to be negligible.

[0126] (3) The electron density N of each layer is calculated by the ionospheric IRI model e and electron temperature T e ;

[0127] (4) Combined with the NRLMSISE-00 atmospheric model to obtain the oxygen atomic density N that varies with altitude O , oxygen molecule density Nitrogen molecular density

[0128] (5) Substituting the parameters in steps (3) and (4) into the calculation formula of the ionospheric collision frequency, the collision frequency v that varies with altitude can be obtained. c ;

[0129] (6) The obtained v c and N e Substitute the data into the calculation formula of ionospheric surface impedance to calculate the equivalent surface impedance at the ionospheric height h;

[0130] (7) Further, update the ionospheric surface impedance Δ i .

[0131] This completes the description of the calculation method of the ionospheric surface impedance.

[0132] In summary, the method for simulating Earth-ionosphere waveguide propagation based on a physical information neural network provided by the present invention introduces a physical information neural network to perform potential function approximation based on the waveguide mode equation of electromagnetic waves in the Earth-ionosphere waveguide, and further solves the electromagnetic field components in combination with the physical model, thereby quickly and accurately obtaining the propagation results of electromagnetic waves. This process significantly improves the calculation speed and overcomes the time and resource limitations of traditional methods. This method draws on the differential equation solving method and calculates the electromagnetic field through PINN, thereby simplifying the complexity of traditional methods such as FDTD in the Earth-ionosphere waveguide. Traditional methods require a large amount of computing resources and time steps, while the PINN method effectively reduces the computational complexity and improves the simulation efficiency by combining data-driven and physical constraints.

[0133] By integrating physical models and data dual drivers, the present invention breaks through the complex solution of traditional Earth-ionosphere waveguide propagation and obtains a method for quickly and concisely solving the Earth-ionosphere waveguide propagation model. This method is based on the PINN model and can predict electromagnetic wave propagation in real time after one training without repeating the complex calculation process.

[0134] The method provided in the embodiments of the present invention can be applied to electronic devices. Specifically, the electronic devices can be desktop computers, portable computers, smart mobile terminals, servers, etc. This is not limited here; any electronic device that can implement the present invention falls within the scope of protection of the present invention.

[0135] Based on the same inventive concept, an embodiment of the present invention further provides a device for simulating Earth-ionosphere waveguide propagation based on a physical information neural network, comprising: a planar Earth-ionosphere waveguide propagation model, a data set, a physical information neural network, and a simulation module;

[0136] In the planar Earth-ionosphere waveguide propagation model, a cylindrical coordinate system is used to describe the propagation position of VLF waves in the Earth-ionosphere waveguide;

[0137] The data samples in the dataset are represented as Among them, ρ, z is the radial distance, azimuth, and altitude of a point between the Earth and the ionosphere in the cylindrical coordinate system; E ρ is the radial component of the electric field intensity in the cylindrical coordinate system, E z is the axial component of the electric field intensity in the cylindrical coordinate system, is the azimuthal component of the magnetic field intensity in the cylindrical coordinate system;

[0138] The data set is used to train the physical information neural network; the physical information neural network is used to simulate a potential function to predict the electromagnetic field components using the simulated potential function; the potential function is about A function for calculating electromagnetic field components; wherein, when training the physical information neural network, a loss function is constructed by combining physical information constraints and prediction errors to train the physical information neural network;

[0139] The simulation module is used to simulate the Earth-ionosphere waveguide propagation using a planar Earth-ionosphere waveguide propagation model and a trained physical information neural network.

[0140] Optionally, the method for obtaining the dataset includes:

[0141] Construct a data space; the data points in the data space are represented as

[0142] The dataset is obtained by random sampling and Latin hypercube sampling of the data space.

[0143] Optionally, the physical information constraints include: Helmholtz equation constraints, impedance boundary condition constraints, and Laplace operator constraints;

[0144] The Helmholtz equation constraint is: the potential function simulated by the physical information neural network must satisfy the Helmholtz equation;

[0145] The impedance boundary condition constraint is the impedance boundary condition satisfied by the electromagnetic field at the ground;

[0146] The Laplace operator constraint is: the bit function simulated by the physical information neural network must satisfy the Laplace operator.

[0147] Optionally, the prediction error is a mean square error between a predicted electromagnetic field component and a true electromagnetic field component; the predicted electromagnetic field component is calculated based on a potential function simulated by a physical information neural network.

[0148] Optionally, the loss function is:

[0149] Loss = loss H +loss L +loss B +loss D ;

[0150] Among them, loss H is the loss of the Helmholtz equation constraint, loss L is the loss of the Laplace operator constraint, loss B is the loss constrained by the impedance boundary condition, loss D is the prediction error, and Loss is the total loss.

[0151] Optionally, the loss with respect to the Helmholtz equation constraint is calculated as follows:

[0152]

[0153] Where k is the sampling point number, m = N d +N b , N d is the number of sampling points in the data set, N b is the number of sampling points for boundary conditions; Represents the bit function simulated by the physical information neural network trained using the k-th sampling point, Loss H is the loss due to the Helmholtz equation constraint.

[0154] Optionally, the loss with respect to the Laplacian constraint is calculated as follows:

[0155]

[0156] Where k is the sampling point number, K is the wave number, and m = N d +N b , N d is the number of sampling points in the data set, N b is the number of sampling points for boundary conditions; Represents the bit function simulated by the physical information neural network trained using the k-th sampling point, Loss L is the loss constrained by the Laplace operator.

[0157] Optionally, the loss with respect to the impedance boundary condition constraint is calculated as follows:

[0158] Loss B =Loss B1 +Loss B2 ,

[0159]

[0160] Among them, k is the sampling point number, N b is the number of sampling points for boundary conditions; represents the bit function simulated by the physical information neural network trained using the k-th sampling point, represents E calculated based on the Hertzian vector function predicted by the physical information neural network for the kth sampling point ρ , Represents the Hertz vector function calculated based on the physical information neural network prediction for the kth sampling point Δ g is the normalized surface impedance of the ground, Δ i is the surface impedance of the ionosphere, η0 is the wave impedance in the air, n is the number of sampling points in the region where ρ→0 and |z-z0|<δ, z0 is the height of the excitation source of the VLF electromagnetic wave, δ is a preset infinitesimal positive threshold; M is the magnetic dipole moment of the dipole, and the excitation source is the dipole.

[0161] Optionally, the simulation module is specifically configured to perform the following operations:

[0162]

[0163] in, represents the trained physical information neural network, θ represents the parameters of the physical information neural network; ω is the angular frequency, i is the imaginary unit, ε0 is the dielectric constant, and K represents the wave number.

[0164] It should be noted that, for the device embodiment, since it is basically similar to the method embodiment, the description is relatively simple, and the relevant parts can be referred to the partial description of the method embodiment.

[0165] It should be noted that the terms "first," "second," and the like are used to distinguish similar objects and are not necessarily used to describe a particular order or precedence. It should be understood that the terms used in this manner are interchangeable where appropriate, so that the embodiments of the present invention described herein can be implemented in sequences other than those illustrated or described herein. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with the present invention. Instead, they are merely examples of devices and methods consistent with some aspects of the present invention.

[0166] In the description of this specification, the reference terms "one embodiment," "some embodiments," "example," "specific example," or "some examples" mean that the specific features or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features or characteristics described can be combined in any suitable manner in any one or more embodiments or examples. In addition, those skilled in the art can combine and combine different embodiments or examples described in this specification.

[0167] Although the present invention is described herein in conjunction with various embodiments, in the process of implementing the claimed invention, those skilled in the art can understand and implement other variations of the disclosed embodiments by viewing the drawings and the disclosed content. In the description of the present invention, the word "comprising" does not exclude other components or steps, "one" or "a" does not exclude multiple situations, and "multiple" means two or more, unless otherwise clearly and specifically defined. In addition, certain measures are recorded in different embodiments, but this does not mean that these measures cannot be combined to produce good results.

[0168] The above is a further detailed description of the present invention in conjunction with specific preferred embodiments, and the specific implementation of the present invention should not be considered to be limited to these descriptions. For those skilled in the art of the present invention, without departing from the concept of the present invention, several simple deductions or substitutions can be made, which should be considered to fall within the scope of protection of the present invention.

Claims

1. A method for simulating Earth-ionosphere waveguide propagation based on a physical information neural network, characterized in that: The method comprises: A planar Earth-ionosphere waveguide propagation model is defined; in the planar Earth-ionosphere waveguide propagation model, a cylindrical coordinate system is used to describe the propagation position of the VLF wave in the Earth-ionosphere waveguide; Get a data set; the data samples in the data set are represented as Among them, ρ, z is the radial distance, azimuth, and altitude of a point between the Earth and the ionosphere in the cylindrical coordinate system; E ρ is the radial component of the electric field intensity in the cylindrical coordinate system, E z is the axial component of the electric field intensity in the cylindrical coordinate system, is the azimuthal component of the magnetic field intensity in the cylindrical coordinate system; The data set is used to train a physical information neural network; the physical information neural network is used to simulate a bit function, and the bit function is about Hertz vector function; wherein, when training the physical information neural network, a loss function is constructed by combining physical information constraints and prediction errors to train the physical information neural network; The planar Earth-ionosphere waveguide propagation model and the trained physical information neural network are used to simulate Earth-ionosphere waveguide propagation.

2. The method for simulating Earth-ionosphere waveguide propagation based on a physical information neural network according to claim 1, characterized in that: The acquiring of the data set includes: Construct a data space; the data points in the data space are represented as The data set is obtained by performing random sampling and Latin hypercube sampling on the data space.

3. The method for simulating Earth-ionosphere waveguide propagation based on physical information neural network according to claim 2, characterized in that: The physical information constraints include: Helmholtz equation constraints, impedance boundary condition constraints and Laplace operator constraints; The Helmholtz equation constraint is that the potential function simulated by the physical information neural network must satisfy the Helmholtz equation; The impedance boundary condition constraint is an impedance boundary condition satisfied by the electromagnetic field at the ground; The Laplace operator constraint is that the bit function simulated by the physical information neural network must satisfy the Laplace operator.

4. The method for simulating Earth-ionosphere waveguide propagation based on physical information neural network according to claim 1, characterized in that: The prediction error is the mean square error between the predicted electromagnetic field component and the actual electromagnetic field component; the predicted electromagnetic field component is calculated based on the potential function simulated by the physical information neural network.

5. The method for simulating Earth-ionosphere waveguide propagation based on physical information neural network according to claim 3, characterized in that: The loss function is: Loss=loss H +loss L +loss B +loss D ; Among them, loss H is the loss with respect to the Helmholtz equation constraint, loss L is the loss of the Laplace operator constraint, loss B is the loss constrained by the impedance boundary condition, loss D is the prediction error, and loss is the total loss.

6. The method for simulating Earth-ionosphere waveguide propagation based on physical information neural network according to claim 5, characterized in that: The loss with respect to the Helmholtz equation constraint is calculated as follows: Where k is the sampling point number, K is the wave number, and m=N d +N b , N d is the number of sampling points in the data set, N b is the number of sampling points for boundary conditions; represents the bit function simulated by the physical information neural network trained using the k-th sampling point, is the Laplace operator, Loss H is the loss due to the Helmholtz equation constraint.

7. The method for simulating Earth-ionosphere waveguide propagation based on physical information neural network according to claim 5, characterized in that: The loss with respect to the Laplacian constraint is calculated as follows: Where k is the sampling point number, K is the wave number, and m=N d +N b , N d is the number of sampling points in the data set, N b is the number of sampling points for boundary conditions; Represents the bit function simulated by the physical information neural network trained using the k-th sampling point, Loss L is the loss constrained by the Laplace operator.

8. The method for simulating Earth-ionosphere waveguide propagation based on physical information neural network according to claim 5, characterized in that: The loss constrained by the impedance boundary condition is calculated by the following formula: Loss B =Loss B1 +Loss B2 , Among them, k is the sampling point number, N b is the number of sampling points for boundary conditions; represents the bit function simulated by the physical information neural network trained using the k-th sampling point, represents E calculated based on the Hertzian vector function predicted by the physical information neural network for the kth sampling point ρ , Represents the Hertz vector function calculated based on the physical information neural network prediction for the kth sampling point Δ g is the normalized surface impedance of the ground, Δ i is the surface impedance of the ionosphere, η0 is the wave impedance in the air, n is the number of sampling points in the region where ρ→0 and |z-z0|<δ, z0 is the height of the excitation source of the VLF electromagnetic wave, δ is a preset infinitesimal positive threshold, M is the magnetic dipole moment of the dipole, and the excitation source is a dipole.

9. The method for simulating Earth-ionosphere waveguide propagation based on physical information neural network according to claim 1, characterized in that: The method of simulating Earth-ionosphere waveguide propagation by using the planar Earth-ionosphere waveguide propagation model and the trained physical information neural network includes: in, represents the trained physical information neural network, θ represents the parameters of the physical information neural network; ω is the angular frequency, i represents the imaginary part, ε0 is the dielectric constant, and K represents the wave number.

10. A device for simulating Earth-ionosphere waveguide propagation based on a physical information neural network, characterized in that: include: Planar Earth-ionosphere waveguide propagation models, datasets, physical information neural networks, and simulation modules; In the planar Earth-ionosphere waveguide propagation model, a cylindrical coordinate system is used to describe the propagation position of the VLF wave in the Earth-ionosphere waveguide; The data samples in the dataset are represented as Among them, ρ, z is the radial distance, azimuth, and altitude of a point between the Earth and the ionosphere in the cylindrical coordinate system; E ρ is the radial component of the electric field intensity in the cylindrical coordinate system, E z is the axial component of the electric field intensity in the cylindrical coordinate system, is the azimuthal component of the magnetic field intensity in the cylindrical coordinate system; The data set is used to train the physical information neural network; the physical information neural network is used to simulate a bit function, the bit function is about Hertz vector function; wherein, when training the physical information neural network, a loss function is constructed by combining physical information constraints and prediction errors to train the physical information neural network; The simulation module is used to simulate the Earth-ionosphere waveguide propagation by using the planar Earth-ionosphere waveguide propagation model and the trained physical information neural network.

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