Waveguide mode calculation method based on PINNs

Abstract

The invention relates to a waveguide mode solution calculation method based on PINNs, and belongs to the technical field of waveguide mode solution, and the method comprises the steps: taking a physical information neural network 4DPINNs based on fourth derivative constraint as a network model of waveguide mode calculation, and constructing a loss function of model training; the 4DPINNs network is constructed into a first network architecture or a second network architecture; wherein in the first network architecture, corresponding electric field distribution is calculated and output under the condition of an effective refractive index of a given waveguide mode; in the second network architecture, the effective refractive index of a given waveguide mode is used as a training parameter, and the effective refractive index and corresponding electric field distribution are output at the same time; and training the first network architecture or the second network architecture by using a training set formed by discrete data points of the waveguide until the total loss function converges, thereby obtaining a full-vector waveguide mode solution. According to the method, the key problems of geometric modeling limitation and high matrix solving difficulty of a traditional numerical method are solved, and high-precision solving of the full-vector optical mode is realized.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of waveguide mode solving, in particular to a waveguide mode calculation method based on PINNs. BACKGROUND

[0002] Optical mode solving plays an important role in the field of integrated photonics, photonic chips, etc., and can realize important calculations such as mode analysis, field-circuit coupling simulation, and optical efficiency solving. Traditional methods such as finite difference method have the limitation of poor conformal ability, and the finite element method faces challenges in solving indefinite matrix generalized eigenvalues. Physics-informed neural networks (PINNs) embed physical laws into neural network architecture, bringing innovation to the field of photonics. The current PINNs waveguide mode solving method based on scalars ignores the vector characteristics of electromagnetic fields, resulting in analysis limited to simple closed waveguide models and difficult to generalize to more complex anisotropic waveguides. The scalar field distribution obtained by final solving needs to be calculated multiple times to obtain the tangential field distribution required for subsequent calculations. SUMMARY

[0003] In view of the above analysis, the present application aims to disclose a waveguide mode calculation method based on PINNs; to realize high-precision solving of full-vector optical modes; to break through the key problems of geometric modeling limitation and matrix solving difficulty of traditional numerical methods,

[0004] The waveguide mode solution calculation method based on PINNs disclosed by the present application comprises:

[0005] Step S1, using a physics-informed neural network based on fourth-order derivative constraints (4DPINNs) as the network model for waveguide mode calculation, constructing a loss function for model training; the loss function includes a differential term, a boundary loss term and an initial loss term; wherein the differential term is constructed according to the fourth-order derivative operator matrix derived from Maxwell's equations;

[0006] Step S2, constructing the 4DPINNs network into a first network architecture or a second network architecture;

[0007] In the first network architecture, the corresponding electric field distribution is calculated under the condition of the given effective refractive index of the waveguide mode; in the second network architecture, the effective refractive index of the given waveguide mode is taken as a training parameter, and the effective refractive index and the corresponding electric field distribution are output at the same time;

[0008] Step S3, using a training set composed of discrete data points of the waveguide to train the first network architecture or the second network architecture until the total loss function converges, thereby obtaining the full-vector waveguide mode solution.

[0009] Further, the construction process of the loss function differential term L includes:

[0010] 1) Expanding the Maxwell curl equation in the transverse plane and eliminating the longitudinal field component, a coupled second-order differential equation set containing only the transverse electric field component is obtained;

[0011] 2) Coupling the coupled second-order differential equation set to eliminate the magnetic field component, a fourth-order derivative operator matrix Ω is obtained;

[0012] 3) Using the fourth-order derivative operator matrix Ω to construct the loss function term of the transverse electric field component in the x and y directions, and synthesizing to obtain the loss function differential term.

[0013] Further, the loss function differential term L pde is:

[0014]

[0015] L pde = L1+ L2

[0016] Where,

[0017]

[0018] respectively, A x , A y are the components of the transverse electric field in the x and y directions, is the effective refractive index.

[0019] Further, according to the perfect electric conductor boundary condition:

[0020] A x | bottom = A x | top = A y | left = A y | right = 0

[0021]

[0022] The boundary loss term L bc established is:

[0023]

[0024] Further, the initial condition loss term L ic is:

[0025] L ic = A x (xi ,y i )-a xi +A y (x i ,y i )-a yi =0

[0026] wherein a xi is the initial value of the x-component electric field, a yi is the initial value of the y-component electric field; A x (x i ,y i ), A y (x i ,y i ) are the values of the electric field components in the x and y directions at the coordinate point (x i ,y i ), respectively.

[0027] Further, the total loss function of the network is:

[0028] L=L pde +αL bc +βL ic

[0029] wherein, α, β are weight terms.

[0030] Further, the first network architecture established comprises:

[0031] an input layer comprising 2 neurons corresponding to normalized coordinates x', y', respectively;

[0032] a hidden layer comprising 5 fully connected layers, each layer having 16 neurons, and the activation function being tanh;

[0033] an output layer comprising 2 neurons outputting A x , A y , respectively.

[0034] The effective refractive index of the waveguide mode is given as a fixed hyperparameter into the network.

[0035] Further, the training of the first network architecture, the electric field distribution characteristics corresponding to the waveguide mode are obtained in advance, and the initial conditions are matched with the corresponding characteristics; the fixed single-point method or the random single-point method is used for training, wherein,

[0036] The fixed single-point method determines the point with field strength of 1 according to the waveguide mode, and the network training learning rate is fixed at 0.01;

[0037] Random single-point method, a point is randomly selected in the waveguide space and is assigned a value of 1 to realize the initial condition setting; whether the current training is effective is judged by the convergence condition of the loss function of each training; if the random initial point is incorrect, the loss function converges badly, and the initial point is randomly selected and retrained.

[0038] Further, the second network architecture established comprises:

[0039] The input layer comprises two neurons corresponding to normalized coordinates x' and y' respectively;

[0040] The hidden layer comprises five full connection layers, each layer comprising 16 neurons, and the activation function is tanh;

[0041] The output layer comprises two neurons, which respectively output A x and A y .

[0042] The effective refractive index of a given waveguide mode is taken as a training parameter, the initial value is selected according to the mode order between 0-1, and the effective refractive index corresponding to the waveguide mode is output by updating the adaptive learning rate during the training process.

[0043] Further, the variable learning rate method is used to control the network convergence in the training of the second network architecture; for the low-order mode of the waveguide with a large effective refractive index, a smaller learning rate is used, and as the waveguide mode changes from low order to high order, the effective refractive index gradually decreases, and the learning rate gradually increases.

[0044] In order to prevent the effective refractive index from suddenly changing during the training process, the adaptive learning rate method is used, and the learning rate is multiplied by the decay coefficient β every specific number of generations N:

[0045]

[0046] Wherein lr(n) represents the learning rate at the nth training, is the rounding symbol.

[0047] The present application can realize one of the following beneficial effects:

[0048] The application discloses a waveguide mode calculation method based on PINNs, which breaks through the key problems of geometric modeling limitation and large matrix solving difficulty of traditional numerical methods, realizes high-precision solving of full-vector optical modes, eliminates the neglect of vector field characteristics by existing physical information neural networks, establishes a synchronous solving mechanism of propagation constant and tangential component of electromagnetic field, provides a solving scheme that can be directly integrated into the design process of photonic circuits, and finally provides help for industrial application of photonic integrated circuits, quantum optical devices and optical communication systems.

[0049] Among them, the specific effects include:

[0050] (1) Using PINNs to solve full vector waveguide modes, higher accuracy and stability than numerical methods.

[0051] (2) A random initial point strategy is given, which can greatly increase the generality of the model.

[0052] (3) The propagation constant and field distribution can be solved simultaneously, and it can be applied to solve any waveguide with unknown modes. BRIEF DESCRIPTION OF DRAWINGS

[0053] The accompanying drawings are included to provide a further understanding of the application and are incorporated in and constitute a part of this application, illustrate embodiments of the application and are used to explain the principles of the application.

[0054] Figure 1 A flow chart of the waveguide mode solution calculation method based on PINNs in the embodiment of the application;

[0055] Figure 2 A fixed initial point mode solution result schematic diagram in the embodiment of the application;

[0056] Figure 3 A random initial point calculation result schematic diagram in the embodiment of the application;

[0057] Figure 4 A result schematic diagram of simultaneously solving the propagation constant and field distribution in the embodiment of the application. DETAILED DESCRIPTION

[0058] The preferred embodiments of the application will be described in detail below with reference to the accompanying drawings, wherein the drawings constitute a part of this application and are used to illustrate the principles of the embodiments of the application.

[0059] One embodiment of the application discloses a waveguide mode solution calculation method based on PINNs, characterized in that it comprises:

[0060] Step S1, using a physical information neural network based on fourth-order derivative constraint (4DPINNs) as a network model for waveguide mode calculation, constructing a loss function for model training; the loss function comprises a differential term, a boundary loss term and an initial loss term; wherein the differential term is constructed according to a fourth-order derivative operator matrix derived from Maxwell's equations;

[0061] Step S2, constructing the 4DPINNs network into a first network architecture or a second network architecture;

[0062] In the first network architecture, the corresponding electric field distribution is calculated under the condition of the given effective refractive index of the waveguide mode; in the second network architecture, the effective refractive index of the given waveguide mode is taken as a training parameter, and the effective refractive index and the corresponding electric field distribution are output simultaneously.

[0063] Step S3, using the training set composed of discrete data points of the waveguide, training the first network architecture or the second network architecture until the total loss function converges, thereby obtaining the full vector waveguide mode solution.

[0064] Specifically, in step S1, the construction process of the loss function differential term includes:

[0065] 1) Expand the Maxwell curl equation in the transverse plane and eliminate the longitudinal field component to obtain a coupled second-order differential equation set containing only the transverse electric field component;

[0066] Expand the curl equation of the Maxwell equation set and replace the variables:

[0067]

[0068] Where i is the imaginary unit, k0 is the wave number in vacuum, η0 is the impedance, μ r is the relative magnetic permeability, μ xx , μ yy , μ zz are the magnetic permeability components in x, y, z directions; is the electric field, is the magnetic field; Since the electromagnetic field is continuous in the z direction, the electromagnetic field can be separated into variables: k z is the z propagation constant, is the electric field after separation of variables, A x , A y , A z are the electric field components in x, y, z directions, is the magnetic field after separation of variables, B x , B y , B z are the electric field components in x, y, z directions, is the effective refractive index; x' = k0x, y' = k0y, z' = k0z are normalized coordinates.

[0069] Another equation can also be expanded as:

[0070]

[0071] Where ε r is the relative dielectric constant, ε xx , ε yy , ε zz are the dielectric constant components in x, y, z directions.

[0072] After eliminating the z component, the above (1) and (2) can be written as a matrix form of a coupled second-order differential equation set containing only the transverse electric field component:

[0073]

[0074] 2) The coupled second-order differential equation group is coupled to eliminate the magnetic field components to obtain a fourth-order derivative operator matrix Ω;

[0075] Considering the material with dielectric constant and magnetic permeability of 1, the total matrix Ω is obtained by eliminating the magnetic field components in the coupled second-order differential equation group:

[0076]

[0077] The partial derivatives of the normalized coordinates x' and y' with respect to the transverse electric field components are respectively:

[0078] 3) The loss function terms of the transverse electric field components in the x and y directions are constructed using the fourth-order derivative operator matrix Ω, and are combined to obtain the loss function differential term;

[0079] The loss function differential term L pde is:

[0080] L pde = L1+ L2

[0081]

[0082] Specifically, in step S1, the loss function boundary loss term is established according to the perfect electric conductor (PEC) boundary condition;

[0083] According to the perfect electric conductor (PEC) boundary condition:

[0084] A x | bottom = A x | top = A y | left = A y | right = 0

[0085]

[0086] The established boundary loss term L bc is:

[0087]

[0088] In step S1, the loss function initial loss term is:

[0089] L ic = A x (x i ,y i )-axi +A y (x i ,y i )-a yi =0

[0090] where a xi is the initial value of the x-component electric field, a yi is the initial value of the y-component electric field; A x (x i ,y i ), A y (x i ,y i ) are the values of the x and y direction electric field components at the coordinate point (x i ,y i ), respectively.

[0091] In summary, the loss function of model training is:

[0092] L=L pde +αL bc +βL ic

[0093] where, α, β are weight terms. The values are α=0.5, β=0.5.

[0094] Specifically, in step S2, the first network architecture established includes:

[0095] The input layer includes 2 neurons, corresponding to normalized coordinates x' and y', respectively;

[0096] The hidden layer includes 5 fully connected layers, each with 16 neurons, and the activation function is tanh;

[0097] The output layer includes 2 neurons, respectively output A x , A y ;

[0098] The effective refractive index of the given waveguide mode is passed into the network as a fixed hyperparameter.

[0099] Specifically, in step S2, the second network architecture established includes:

[0100] The input layer includes 2 neurons, corresponding to normalized coordinates x' and y', respectively;

[0101] The hidden layer includes 5 fully connected layers, each with 16 neurons, and the activation function is tanh;

[0102] The output layer includes 2 neurons, respectively output A x , A y ;

[0103] The effective refractive index of a given waveguide mode is taken as a training parameter, the initial value is selected according to the mode order between 0-1, and is updated by an adaptive learning rate during the training process, and the effective refractive index corresponding to the output waveguide mode is output.

[0104] Specifically, in the training process of step S3, the data set used by the network is first input as the discrete data points of the waveguide. Usually, several points are randomly sampled in the waveguide solving space as data, and the waveguide space can also be discretized according to a triangular grid or a square grid, and the corresponding grid node coordinates are selected as input data. The advantage of the square grid is that the data structure is simple, and the loss function form is simple, but the solving accuracy is low for non-uniform waveguides or irregularly shaped waveguides. The triangular grid has high conformal accuracy, but the loss function derivation is complex, and the final solving result needs to be mapped to a square grid through coordinate transformation to be used by the finite difference time domain method, the transfer matrix algorithm, etc.

[0105] The simplest square grid is used in the embodiment of the application, the waveguide size to be solved is a=10mm, b=4mm, and the waveguide space is uniformly sampled, with 100 points in the x direction and 40 points in the y direction, a total of 4000 data points, and the data set dimension is (4000, 2).

[0106] For the training process of the first network architecture, the electric field distribution characteristics corresponding to the waveguide mode are obtained in advance, and the initial conditions are matched with the corresponding characteristics; a fixed single-point method or a random single-point method is used for training, wherein,

[0107] In the fixed single-point method, the point with a field strength of 1 is determined according to the waveguide mode, and the network training learning rate is fixed at 0.01;

[0108] In the random single-point method, an initial condition is set by randomly selecting a point in the waveguide space and assigning a value of 1; whether the current training is effective is determined by the loss function convergence of each training; if the incorrect point is randomly selected, the loss function convergence is poor, and the initial point is randomly selected for retraining.

[0109] In this embodiment, the solving of the five TE modes in Table 1 is taken as an example.

[0110] In the fixed single-point method, the field strength of the TE10 mode at the center point of the waveguide is 1, and the initial condition needs to be set as E(5, 2)=1. The network training learning rate is fixed at 0.01.

[0111] In the random single-point method, the instability is strong, and whether the current training is effective is determined by the loss function convergence of each training. If the incorrect point is randomly selected, the loss function convergence is poor, and the initial point is randomly selected for retraining.

[0112] The fixed single-point method requires prior knowledge of the field distribution characteristics of the corresponding mode, which greatly limits its applicability. In contrast, although the random single-point method has larger error than the fixed single-point method, it can exhibit more extensive applicability while maintaining comparable accuracy, and can be applied to more complex waveguide mode solving.

[0113] Table 1

[0114]

[0115] For the second network architecture training process, 4D PINNs are used to simultaneously calculate and the electric field distribution. In the network, is taken as a trainable parameter. The framework automatically calculates the derivative of the electric field output corresponding to , thereby realizing gradient descent optimization. The eigenvalue problem is different from the traditional forward problem and inverse problem, as it does not require external data input, and its trainable parameters also contain unknown quantities. Traditional electromagnetic field problems obtain a unique solution by solving Maxwell's equations under given boundary conditions, while the eigenvalue problem has infinitely many solutions.

[0116] Meanwhile, since the electric field characteristics corresponding to the waveguide mode cannot be known in advance, the present application uses a variable learning rate method to control the network convergence. For the initialization strategy of , there are mainly two kinds: starting from the maximum value and decreasing or starting from the minimum value and increasing. The present application solves the example of a rectangular metal waveguide, so the value range of γ can be limited to 0 to 1.

[0117] The learning rate, learning decay algebra, and decay coefficient used for the five modes shown in Table 1 are shown in Table 2;

[0118] For the above five modes, the variable learning rate method is used to control the network convergence in the second network architecture training; for low-order modes of waveguides with large effective refractive index, a smaller learning rate is used, and as the waveguide gradually changes from low-order mode to high-order mode, the effective refractive index gradually decreases, and the learning rate is gradually increased;

[0119] For low-order modes such as TE10, TE20( , a smaller learning rate can achieve rapid convergence due to the relatively compact parameter space. Conversely, high-order modes such as TE11, TE30, and TE21( , a larger learning rate lr is required to quickly escape from local optima.

[0120] Table 2

[0121]

[0122] To prevent the effective refractive index from sudden changes during the training process, an adaptive learning rate method is used. After a certain number of iterations N, the learning rate is multiplied by the decay coefficient β:

[0123]

[0124] where lr(n) represents the learning rate at the nth training, is the rounding symbol.

[0125] The initial learning rate for the TE10 mode is 0.002, as it can quickly converge to the corresponding value. A smaller decay coefficient β at this time can achieve stable optimization. For the TE11 mode, the learning rate needs to be increased to 0.01 due to the significant reduction compared to the TE20 mode. The initial learning rate for the TE30 and TE21 modes needs to be higher, and it is set to 0.02 and 0.05 in this invention to accelerate convergence. To ensure stable convergence near the true value, the decay coefficient β needs to be increased, and the learning rate decay interval needs to be shortened accordingly to improve the robustness of high-order mode convergence.

[0126] The following examples are given in this embodiment to verify the effectiveness of the scheme in this embodiment.

[0127] Example 1:

[0128] In the case of solving the corresponding electric field distribution for the given waveguide mode effective refractive index , a two-dimensional metal waveguide with a length of 10 mm and a width of 4 mm is calculated at a characteristic frequency of 50 GHz.

[0129] The calculation results using the fixed single-point method are shown in Figure 2 , Figure 2 a is the calculation result of the 4DPINNs, Figure 2 b is the analytical result, Figure 2 c is the absolute value error of the 4PINNs and the analytical result.

[0130] The calculation results using the random single-point method are shown in Figure 3 , Figure 3 a is the statistical histogram of the absolute value error of multiple randomizations, most of the errors are within 0.1. This method can be used to judge the quality of this randomization by the convergence curve shown in Figure 3 b. Figure 3 c shows the results of four randomizations. Figure 3 d shows the comparison between the random initial point and the fixed initial point method. The random initial point greatly improves the model's universality while maintaining high accuracy.

[0131] Example 2: ​

[0132] The effective refractive index of a given waveguide mode is taken as the training parameter using the variable learning rate method, and the effective refractive index and the corresponding electric field distribution are output; for low-order modes, the propagation constant is large, and a low initial learning rate and a low decay coefficient method is used, for high-order modes, the propagation constant is small, and a high initial learning rate and a high decay coefficient method is used, and the calculation results are as shown in Figure 4 Figure 4 a is the solution result of each mode at a frequency of 50GHz, Figure 4 b is the calculation result in a wide band.

[0133] In summary, the waveguide mode calculation method based on PINNs of the embodiment breaks through the key problems of the geometric modeling limitation of the traditional numerical method and the difficulty of matrix solution, and realizes high-precision solution of full-vector optical modes. And it eliminates the neglect of the existing physical information neural network for the characteristics of the vector field, establishes a synchronous solution mechanism of the propagation constant and the tangential component of the electromagnetic field, provides a solution scheme that can be directly integrated into the photonic circuit design process, and finally provides help for the industrialization application of photonic integrated circuits, quantum optical devices and optical communication systems.

[0134] Among the specific effects, the following are included:

[0135] (1) PINNs are used to solve full-vector waveguide modes, which have higher accuracy and stability than numerical methods.

[0136] (2) A random initial point strategy is given, which can greatly increase the generality of the model.

[0137] (3) The propagation constant and field distribution can be solved simultaneously, and it can be applied to solving any waveguide with unknown modes.

[0138] The above is only a preferred specific embodiment of the present application, but the protection scope of the present application is not limited thereto, any person skilled in the art can easily think of changes or replacements within the technical scope disclosed by the present application, which should be covered within the protection scope of the present application.​

Claims

1. A waveguide mode solution calculation method based on PINNs, characterized in that, include: Step S1: Using the 4DPINNs (Physical Information Neural Networks) based on fourth-order derivative constraints as the waveguide mode, construct the loss function for model training; the loss function includes a differential term, a boundary loss term, and an initial loss term; wherein the differential term is constructed based on the fourth-order derivative operator matrix derived from Maxwell's equations; Step S2: Construct the 4DPINNs network into a first network architecture or a second network architecture; In the first network architecture, the corresponding electric field distribution is calculated and output under the given effective refractive index of the waveguide mode; in the second network architecture, the effective refractive index of the given waveguide mode is used as the training parameter, and the effective refractive index and the corresponding electric field distribution are output simultaneously. Step S3: Use the training set composed of discrete data points of the waveguide to train the first network architecture or the second network architecture until the total loss function converges, thereby obtaining the full vector waveguide mode solution.

2. The waveguide mode calculation method based on PINNs according to claim 1, characterized in that, The process of constructing the differential term of the loss function includes: 1) Expand Maxwell's curl equations in the transverse plane and eliminate the longitudinal field components to obtain a set of coupled second-order differential equations containing only transverse electric field components. 2) Couple the system of second-order differential equations into two components with eliminated magnetic fields to obtain the fourth-order derivative operator matrix Ω; 3) Construct loss function terms for the transverse electric field components in the x and y directions using the fourth-order derivative operator matrix Ω, and synthesize them to obtain the differential terms of the loss function.

3. The waveguide mode calculation method based on PINNs according to claim 2, characterized in that, Loss function differential term L pde for: L pde =L1+L2 in, Take the partial derivatives with respect to the normalized coordinates x' and y' of the transverse electric field component, respectively, and A x A y These are the components of the transverse electric field in the x and y directions, respectively. The effective refractive index.

4. The waveguide mode calculation method based on PINNs according to claim 3, characterized in that, Based on the boundary conditions of a perfect electrical conductor: A x | bottom =A x | top =A y | left =A y | right =0 Established boundary loss term L bc for:

5. The waveguide mode calculation method based on PINNs according to claim 4, characterized in that, Initial condition loss term L ic for: L ic =A x (x i ,and i )-to xi +A y (x i ,and i )-to yi =0 Where a xi Let a be the initial value of the electric field of the x-component. yi A represents the initial value of the y-component electric field. x (x i ,y i A) y (x i ,y i ) are the coordinates of the points (x) i ,y i The electric field components in the x and y directions on the ) 6. The waveguide mode calculation method based on PINNs according to claim 5, characterized in that, The total loss function of the network is: L=L pde +αL bc +βL ic Where α and β are weight terms.

7. The waveguide mode calculation method based on PINNs according to claim 6, characterized in that, The first network architecture established includes: The input layer consists of two neurons, corresponding to normalized coordinates x' and y' respectively; The hidden layer consists of 5 fully connected layers, each with 16 neurons, and the activation function is tanh. The output layer consists of two neurons, each outputting A. x A y ; Given the effective refractive index of a waveguide mode It is passed into the network as a fixed hyperparameter.

8. The waveguide mode calculation method based on PINNs according to claim 7, characterized in that, The first network architecture is trained by pre-acquiring the electric field distribution characteristics corresponding to the waveguide modes and matching the initial conditions with the corresponding characteristics; training is performed using either a fixed single-point method or a random single-point method, wherein... The fixed single-point method determines the initial condition based on the waveguide mode, with a point having a field strength of 1, and the network training learning rate is fixed at 0.

01. The random single-point method sets the initial condition by randomly selecting a point in the waveguide space and assigning it a value of 1. The effectiveness of the training is judged by the convergence of the loss function in each training session. If an incorrect point is randomly selected and the convergence of the loss function is poor, the initial point is randomly selected again and the training is repeated.

9. The waveguide mode calculation method based on PINNs according to claim 6, characterized in that, The established second network architecture includes: The input layer consists of two neurons, corresponding to normalized coordinates x' and y' respectively; The hidden layer consists of 5 fully connected layers, each with 16 neurons, and the activation function is tanh. The output layer consists of two neurons, each outputting A. x A y ; The effective refractive index of a given waveguide mode is used as a training parameter. The initial value is selected between 0 and 1 according to the mode order and is updated through an adaptive learning rate during training. The effective refractive index corresponding to the waveguide mode is output.

10. The waveguide mode calculation method based on PINNs according to claim 9, characterized in that, The second network architecture training uses a variable learning rate method to control network convergence; for waveguides with high effective refractive index and low-order modes, a smaller learning rate is used, and as the waveguide changes from low-order mode to high-order mode, the effective refractive index gradually decreases, and the learning rate is gradually increased. To prevent abrupt changes in the effective refractive index during training, an adaptive learning rate method is used. Every N iterations, the learning rate is multiplied by a decay factor β. Where lr(n) represents the learning rate during the nth training iteration. The integer symbol.

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