Calculation method of space charge field and beam coupling impedance in vacuum chamber based on PINN

Abstract

The present invention proposes a method for calculating the space charge field and beam coupling impedance of a vacuum chamber based on PINN, including: setting a calculation domain on the cross section of the cavity and generating random sampling points; using the sampling points as input parameters of the PINN neural network; constructing the hidden layer of the neural network; obtaining the space charge field output value θ of the neural network as a vector containing all weights w and bias b in the neural network to be trained; defining the loss function L of the neural network; minimizing the loss L by the L-BFGS algorithm based on gradient optimization, training the constructed neural network to find the optimal parameter θ; judging whether the loss L is less than a threshold, if not, returning to step four to continue training, if yes, continuing to step eight; beam coupling impedance calculation based on the space charge field output value of the neural network with the optimal parameter θ. The present invention can effectively avoid the situation that the standard FIT of structured grids has step errors of curved boundaries and the precise boundary modeling of unstructured grids leads to increased computing power.

Description

Technical Field

[0001] The invention belongs to the technical field of beam dynamics, and in particular relates to a method for calculating a vacuum chamber space charge field and beam coupling impedance based on PINN. Background Art

[0002] The electromagnetic coupling of charged particle beams and vacuum chambers is an important part of the study of beam dynamics in the design of electromagnetic separators for high-abundance stable isotopes. This coupling effect is quantified in the frequency domain using the concept of beam coupling impedance. Analytical and numerical methods are widely used for impedance calculations of vacuum chambers in various accelerators. In FIT and FEM, the computational domain is decomposed into volume meshes. When using volume meshes for impedance calculations, standard FIT with structured meshes has step errors on curved boundaries, while FEM with unstructured meshes can accurately model boundaries. Finite element models using unstructured meshes can perform accurate boundary modeling, but due to the large field variations near the space charge of a relativistic beam traveling through a vacuum chamber, it may be necessary to generate a dense mesh, resulting in a large increase in computing power. Summary of the invention

[0003] In order to solve the problems existing in the prior art, a calculation method for the space charge field and beam coupling impedance of the vacuum chamber based on PINN is proposed. The purpose is to use a new meshless method to solve the motion field of the space charge in the vacuum chamber of the accelerator high-abundance stable isotope electromagnetic separation device, and to solve the problem that when using volume grids for impedance calculation, the standard FIT with structured grids has step errors of curved boundaries, while the FEM with unstructured grids needs to generate dense grids due to the large field changes near the space charge of the relativistic beam passing through the vacuum chamber, which leads to a large increase in computing power.

[0004] The present invention proposes the following technical solutions to solve the technical problems:

[0005] A method for calculating space charge coupling impedance of a vacuum chamber based on PINN is characterized by:

[0006] Step 1: Set the calculation domain on the cavity cross section and generate random sampling points;

[0007] Step 2: Use the sampling point as the input parameter of the PINN neural network;

[0008] Step 3: construct the hidden layer of the neural network;

[0009] Step 4: Get the spatial charge field output value of the neural network Where θ is a vector containing all weights w and biases b in the neural network to be trained;

[0010] Step 5: Define the loss function L of the neural network;

[0011] Step 6: Minimize the loss L by using the L-BFGS algorithm based on gradient optimization and train the constructed neural network to find the optimal parameter θ;

[0012] Step 7: Determine whether the loss L is less than the threshold. If not, return to step 4 to continue training. If yes, continue to step 8.

[0013] Step 8. Output value of the neural network space charge field based on the optimal parameter θ Calculation of beam coupling impedance.

[0014] Furthermore, the random sampling points generated in step 1 are N PDE and N BC , are the number of sampling points on the XY plane of the computational domain and the number of sampling points on the XY plane of the boundary condition.

[0015] Furthermore, the random sampling points generated in step 1 are generated within the vacuum domain, and no sampling points are generated outside the chamber.

[0016] Furthermore, the hidden layer of step three is three layers, with 20 neurons in each layer; a fully connected neural network and a tanh activation function are used.

[0017] Furthermore, the loss function L defined in step 5 is defined as:

[0018] L=L PDE +L BC (10)

[0019] Formula (10) indicates that the total loss function is the sum of the PDE loss and the boundary condition BC loss;

[0020] L represents the total loss function; L PDE is a function related to the partial differential equation; L BC is a function related to the boundary conditions;

[0021] in:

[0022]

[0023] Formula (11) means that in the PDE calculation area, the squares of the residuals of all sampling points are summed and the average value is taken;

[0024] Among them, L PDE represents the loss function associated with PDE; N PDE is the number of sampling points in the PDE calculation area; p is the sampling point, L PDEis the loss function associated with the PDE of formula (6), and it minimizes (L PDE →0) is performed according to equation (6) on a finite set of sampling points in the computational domain;

[0025]

[0026] Formula (12) indicates that within the boundary condition BC calculation area, the squares of the electric field components of all sampling points are summed and the average value is taken;

[0027] Among them, L BC represents the loss function associated with the boundary conditions; N BC is the number of sampling points in the computational domain of the boundary condition; is the electric field component under boundary conditions; p is the sampling point, L BC is the loss function associated with PEC-BC in formula (9), which minimizes (L BC →0) is performed according to equation (9) at a finite set of sampling points on the boundary surface;

[0028]

[0029] in, It is a combination of the Laplace operator and the wave number;

[0030] is the electric field component

[0031] is a term related to the charge density.

[0032] Furthermore, the L PDE is the loss function associated with the PDE of formula (6), which is derived as follows:

[0033] 1) Determine the relationship between charge density and current density

[0034] Consider a relativistic charged particle beam with a total charge Q moving in an infinitely long vacuum chamber. Assume that the beam has a rigid charge density distribution ρ and a constant longitudinal velocity v. z =ve z =βce z , where β = v / c, c is the speed of light in vacuum, e z is the unit vector of the beam motion direction (z direction); the beam current density J has only the longitudinal component J z =ρv. The charge and current density obey the following continuity equation

[0035]

[0036] The continuity equation of formula (1) shows that in any volume element, the static change of charge is equal to the difference between the current flowing into and out of the volume element; if the charge density in the volume element increases per unit time It means that current flows into the volume; if the charge density in the volume element decreases per unit time This means that current is flowing out of the volume;

[0037] in: is the charge density change term, which indicates the rate of change of charge density ρ per unit volume with time t; a positive value indicates an increase in charge density, and a negative value indicates a decrease in charge density; represents the current density term, which represents the divergence of the current density; a positive value indicates that the current flows out of the volume element, and a negative value indicates that the current decreases from the volume element;

[0038] 2) Charge density propagates in the z direction in the form of waves

[0039] ρ(x,y,z,t)=ρ0(x,y)e j(ωt-kz) (2)

[0040] Equation (2) gives the distribution of the charge density in space and time; in particular, they propagate as waves in the z direction;

[0041] Among them, ρ(x,y,z,t) is the charge density expression, which describes the distribution of charge density in (x,y,z) and time; ρ0(x,y) is the transverse (x,y) distribution function of charge density, which represents the charge density at any position in the z direction, which depends only on x and y; e j(ωt-kz) is a complex exponential term, which represents the change of charge density in z direction and time; j is the imaginary unit, ω is the angular frequency, and k is the wave number; J = ρv is the definition of current density, where v is the velocity vector of the charge;

[0042] 3) The electric field propagates in the z direction in the form of waves

[0043] E z (x,y,z,t)=E z (x,y,ω)e j(ωt-kz) (3)

[0044] Formula (3) gives the distribution of the component of the electric field in the z direction, which also propagates in the z direction in the form of waves;

[0045] Among them, E z (x, y, z, t) is the component of the electric field in the z direction, describing the components of the electric field in space x, y, z and time t; E z (x, y, ω) indicates that the lateral distribution of the electric field in the z direction depends only on x, y and frequency ω; ej(ωt-kz) is a complex exponential term, which represents the change of charge density in the z direction and time; j is the imaginary unit, ω is the angular frequency, and k is the wave number; this term represents a wave that propagates in time and space;

[0046] 4) The total charge Q obtained by integrating the charge density on the x and y planes

[0047] ∫ρ0(x,y)dxdy=Q (4)

[0048] Formula (4) represents the total charge obtained by integrating the transverse charge density;

[0049] Among them, ρ0(x,y)dxdy is the transverse charge density distribution function, which represents the distribution of charge in the x and y directions; Q is the total charge;

[0050] 5) Substitute formulas (1)-(3) into the wave equation of the longitudinal component of the electric field:

[0051]

[0052] Formula (5) describes the electric field E z The equation for fluctuations in the z-direction combines the time-dependent variations of charge density and current density;

[0053] in, YesE z The Laplace operator of E z Second derivative in space;

[0054] YesE z The time derivative of E z Changes in amount over time;

[0055] is the time-varying term of charge density;

[0056] is the term of the change of current density with time;

[0057] 6) Determine the partial differential equation for the electric field

[0058] Derived L PDE

[0059]

[0060] Formula (6) is a partial differential equation for the distribution of the electric field in the transverse plane, taking into account the influence of the transverse charge density;

[0061] in, Equivalent to All of them are Laplace operators;

[0062] YesE z Second-order derivative terms in the x and y directions;

[0063] is a term related to the wave number k and the relative factor γ;

[0064] is a term related to the transverse charge density ρ0.

[0065] Furthermore, the formula (9) is derived as follows:

[0066] 1) The charge distribution on the transverse plane is an elliptical double Gaussian distribution

[0067]

[0068] Formula (7) indicates that the distribution of charge in the transverse plane is an elliptical Gaussian distribution. The charge density is the largest at the center. The farther away from the center, the more the charge density exponential decays.

[0069] Where ρ0 represents the elliptical double Gaussian charge density distribution;

[0070] Q is the total charge;

[0071] σ x and σ y are the half-height widths of the charge in the x and y directions, respectively;

[0072] (x c ,y c ) is the center position of the charge distribution on the plane;

[0073] 2) Radially symmetric Gaussian distribution simplifies the elliptical Gaussian distribution

[0074]

[0075] Formula (8) represents the radially symmetric Gaussian distribution, which simplifies the elliptical Gaussian distribution to make it easier to handle. r is the radial width at half maximum;

[0076] 3) Determine the boundary conditions of a perfect conductor

[0077] In order to obtain Ez in the cavity cross section, solve equation (6) and adopt the boundary condition of an ideal conductor:

[0078] n×E=0→E z =0 (9)

[0079] Formula (9) is the perfect conductor boundary condition, which explains the behavior of the electric field on the PEC surface. The tangential component of the electric field must be zero on the PEC surface, which is an important condition in electromagnetic theory; n×E=0 or E z =0, n is the surface normal vector. On the PEC surface, the tangential component of the electric field is 0, that is, the electric field cannot have a tangential component on the PEC surface.

[0080] Furthermore, the output value of the neural network space charge field based on the optimal parameter θ in step eight is The beam coupling impedance is calculated as follows:

[0081] 1) Calculation of space charge field Impedance at any point, Z ‖ Defined as

[0082]

[0083] 2) Calculate the space charge field The average impedance of Defined as

[0084]

[0085] in, is its conjugate complex number.

[0086] Advantages and effects of the present invention

[0087] 1. This method can effectively avoid the step error of curved boundaries in standard FIT of structured grids and the situation that precise boundary modeling of unstructured grids requires the generation of dense grids, which leads to a large increase in computing power.

[0088] 2. This method is easy to use. Theoretically, it can be used to numerically calculate various vacuum chamber space charge fields and beam coupling impedances in stable isotope electromagnetic separators. And the overall trend of PINN in terms of accuracy is that as long as there is a sufficiently expressive NN architecture and a sufficient number of sampling points, good prediction accuracy can be obtained.

[0089] 3. The neural network simulation results of the space charge impedance of the square chamber are compared with the analytical solution, and the error is within 0.2%.

[0090] 4. This method can be used for the numerical calculation of the space charge field and beam coupling impedance of the vacuum chamber of different chamber shapes in the stable isotope electromagnetic separator. BRIEF DESCRIPTION OF THE DRAWINGS

[0091] Figure 1 A relativistic charged particle beam with transverse non-uniform charge density moves at a uniform speed on the axis of an infinitely long vacuum chamber with an elliptical cross section;

[0092] Figure 2 A physical information neural network is proposed for the space charge effect of electromagnetic separators for high-abundance stable isotopes. A partial differential equation with Lorentz factors, elliptic double Gaussian charge density and boundary conditions is embedded in the loss function. DETAILED DESCRIPTION

[0093] Design principle of the present invention

[0094] 1. Innovation of the present invention: The innovation lies in replacing the grid in the computational domain with sampling points. Specifically: In this embodiment, a physical information-based neural network PIN is constructed based on the physical model of the beam coupling effect caused by the space charge field. A partial differential equation (PDE) with charge density distribution and boundary conditions is embedded in the constructed physical information-based neural network. Due to the use of a neural network, the method of the present invention does not contain a grid in the computational domain. Compared with FIT and FET, this method provides the advantage of simulating space charge effects. A numerical calculation method for the space charge field and beam coupling impedance of a stable isotope electromagnetic separator vacuum chamber based on a physical information neural network (PINN) is proposed.

[0095] 2. Core issues and key points of the present invention

[0096] The core issue is how to establish a loss function. There are several key points around establishing a loss function:

[0097] The first key point is to simplify the continuity equation of formula (1): The first simplification is to convert the physical model from flow density to current density: The first term of the continuity equation is generally the partial derivative of density with respect to time, and the second term is flow density. This embodiment converts the flow density of the second term into current density according to the needs of the project. In this way, the present invention establishes the relationship between charge density and current density through formula (1); The second simplification is not to consider the distribution of the field on the charge. Generally speaking, if self-consistency is considered, the distribution of the field on the charge must be considered. In this way, the changes in partial x, partial y, partial z and partial t need to be considered in formula (1). However, the present invention does not consider these changes, but only makes an approximation of the relativistic speed, so the formula (1) that only shows the relationship between charge density and current density is obtained;

[0098] The second key point: Formula (2) and Formula (3) are optimized: The first optimization: Fourier transform the charge density (e j(ωt-kz) ), through Fourier transform, the initial curve is transformed into a regular wave form. Otherwise, if it is not transformed into a wave form, it will not be able to proceed. The following formula (6) must use the Fourier transform of formula (2) and formula (3)

[0099] The third key point: Formula (4) establishes a complete path for engineering applications. The specific method is to integrate the entire path of light speed motion after sampling the points, and then get a total charge. If this process is missing, all the subsequent things cannot be done.

[0100] The fourth key point: Formula (5) is derived from Formula (1) to Formula (3);

[0101] The fifth key point is that the process of deriving formula (6) from formula (5) is optimized. The optimization is to define the definition of an ideal conductor through formula (9) before deriving formula (6) from formula (5). Formula (6) is obtained only after the definition of an ideal conductor is performed through formula (9).

[0102] Summary: The core of the technical means of the present invention is to define the loss function. Formula (6) is related to the loss function. Formula (6) is derived from formula (5), and formula (5) is derived from formulas (1) to (3). At the same time, before formula (5) is derived into formula (6), the ideal conductor is defined by formula (9).

[0103] The sixth key point: Simplification of the physical model of formula (7). Formula (7) indicates that the distribution of charge in the transverse plane is an elliptical Gaussian distribution. If it is calculated according to formula (7), it is very likely that the calculation will not work. Therefore, formula (7) is simplified in two aspects and becomes formula (8). The first simplification of formula (7) is to replace σ x Equal to σ y , thus simplifying the denominator of formula (7), from 2πσ in formula (7) x σ y It becomes 2σ of formula (8) x 2 ; The second simplification of formula (7): make another adjustment to the elliptical Gaussian distribution, transform the elliptical Gaussian distribution into a circular Gaussian distribution, and

[0104] (xx c ) 2 -(yy c ) 2 becomes r 2 The above two simplifications form formula (8).

[0105] The seventh key point: the derivation from formula (5) to formula (6): although formula (5) has been derived, it cannot be used. Formula (5) must be used to derive formula (6);

[0106] 1) Substitute (2) into (5)

[0107]

[0108] Calculating partial derivatives

[0109]

[0110] 2) Substitute (3) into the above formula

[0111] The left side of the equation is

[0112] 3) Calculate the parts of the left side

[0113] Laplace term:

[0114] 4) Find the partial derivative with respect to time:

[0115] 5) Substitute into (5)

[0116]

[0117] Eliminate e j(ωt-kz) , which can be simplified to

[0118]

[0119] And ω=kv, γ=(1-β 2 ) -1 / 2 ,

[0120] We can get formula (6)

[0121]

[0122] Based on the above invention principle, the present invention designs a method for calculating the space charge coupling impedance of a vacuum chamber based on PINN. Figure 1 , Figure 2 As shown, its characteristics are:

[0123] Step 1: Set the calculation domain on the cavity cross section and generate random sampling points;

[0124] Step 2: Use the sampling point as the input parameter of the PINN neural network;

[0125] Step 3: construct the hidden layer of the neural network;

[0126] Step 4: Get the spatial charge field output value of the neural network Where θ is a vector containing all weights w and biases b in the neural network to be trained;

[0127] Step 5: Define the loss function L of the neural network;

[0128] Step 6: Minimize the loss L by using the L-BFGS algorithm based on gradient optimization and train the constructed neural network to find the optimal parameter θ;

[0129] Step 7: Determine whether the loss L is less than the threshold. If not, return to step 4 to continue training. If yes, continue to step 8.

[0130] Step 8. Output value of the neural network space charge field based on the optimal parameter θ Calculation of beam coupling impedance.

[0131] Furthermore, the random sampling points generated in step 1 are N PDE and N BC , are the number of sampling points on the XY plane of the computational domain and the number of sampling points on the XY plane of the boundary condition.

[0132] Furthermore, the random sampling points generated in step 1 are generated within the vacuum domain, and no sampling points are generated outside the chamber.

[0133] Furthermore, the hidden layer of step three is three layers, with 20 neurons in each layer; a fully connected neural network and a tanh activation function are used.

[0134] Furthermore, the loss function L defined in step 5 is defined as:

[0135] L=L PDE +L BC (10)

[0136] Formula (10) indicates that the total loss function is the sum of the PDE loss and the boundary condition BC loss;

[0137] L represents the total loss function; L PDE is a function related to the partial differential equation; L BC is a function related to the boundary conditions;

[0138] in:

[0139]

[0140] Formula (11) means that in the PDE calculation area, the squares of the residuals of all sampling points are summed and the average value is taken;

[0141] Among them, L PDE represents the loss function associated with PDE; N PDE is the number of sampling points in the PDE calculation area; p is the sampling point, L PDE is the loss function associated with the PDE of formula (6), and it minimizes (L PDE→0) is performed according to equation (6) on a finite set of sampling points in the computational domain;

[0142]

[0143] Formula (12) indicates that within the boundary condition BC calculation area, the squares of the electric field components of all sampling points are summed and the average value is taken;

[0144] Among them, L BC represents the loss function associated with the boundary conditions; N BC is the number of sampling points in the computational domain of the boundary condition; is the electric field component under boundary conditions; p is the sampling point, L BC is the loss function associated with PEC-BC in formula (9), which minimizes (L BC →0) is performed according to equation (9) at a finite set of sampling points on the boundary surface;

[0145]

[0146] in, It is a combination of the Laplace operator and the wave number;

[0147] is the electric field component

[0148] is a term related to the charge density.

[0149] Furthermore, the L PDE is the loss function associated with the PDE of formula (6), which is derived as follows:

[0150] 1) Determine the relationship between charge density and current density

[0151] Consider a relativistic charged particle beam with a total charge Q moving in an infinitely long vacuum chamber. Assume that the beam has a rigid charge density distribution ρ and a constant longitudinal velocity v. z =ve z =βce z , where β = v / c, c is the speed of light in vacuum, e z is the unit vector of the beam motion direction (z direction); the beam current density J has only the longitudinal component J z =ρv. The charge and current density obey the following continuity equation

[0152]

[0153] The continuity equation of formula (1) shows that in any volume element, the static change of charge is equal to the difference between the current flowing into and out of the volume element; if the charge density in the volume element increases per unit time It means that current flows into the volume; if the charge density in the volume element decreases per unit time This means that current is flowing out of the volume;

[0154] in: is the charge density change term, which indicates the rate of change of charge density ρ per unit volume with time t; a positive value indicates an increase in charge density, and a negative value indicates a decrease in charge density; represents the current density term, which represents the divergence of the current density; a positive value indicates that the current flows out of the volume element, and a negative value indicates that the current decreases from the volume element;

[0155] 2) Charge density propagates in the z direction in the form of waves

[0156] ρ(x,y,z,t)=ρ0(x,y)e j(ωt-kz) (2)

[0157] Equation (2) gives the distribution of the charge density in space and time; in particular, they propagate as waves in the z direction;

[0158] Among them, ρ(x,y,z,t) is the charge density expression, which describes the distribution of charge density in (x,y,z) and time; ρ0(x,y) is the transverse (x,y) distribution function of charge density, which represents the charge density at any position in the z direction, which depends only on x and y; e j(ωt-kz) is a complex exponential term, which represents the change of charge density in z direction and time; j is the imaginary unit, ω is the angular frequency, and k is the wave number; J = ρv is the definition of current density, where v is the velocity vector of the charge;

[0159] 3) The electric field propagates in the z direction in the form of waves

[0160] E z (x,y,z,t)=E z (x,y,ω)e j(ωt-kz) (3)

[0161] Formula (3) gives the distribution of the component of the electric field in the z direction, which also propagates in the z direction in the form of waves;

[0162] Among them, E z (x, y, z, t) is the component of the electric field in the z direction, describing the components of the electric field in space x, y, z and time t; E z (x, y, ω) indicates that the lateral distribution of the electric field in the z direction depends only on x, y and frequency ω; ej(ωt-kz) is a complex exponential term, which represents the change of charge density in the z direction and time; j is the imaginary unit, ω is the angular frequency, and k is the wave number; this term represents a wave that propagates in time and space;

[0163] 4) The total charge Q obtained by integrating the charge density on the x and y planes

[0164] ∫ρ0(x,y)dxdy=Q (4)

[0165] Formula (4) represents the total charge obtained by integrating the transverse charge density;

[0166] Among them, ρ0(x,y)dxdy is the transverse charge density distribution function, which represents the distribution of charge in the x and y directions; Q is the total charge;

[0167] 5) Substitute formulas (1)-(3) into the wave equation of the longitudinal component of the electric field:

[0168]

[0169] Formula (5) describes the electric field E z The equation for fluctuations in the z-direction combines the time-dependent variations of charge density and current density;

[0170] in, YesE z The Laplace operator of E z Second derivative in space;

[0171] YesE z The time derivative of E z Changes in amount over time;

[0172] is the time-varying term of charge density;

[0173] is the term of the change of current density with time;

[0174] 6) Determine the partial differential equation for the electric field

[0175] Derived L PDE

[0176]

[0177] Formula (6) is a partial differential equation for the distribution of the electric field in the transverse plane, taking into account the influence of the transverse charge density;

[0178] in, Equivalent to All of them are Laplace operators;

[0179] YesE z Second-order derivative terms in the x and y directions;

[0180] is a term related to the wave number k and the relative factor γ;

[0181] is a term related to the transverse charge density ρ0.

[0182] Furthermore, the formula (9) is derived as follows:

[0183] 1) The charge distribution on the transverse plane is an elliptical double Gaussian distribution

[0184]

[0185] Formula (7) indicates that the distribution of charge in the transverse plane is an elliptical Gaussian distribution. The charge density is the largest at the center. The farther away from the center, the more the charge density exponential decays.

[0186] Where ρ0 represents the elliptical double Gaussian charge density distribution;

[0187] Q is the total charge;

[0188] σ x and σ y are the half-height widths of the charge in the x and y directions, respectively;

[0189] (x c ,y c ) is the center position of the charge distribution on the plane;

[0190] 2) Radially symmetric Gaussian distribution simplifies the elliptical Gaussian distribution

[0191]

[0192] Formula (8) represents the radially symmetric Gaussian distribution, which simplifies the elliptical Gaussian distribution to make it easier to handle. r is the radial width at half maximum;

[0193] 3) Determine the boundary conditions of a perfect conductor

[0194] In order to obtain Ez in the cavity cross section, solve equation (6) and adopt the boundary condition of an ideal conductor:

[0195] n×E=0→E z =0 (9)

[0196] Formula (9) is the perfect conductor boundary condition, which explains the behavior of the electric field on the PEC surface. The tangential component of the electric field must be zero on the PEC surface, which is an important condition in electromagnetic theory; n×E=0 or E z =0, n is the surface normal vector. On the PEC surface, the tangential component of the electric field is 0, that is, the electric field cannot have a tangential component on the PEC surface.

[0197] Furthermore, the output value of the neural network space charge field based on the optimal parameter θ in step eight is The beam coupling impedance is calculated as follows:

[0198] 1) Calculation of space charge field Impedance at any point, Z || Defined as

[0199]

[0200] 2) Calculate the space charge field The average impedance of Defined as

[0201]

[0202] in, is its conjugate complex number.

[0203] It should be emphasized that the above specific embodiments are merely explanations of the present invention, and they are not limitations of the present invention. After reading this specification, those skilled in the art can make modifications to the above embodiments without any creative contribution as needed, but as long as they are within the scope of the claims of the present invention, they are protected by patent law.

Claims

1. A method for calculating the space charge field and beam coupling impedance of a vacuum chamber based on PINN, characterized in that: Step 1: Set the calculation domain on the cavity cross section and generate random sampling points; Step 2: Use the sampling point as the input parameter of the PINN neural network; Step 3: construct the hidden layer of the neural network; Step 4: Get the spatial charge field output value of the neural network Where θ is a vector containing all weights w and biases b in the neural network to be trained; Step 5: Define the loss function L of the neural network; Step 6: Minimize the loss L by using the L-BFGS algorithm based on gradient optimization and train the constructed neural network to find the optimal parameter θ; Step 7: Determine whether the loss L is less than the threshold. If not, return to step 4 to continue training. If yes, continue to step 8. Step 8. Output value of the neural network space charge field based on the optimal parameter θ Calculation of beam coupling impedance; The loss function L defined in step 5 is defined as: L=L PDE +L BC (10) Formula (10) indicates that the total loss function is the sum of the PDE loss and the boundary condition BC loss; L represents the total loss function; L PDE is a function related to the partial differential equation; L BC is a function related to the boundary conditions; in: Formula (11) means that in the PDE calculation area, the squares of the residuals of all sampling points are summed and the average value is taken; Among them, L PDE represents the loss function associated with PDE; N PDE is the number of sampling points in the PDE calculation area; p is the sampling point, L PDE is the loss function associated with the PDE of formula (6), and it minimizes (L PDE →0) is performed according to equation (6) on a finite set of sampling points in the computational domain; Formula (12) indicates that within the boundary condition BC calculation area, the squares of the electric field components of all sampling points are summed and the average value is taken; Among them, L BC represents the loss function associated with the boundary conditions; N BC is the number of sampling points in the computational domain of the boundary condition; is the electric field component under boundary conditions; p is the sampling point, L BC is the loss function associated with PEC-BC in formula (9), which minimizes (L BC →0) is performed according to equation (9) at a finite set of sampling points on the boundary surface; in, It is a combination of the Laplace operator and the wave number; is the electric field component is a term related to charge density; ρ0 represents an elliptical double-Gaussian charge density distribution.

2. The method for calculating the space charge field and beam coupling impedance of a vacuum chamber based on PINN according to claim 1, characterized in that: The random sampling point generated in step 1 is N PDE and N BC , are the number of sampling points on the XY plane of the computational domain and the number of sampling points on the XY plane of the boundary condition.

3. The method for calculating the space charge field and beam coupling impedance of a vacuum chamber based on PINN according to claim 1, characterized in that: The random sampling points generated in step 1 are generated within the vacuum domain, and no sampling points are generated outside the chamber.

4. The method for calculating the space charge field and beam coupling impedance of a vacuum chamber based on PINN according to claim 1, characterized in that: The hidden layer of step three is three layers, each layer has 20 neurons; a fully connected neural network and a tanh activation function are used.

5. The method for calculating the space charge field and beam coupling impedance of a vacuum chamber based on PINN according to claim 1, characterized in that: The L PDE is the loss function associated with the PDE of formula (6), which is derived as follows: 1) Determine the relationship between charge density and current density Consider a relativistic charged particle beam with a total charge Q moving in an infinitely long vacuum chamber. Assume that the beam has a rigid charge density distribution ρ and a constant longitudinal velocity v. z =ve z =βce z , where β = v / c, c is the speed of light in vacuum, e z is the unit vector in the direction of beam motion, i.e., the z direction; the beam current density J has only the longitudinal component J z =ρv, the charge and current density obey the following continuity equation The continuity equation of formula (1) shows that in any volume element, the static change of charge is equal to the difference between the current flowing into and out of the volume element; If the charge density per unit time in a volume element increases It means that current flows into the volume; if the charge density in the volume element decreases per unit time This means that current is flowing out of the volume; in: is the charge density change term, which indicates the rate of change of charge density ρ per unit volume with time t; a positive value indicates an increase in charge density, and a negative value indicates a decrease in charge density; ▽·J is the current density term, which indicates the dispersion of current density; a positive value indicates that current flows out of the volume element, and a negative value indicates that current decreases from the volume element; 2) Charge density propagates in the z direction in the form of waves ρ(x,y,z,t)=ρ0(x,y)e j(ωt-kz) (2) Formula (2) gives the distribution of charge density in space and time; they propagate in the z direction in the form of waves; Among them, ρ(x,y,z,t) is the charge density expression, which describes the distribution of charge density in (x,y,z) and time; ρ0(x,y) is the transverse (x,y) distribution function of charge density, which represents the charge density at any position in the z direction, which depends only on x and y; e j(ωt-kz) is a complex exponential term, which represents the change of charge density in z direction and time; j is the imaginary unit, ω is the angular frequency, and k is the wave number; J = ρv is the definition of current density, where v is the velocity vector of the charge; 3) The electric field propagates in the z direction in the form of waves E z (x,y,z,t)=E z (x,y,ω)e j(ωt-kz) (3) Formula (3) gives the distribution of the component of the electric field in the z direction, which also propagates in the z direction in the form of waves; Among them, E z (x, y, z, t) is the component of the electric field in the z direction, describing the components of the electric field in space x, y, z and time t; E z (x, y, ω) indicates that the lateral distribution of the electric field in the z direction depends only on x, y and frequency ω; e j(ωt-kz) is a complex exponential term, which represents the change of charge density in the z direction and time; j is the imaginary unit, ω is the angular frequency, and k is the wave number; this term represents a wave that propagates in time and space; 4) The total charge Q obtained by integrating the charge density on the x and y planes ∫ρ0(x,y)dxdy=Q (4) Formula (4) represents the total charge obtained by integrating the transverse charge density; Among them, ρ0(x,y)dxdy is the transverse charge density distribution function, which represents the distribution of charge in the x and y directions; Q is the total charge; 5) Substitute formulas (1)-(3) into the wave equation of the longitudinal component of the electric field: Formula (5) describes the electric field E z The equation for fluctuations in the z-direction combines the time-dependent variations of charge density and current density; Among them, 2 E z YesE z The Laplace operator of E z Second derivative in space; YesE z The time derivative of E z Changes in amount over time; is the time-varying term of charge density; is the term of the change of current density with time; 6) Determine the partial differential equation for the electric field Derived L PDE Formula (6) is a partial differential equation for the distribution of the electric field in the transverse plane, taking into account the influence of the transverse charge density; in, Equivalent to ▽ 2 , all belong to Laplace operators; YesE z Second-order derivative terms in the x and y directions; is a term related to the wave number k and the relative factor γ; is a term related to the transverse charge density ρ0.

6. The method for calculating the space charge field and beam coupling impedance of a vacuum chamber based on PINN according to claim 1, characterized in that: The formula (9) is derived as follows: 1) The charge distribution on the transverse plane is an elliptical double Gaussian distribution Formula (7) indicates that the charge distribution in the transverse plane is an elliptical Gaussian distribution, the charge density is the largest at the center, and the farther away from the center, the more the charge density exponential decays; Where ρ0 represents the elliptical double Gaussian charge density distribution; Q is the total charge; σ x and σ y are the half-height widths of the charge in the x and y directions, respectively; (x c ,y c ) is the center position of the charge distribution on the plane; 2) Radially symmetric Gaussian distribution simplifies the elliptical Gaussian distribution Formula (8) represents the radially symmetric Gaussian distribution, which simplifies the elliptical Gaussian distribution and makes it easier to handle. r is the radial width at half maximum; 3) Determine the boundary conditions of a perfect conductor In order to obtain the E in the cavity cross section z , solve equation (6) and adopt the boundary conditions of an ideal conductor: n×E=0→E z =0 (9) Formula (9) is the perfect electric conductor boundary condition, which describes the behavior of the electric field on the PEC surface. The tangential component of the electric field must be zero on the PEC surface, which is an important condition in electromagnetic theory. n×E=0orE z =0, n is the surface normal vector. On the PEC surface, the tangential component of the electric field is 0, that is, the electric field cannot have a tangential component on the PEC surface.

7. The method for calculating the space charge field and beam coupling impedance of a vacuum chamber based on PINN according to claim 1, characterized in that: The output value of the neural network space charge field based on the optimal parameter θ in step eight The beam coupling impedance is calculated as follows: 1) Calculation of space charge field Impedance at any point, Z || Defined as 2) Calculate the space charge field The average impedance of Defined as in, is its conjugate complex number.

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