Physics‐informed neural network method for modelling beam‐wall interactions
2022; Institution of Engineering and Technology; Volume: 58; Issue: 10 Linguagem: Inglês
10.1049/ell2.12469
ISSN1350-911X
Autores Tópico(s)Structural Health Monitoring Techniques
ResumoElectronics LettersEarly View LetterOpen Access Physics-informed neural network method for modelling beam-wall interactions Kazuhiro Fujita, Corresponding Author Kazuhiro Fujita kfujita@sit.ac.jp orcid.org/0000-0002-6726-2035 Department of Information Systems, Saitama Institute of Technology, Fukaya, Japan Email: kfujita@sit.ac.jpSearch for more papers by this author Kazuhiro Fujita, Corresponding Author Kazuhiro Fujita kfujita@sit.ac.jp orcid.org/0000-0002-6726-2035 Department of Information Systems, Saitama Institute of Technology, Fukaya, Japan Email: kfujita@sit.ac.jpSearch for more papers by this author First published: 13 March 2022 https://doi.org/10.1049/ell2.12469AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract A mesh-free approach for modelling beam-wall interactions in particle accelerators is proposed. The key idea of our method is to use a deep neural network as a surrogate for the solution to a set of partial differential equations involving the particle beam, and the surface impedance concept. The proposed approach is applied to the coupling impedance of an infinitely long vacuum chamber with a thin conductive coating, and also verified in comparison with traditional numerical methods. Introduction A relativistic beam of charged particles traversing in a particle accelerator can interact with vacuum chambers with resistive walls [1]. To estimate this interaction effect, which can limit the performance of a particle accelerator, the coupling impedance [2] is used. A typical way to obtain the coupling impedance is to use purely numerical methods, for example, [3-6]. When calculating the resistive wall impedance with such methods, one needs to address the skin effect [1] properly. To avoid meshing very small skin depths at high frequencies and/or calculating fields inside conductors, the surface impedance concept has been used. To obtain a numerical solution, traditional numerical methods require the discretization of domains or boundaries for an accelerator vacuum chamber of interest. We propose a novel mesh-free approach for modelling the resistive wall impedance of infinitely long accelerator vacuum chambers. The key idea of our method is to use a deep neural network (DNN) as an approximate solution to a set of partial differential equations (PDEs) involving the particle beam, and the surface impedance concept. The mesh-free feature originates from the use of the DNN, which has the universal function approximation capability. Our solution is based on deep learning, inspired by the physics-informed neural network (PINN) [7]. This letter first introduces the PINN into the resistive wall impedance modelling. The main focus is to extend our recent study [8] on the perfectly electric conducting (PEC) wall to the resistive wall. This approach is not yet addressed in other publications [9, 10]. The goal of this letter is to clarify that the PINN can be applied to resistive wall impedance modelling (to show the feasibility of PINN for it). We limit our discussion to this point. Method The coupling impedance Z|| is defined in the frequency domain as [2] Z | | = − E z I \begin{equation} {Z}_{||}=\ -\frac{{E}_{z}}{I} \end{equation} (1)where I = Qv is the total beam current, Q is the total charge, v = vez is the beam velocity, and ez is the unit vector in the direction of beam motion (z-direction). To compute the impedance (1), we need to know the longitudinal component of the electric field Ez for one particular harmonic component with an angular frequency ω = 2πf (or wave number k = ω/v). We assume that the beam has a rigid charge density distribution normalized by Q, and moves along the axis of an infinitely long vacuum chamber. For the above beam-wall system, using the special scaling scheme [Appendix A, 8] for deep learning, we can derive the following PDE ∂ 2 ∂ X 2 + ∂ 2 ∂ Y 2 e z − s 0 2 k 2 γ 2 e z + jB ρ n = 0 \begin{equation} \left(\frac{{\partial}^{2}}{\partial {X}^{2}}+\frac{{\partial}^{2}}{\partial {Y}^{2}}\right){e}_{z}-\frac{{s}_{0}^{2}{k}^{2}}{{\gamma}^{2}}{e}_{z}+\textit{jB}{\rho}_{n}=0 \end{equation} (2)where (X,Y) = (x/s0, y/s0) are Cartesian coordinates scaled with a typical chamber length s0 (e.g., radius, height and width), ez = Ez/E0 is the electric field scaled with E 0 = s 0 2 k q n B ε 0 γ 2 , q n = Q 2 π σ x σ y \begin{equation}{E_0} = \frac{{{s_0}^2k{q_n}}}{{B{\varepsilon _0}{\gamma ^2}}},{q_n} = \frac{Q}{{2\pi {\sigma _x}{\sigma _y}}}\end{equation} (3)and ε0 is the permittivity of vacuum, γ = (1−β2)−1/2 is the relativistic factor, β = v/c, B is an empirical parameter, ρn is the bi-Gaussian charge density scaled as ρ n = e − x − x c 2 2 σ x 2 − y − y c 2 2 σ y 2 \begin{equation}{\rho _n} = {e^{ - \frac{{{{\left( {x - {x_c}} \right)}^2}}}{{2\sigma _x^2}} - \frac{{{{\left( {y - {y_c}} \right)}^2}}}{{2\sigma _y^2}}}}\end{equation} (4)where (σx,σy) is the half value of the Gaussian distribution in the x- and y-direction and (xc,yc) is the centre position in the transverse plane. Here, we replace the original problem with resistive walls by an equivalent problem with the Leontovich boundary condition or surface impedance boundary condition (SIBC) [1] E z = − Z s H t \begin{equation}{E_z} = - {Z_s}{H_t}\end{equation} (5)where Ht is the tangential component of the magnetic field on the surface of the chamber cross section, and Zs is the surface impedance function. In our approach, Equation (6) must be valid. Using Equation (3), we can scale Equation (5) as e z + Z s H t / E 0 = 0 \begin{equation}{e_z} + {Z_s}{H_t}/{\rm{\ }}{E_0} = 0\end{equation} (6)Note that Equation (6) is enforced only on the innermost wall of the chamber. All the domain outside the innermost wall is assumed to be filled by PEC. This can be also regarded as the assumption of infinitely thick PEC wall. Therefore, the field is zero outside the innermost chamber wall. In accelerator physics, this surface impedance concept can be used to reasonably model multiscale features of multiple surface perturbations such as the skin effect and the thin layer in the numerical method [11]. When Zs = 0, Equation (6) can be reduced to just the PEC-BC ez = 0. We also assume the perturbative treatment of magnetic field, as discussed in [1] and used in [12, 13]. This means that the magnetic field on the resistive wall is the same as that of the PEC wall; the longitudinal component of the magnetic field is zero even for non-ultrarealistic beams. It should be mentioned that only the transverse component of the magnetic field on the wall is given in Equation (5). Under the above assumptions, we simplify the situation and facilitate modelling beam-wall interactions. A schematic of the method is illustrated in Figure 1. The PDE (2) and the scaled SIBC (6) are involved in the loss function of a NN using automatic differentiation. This works well especially for smooth transverse charge density as in Equation (4). Note that a neural network is used as a solution surrogate. Unlike the previous study [8], although the space-charge field has only a purely imaginary part, the resistive wall wake field has both real and imaginary parts. Therefore, the constructed NN also has two outputs ( e z r , e z i ) $( {e_z^r,e_z^i} )$ corresponding to the real (r) and imaginary (i) parts of Ez. Our algorithm is summarized in the following list. Set up a computational domain surrounded by the innermost wall of a chamber cross section, the scaled SIBC (6), the physical constant ε0, the beam parameters (Q, v, β, γ), a source domain related to ρn and qn, the wave number k, and the scaling parameters s0 and E0. Assume that the beam traverses inside the chamber and the field is zero outside the computational domain. Generate randomly sampled points (or regular or irregular grid points) within the computational domain surrounded by the innermost wall. Note that no sampling point is generated outside the domain. The generated sampling points are used to train a NN as input. Construct a NN with two outputs e ̂ z r ( x , y ; θ ) , e ̂ z i ( x , y ; θ ) $\hat{e}_z^r( {x,y;\theta } ),\hat{e}_z^i( {x,y;\theta } )$ as a surrogate of the scaled PDE solution e z ( x , y ) = e z r ( x , y ) + j e z i ( x , y ) ${e_z}\ ( {x,y} ) = e_z^r\ ( {x,y} ) + je_z^i( {x,y} )$ , where θ is a vector containing all weights w and bias b in the neural network to be trained, σ denotes an activation function. Define the loss function L defined by Equations (2) and (6) Train the constructed NN to find the best parameters θ by minimizing L via the L-BFGS algorithm [14] as a gradient-based optimizer, until L is smaller than a threshold ε. Obtain the coupling impedance (1) and original outputs (unscaled) from E ̂ z ( x , y ; θ ) = E 0 e ̂ z ( x / s 0 , y / s 0 ; θ ) ${\hat{E}_z}\ ( {x,y;\theta } ) = {E_0}\ {\hat{e}_z}( {x/{s_0},y/{s_0};\theta } )$ using the trained NN. Fig. 1Open in figure viewerPowerPoint Physics-informed neural network. In this method, the loss function L is defined by L = w PDE l PDE + w SIBC l SIBC \begin{equation}L{\rm{\ }} = {w_{{\rm{PDE}}}}{\rm{\ }}{l_{{\rm{PDE}}}} + {w_{{\rm{SIBC}}}}{l_{{\rm{SIBC}}}}\end{equation} (7) l PDE = 1 N PDE ∑ p = 1 N PDE f r x p , y p ; θ 2 + f i x p , y p ; θ 2 \begin{equation}{l_{{\rm{PDE}}}} = \frac{1}{{{N_{{\rm{PDE}}}}}}{\rm{\ }}\mathop \sum \limits_{p{\rm{\ }} = {\rm{\ }}1}^{{N_{{\rm{PDE}}}}} \left[ {{{\left| {{f_r}\left( {{x_p},{y_p};\theta } \right)} \right|}^2} + {{\left| {{f_i}\left( {{x_p},{y_p};\theta } \right)} \right|}^2}} \right]\end{equation} (8) l SIBC = 1 N SIBC ∑ p = 1 N SIBC g r x p , y p ; θ 2 + g i x p , y p ; θ 2 \begin{equation}{l_{{\rm{SIBC}}}} = \frac{1}{{{N_{{\rm{SIBC}}}}}}{\rm{\ }}\mathop \sum \limits_{p{\rm{\ }} = {\rm{\ }}1}^{{N_{{\rm{SIBC}}}}} \left[ {{{\left| {{g_r}\left( {{x_p},{y_p};\theta } \right)} \right|}^2} + {{\left| {{g_i}\left( {{x_p},{y_p};\theta } \right)} \right|}^2}} \right]\end{equation} (9) f r = ∂ 2 ∂ X 2 + ∂ 2 ∂ Y 2 e ̂ z r − s 0 2 k 2 γ 2 e ̂ z r \begin{equation}{f_r} = \left( {\frac{{{\partial ^2}}}{{\partial {X^2}}} + \frac{{{\partial ^2}}}{{\partial {Y^2}}}} \right){\rm{\ }}\hat{e}_z^r - \frac{{s_0^2{k^2}}}{{{\gamma ^2}}}\hat{e}_z^r\end{equation} (10) f i = ∂ 2 ∂ X 2 + ∂ 2 ∂ Y 2 e ̂ z i − s 0 2 k 2 γ 2 e ̂ z i + B ρ n \begin{equation}{f_i} = \left( {\frac{{{\partial ^2}}}{{\partial {X^2}}} + \frac{{{\partial ^2}}}{{\partial {Y^2}}}} \right){\rm{\ }}\hat{e}_z^i - \frac{{s_0^2{k^2}}}{{{\gamma ^2}}}\hat{e}_z^i + B{\rho _n}\end{equation} (11) g r = e ̂ z r + Re Z s H t / E 0 \begin{equation}{g_r} = \hat{e}_z^r{\rm{\ }} + {\rm{Re}}\left( {{Z_s}{H_t}/{E_0}} \right)\end{equation} (12) g i = e ̂ z i + Im Z s H t / E 0 \begin{equation}{g_i} = \hat{e}_z^i{\rm{\ }} + {\rm{Im}}\left( {{Z_s}{H_t}/{E_0}} \right)\end{equation} (13)where p denotes the sampling point. NPDE and NSIBC are the numbers of sampling points in the computational domain and on the boundary surface, respectively. wPDE and wSIBC are the weights of the loss function. lPDE is the loss function related to the scaled PDE (2), and its minimization (lPDE→0) enforces Equation (2) at a set of finite sampling points in the computational domain. lSIBC is the loss function related to the SIBC, and its minimization (lSIBC→0) enforces Equation (6) at a set of finite sampling points on the boundary surface. Throughout this study, we adapted a fully connected neural network and the tanh activation function. We used three hidden layers and 20 neurons per layer. We chose B = 100, (wPDE, wSIBC) = (1,100) and (NPDE, NSIBC) = (2000,200). NPDE random points are generated inside a chamber and NSIBC grid points are generated on the chamber wall. These are similar to those used in [Appendix B, 8]. The prediction accuracy of PINNs and its general trend were already discussed in [7, 8] and demonstrated for different PDEs. Therefore, their properties are believed to be the same in this study. See also [15] for the convergence of PINNs. The following two approaches are considered to give the magnetic field for Equation (6) in the proposed method: The simplest way is to give the magnetic field analytically calculated with available closed-form exact or approximate solutions. This can be easily incorporated in the PINN, but its applicability will be limited only to relatively simple geometries such as round chamber. A more flexible approach is to use a field solution simulated with traditional numerical methods. This will apply to general cross-section vacuum chambers. Although not shown here, the first approach (a) was tested as a preliminary study. As the next step, I use the second one (b) in the next section. Results and discussion To show the feasibility of the proposed method, we apply it to the analysis of a round vacuum chamber with inner radius R = 25 mm and the first layer with a small conductivity σ = 400 S/m and a thickness d = 5 mm followed by a PEC, as shown in Figure 2a. The domain and boundary sampling points were generated as in Figure 2b, where the coordinates are scaled with s0 = R. The surface impedance function on the innermost chamber wall can be given by [12] Z s ω = j η c t tan k c t d , k c t = ω μ ̂ c t ε ̂ c t , η c t = μ ̂ c t ε ̂ c t \begin{equation}{Z_s}{\rm{\ }}\left( \omega \right) = {\rm{\ }}j{\eta _{ct}}\tan {k_{ct}}d,{k_{ct}} = \omega \sqrt {{{\hat{\mu }}_{ct}}{{\hat{\varepsilon }}_{ct}}} ,{\rm{\ }}{\eta _{ct}} = \sqrt {\frac{{{{\hat{\mu }}_{ct}}}}{{{{\hat{\varepsilon }}_{ct}}}}} \end{equation} (14) μ ̂ c t = μ 0 , ε ̂ c t = ε 0 − j σ ω \begin{equation}{\hat{\mu }_{ct}} = {\mu _0},{\hat{\varepsilon }_{ct}} = {\varepsilon _0} - j\frac{\sigma }{\omega }\end{equation} (15)where μ0 is the permeability of vacuum. Note that the real part is different from the imaginary part at low frequencies. In this chamber, a round Gaussian beam with Q = 1 pC, γ = 27.7 and σx = σy = σr = 2.5 mm traverses on its centre. To the best of our knowledge, no closed-form exact solution for the above beam-wall system is available. Therefore, using the boundary element method (BEM) [6] in the frequency domain, we calculate Ht induced by the same round Gaussian beam traversing in the round PEC vacuum chamber of R = 25 mm. This Ht is used for Equation (6) in the proposed method. Fig. 2Open in figure viewerPowerPoint Electric field of a relativistic particle beam in infinitely long round vacuum chamber with a thin conductive coating. (a) chamber geometry, (b) sampling points, (c) Im(Ez) inside the chamber, (d) Re(Ez) and Im(Ez) along y-axis at x = 0. Figures 2c,d show the real and imaginary parts of PINN-simulated field (Ez) in the chamber at f = 3.6×108 Hz. For comparison, the BEM simulation results are also displayed in Figure 2d. The PINN result is in excellent agreement with the BEM ones. It is seen that Im(Ez) has a nonuniform distribution, and its peak is near the chamber centre. This originates from the space charge field, which has the purely imaginary part related to ρn in the left-hand side of Equation (2). By contrast, it is observed in Figure 2d that Re(Ez) is nonzero, and it has a uniform distribution. Note that Re(Ez) does not depend on r, and its value at the axis is the same as the one on the inner chamber wall. Importantly, this demonstrates the behaviour of resistive wall wake field as well explained in [1]. These results indicate that both the resistive wall wake field and space charge field can be simulated with the constructed PINNs, unlike our previous study [8]. Figure 3 shows resistive wall impedances (Z||) computed with the PINN and the CST wakefield solver [4] based on the finite integration technique (FIT). With this solver [4], we performed the 3-D field simulation incorporating the SIBC (5) in the time domain, and obtained the impedance. The contribution of space charge impedance is subtracted, and is not included in the shown impedances. The two computed impedances are in excellent agreement both for the real and imaginary parts. As expected from Equation (14), the frequency dependency of Re(Z||) is different from that of Im(Z||) at low frequencies. This result demonstrates that the constructed PINNs can be successfully applied to the resistive wall impedance modelling. Fig. 3Open in figure viewerPowerPoint Comparison of resistive wall impedances simulated with the CST wakefield solver [4] and the proposed PINN. Since the goal of this letter is to show the feasibility of PINN, we have not discussed the computational efficiency in comparison with other numerical methods. This issue will be our future work. See, for example, [16] for comparison between PINNs and the finite element method. 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