Physics-informed neural network method for modelling beam-wall interactions

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 accelerator vacuum chamber with thin conductive coating, and also verified in comparison with the existing analytical formula.

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 accelerator vacuum chamber with thin conductive coating, and also verified in comparison with the existing analytical formula.
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. However, the volume meshtype methods may suffer from numerical errors [3] in calculating the space-charge impedance. Calculating the resistive-wall impedance requires properly modelling the skin effect [1] in volume meshes. Therefore, it is still challenging to model the wideband impedance of vacuum chambers in accelerator designs.
We propose a novel mesh-free approach for modelling the resistive-wall impedance of 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 the deep learning, inspired by the physics-informed neural network (PINN) [4].
This letter first introduces the PINN into the resistivewall impedance modeling. The main focus is to extend our recent study [5] on the perfectly electric conducting (PEC) wall to the resistive wall. This approach is not yet addressed in other publications [6,7].

Method:
The coupling impedance Z|| is defined in frequency domain as [2] where I=Qv is the total beam current, Q is the total charge, v=vez is the beam velocity, 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 [5] for deep learning, we can derive the following PDE 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 and ε0 is the permittivity of vacuum, γ=(1−β 2 ) −1/2 is the relativistic factor, β=v/c, B is an empirical parameter, is the normalized bi-Gaussian charge density where (σx,σy) is the half value of the Gaussian distribution in the xand y-direction and (xc,yc) is the center 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] = − (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. Using (3), we can scale (5) as Note that (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 [8]. When Zs=0, (6) can be reduced to just the PEC-BC ez=0.
A schematic of the method is illustrated in Fig.1. The PDE (2) and the scaled SIBC (6) are involved into the loss function of a NN using automatic differentiation. This works well especially for smooth transverse charge density as in (4). Note that a neural network is used as a solution surrogate. Unlike the previous study [5], 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 ( , ) corresponding to the real (r) and imaginary (i) parts of Ez. Our algorithm is summarized in the following list.
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.  (2) and (6) 5. Train the constructed NN to find the best parameters θ by minimizing L via the L-BFGS algorithm [9] as a gradient-based optimizer, until L is smaller than a threshold . 6. Obtain the coupling impedance (1) 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 (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 (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. The prediction accuracy of this method depends on the NN architecture and hyperparameters in deep learning, as confirmed in [5]. Its general trend on accuracy [4] is that a good prediction accuracy can be achieved as a sufficiently expressive NN architecture and sufficient numbers of sampling points are given.

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=25mm and the first layer with a small conductivity σ=400S/m and a thickness of 5mm followed by a PEC, as shown Fig.2(a). The surface impedance function on the innermost chamber wall can be given by [10] ( ) = tan , = √̂̂, = √̂̂(14) where μ0 is the permeability of vacuum. Note that the real part is different from the imaginary one in low frequencies.
In this chamber, a round Gaussian beam with Q=1pC, γ=27.7 and σx=σy=σr=2.5mm traverses on its center. Since the closed-form exact solution in this beam-wall system is not available, we verify our simulation with the PINN in comparison with the analytical impedance of a round beam with uniform transverse charge density in the circular chamber [11] involving the same surface impedance as (14). Here we choose the uniform beam radius as rb=1.747σr. This choice allows us to approximate the field of the round Gaussian beam by that of the round uniform beam, up to kσr/γ≃0.5 as used in [5]. This condition reads f<0.66THz in our case.
Here, we assume that the magnetic field on the resistive wall is the same as that on the PEC wall at the same radius R=25mm. This hypothesis is often used for the impedance theory [1,10,11] in accelerator physics. We use the same magnetic field in calculating (6).  Fig.2(b),(c) shows the PINN-simulated result of the field (Ez) in the chamber at f=3.6×10 8 Hz. It is seen that the imaginary part of the field has a nonuniform distribution, and it becomes stronger near the chamber center. This main contribution originates from the space charge field, which has the purely imaginary part related to the charge density in the right-hand side of Eq. (2). By contrast, the real part of the field is nonzero, and it has a uniform distribution. It seems that the real part does not depend on r, and the field value at the axis is same as the one on the inner chamber wall. The similar field behavior is well explained in [1]. This indicates that the proposed method can model both the resistive-wall wake field and space-charge field. Fig.3 shows the simulated impedances of the round Gaussian beam and the analytical impedance of the round uniform beam with rb=1.747σr in the same round chamber. Both for the real and imaginary parts, the simulated and analytical impedances is in excellent agreement. We can also see that the frequency dependency of the real part is different from that of imaginary part.
The results demonstrate that the constructed PINNs can properly model the beam-wall interactions addressed here and accurately simulate the fields and coupling impedance.

Conclusion:
The PINN method with the surface impedance concept has been proposed for the coupling impedance modelling of accelerator vacuum chamber with resistive walls. To verify the method, a round chamber with thin conductive coating is analyzed. The computed coupling impedance agrees well with the approximated analytical impedance. Although only the round geometry with a single layer has been analyzed, the presented method can be extended to multilayer vacuum chambers with other geometries.