Maxwell's equations in Fourier space
The formalism used here is also used in the canonical quantization of the electromagnetic field.41,It is important to distinguish here between physical degrees of freedom and gauge redundancy. In the context of our research, we use this to minimize the number of fields that need to be modeled, while at the same time ensuring that a significant number of Maxwell's equations are automatically satisfied as hard constraints. We start with Maxwell's equations in position space.
$$\begin{aligned} \nabla \cdot \textbf{E}= \frac{\rho }{\varepsilon _0}, \quad \nabla \cdot \textbf{B}= 0, \quad \nabla \times \textbf{E}= -\frac{\partial \textbf{B}}{\partial t}, \quad \nabla \times \textbf{B}= \mu _0 \left( \textbf{J}+ \varepsilon _0 \frac{\partial \textbf{E}}{\partial t} \right) , \end{aligned}$$
(1)
where \(\textbf{E}(\textbf{r},t)\), \(\textbf{B}(\textbf{r},t)\), \(\textbf{J}(\textbf{r},t)\)and \(\rho (\textbf{r},t)\) These are the electric field, magnetic field, current density, and charge density that change over time, respectively.42This performs a 3D spatial Fourier transform on Maxwell's equations defined for a vector field. \(\textbf{F}(\textbf{r},t)\) As
$$\begin{aligned} \hat{\textbf{F}}(\textbf{k},t) = \frac{1}{(2\pi )^{3/2}}\iiint \textbf{F}(\textbf{r},t)e^{-i\textbf{r}\cdot \textbf{k}}d^3\textbf{r}, \end{aligned}$$
(2)
Equation (1) can be rewritten as ( \(\textbf{k}\ne \textbf{0}\)(See Appendix A for a discussion of the zero case.)
$$\begin{aligned} i\textbf{k}\cdot \hat{\textbf{E}}= \frac{\hat{\rho }}{\varepsilon _0}, \quad i\textbf{k}\cdot \hat{\textbf{B}}=0, \quad i\textbf{k}\times \hat{\textbf{E}}= – \frac{\partial \hat{\textbf{B}}}{\partial t}, \quad i\textbf{k}\times \hat{\textbf{B}}=\mu _0\left( \hat{\textbf{J}}+\varepsilon _0\frac{\partial \hat{\textbf{E}}}{\partial t} \right) . \end{aligned}$$
(3)
For each wave vector \(\textbf{k}\)A general vector can be divided into vertical and horizontal components. \(\textbf{k}\):
$$\begin{aligned} \hat{{\textbf {F}}} (\textbf{k}, t) = \hat{{\textbf {F}}}_{\parallel } (\textbf{k} , t) + \hat{{\textbf {F}}}_{\perp } (\textbf{k}, t), \quad \text {hence} \quad \textbf{k}\cdot \hat{{ \textbf {F}}}_{\perp } (\textbf{k}, t) = 0, \quad \textbf{k}\times \hat{{\textbf {F}}}_{\parallel } ( \textbf{k}, t) = {\textbf {0}}. \end{aligned}$$
(Four)
The inverse Fourier transform of equation (4) leads to the Helmholtz decomposition, \({\textbf {F}}({\textbf {r}},t)\)Gauss's law for electric and magnetic fields (Equation (3)) is expressed as follows:
$$\begin{aligned} \hat{{\textbf {E}}}_{\parallel } (\textbf{k}, t) = -i\textbf{k}\frac{ \hat{\rho } (\textbf{k}, t) }{\varepsilon _0 |\textbf{k}|^2}, \quad \hat{{\textbf {B}}}_{\parallel } (\textbf{k}, t) = {\textbf {0}}. \end{aligned}$$
(5)
In Fourier space, the electric and magnetic fields depend on the Fourier transformed scalar and vector potentials, \(\hat{\phi } (\textbf{k}, t)\) and \(\hat{{\textbf {A}}}(\textbf{k}, t)\)given by \(\hat{{\textbf {E}}}(\textbf{k},t) = – i \textbf{k}\hat{\phi }(\textbf{k},t) – \frac{\partial \hat{{\textbf {A}}}(\textbf{k},t)}{\partial t}, \ \hat{{\textbf {B}}}(\textbf{k},t) = i \textbf{k}\times \hat{{\textbf {A}}}(\textbf{k},t)\)The transverse magnetic field components are:
$$\begin{aligned} \hat{{\textbf {E}}}_\perp (\textbf{k},t) = -\frac{\partial \hat{{\textbf {A}}}_\ perp (\textbf{k},t)}{\partial t}, \quad \hat{{\textbf {B}}}_\perp (\textbf{k},t) = i \textbf{k}\ times \hat{{\textbf {A}}}_\perp (\textbf{k},t). \end{aligned}$$
(6)
Using equation (3), Faraday's law is self-evident.
$$\begin{aligned} – i\textbf{k}\times \frac{\partial \hat{\textbf{A}}_\perp (\textbf{k}, t)}{\partial t} = – \frac{\partial }{\partial t} i \textbf{k}\times \hat{\textbf{A}}_\perp (\textbf{k}, t). \end{aligned}$$
(7)
The transverse components of the Ampere-Maxwell equations are:
$$\begin{aligned} \frac{1}{c^2} \frac{\partial ^2 \hat{{\textbf {A}}}_\perp (\textbf{k},t) }{\partial t^2} +|\textbf{k}|^2 \hat{{\textbf {A}}}_\perp (\textbf{k},t) = \mu _0 \hat{{\textbf {J}}}_\perp (\textbf{k},t), \end{aligned}$$
(8)
Using equation (5), it can be shown that the longitudinal component of the Ampere-Maxwell equations is equivalent to the continuity equation, which can also be expressed as: \(\hat{\textbf{J}}_\parallel \) To \(\hat{\rho }\),therefore, \(\hat{\rho }\) and \(\hat{\textbf{J}}_\perp \) You should use it.
From equations (5) and (6), the electromagnetic field can be determined as follows:
$$\begin{aligned} \hat{\textbf{E}} (\textbf{k}, t) = -i\textbf{k}\frac{ \hat{\rho } (\textbf{k}, t) }{\varepsilon _0 |\textbf{k}|^2} – \frac{\partial \hat{{\textbf {A}}}_\perp (\textbf{k}, t)}{\partial t}, \quad \hat{\textbf{B}} (\textbf{k}, t) = i \textbf{k}\times \hat{{\textbf {A}}}_\perp (\textbf{k}, t), \end{aligned}$$
(9)
This is followed by the inverse Fourier transform.
Fourier-Helmholtz-Maxwell neural operators
Given the distribution of charge and current densities, the electromagnetic field can be determined as follows: \(\hat{{\textbf {A}}}_\perp ({\textbf {k}},t)\)according to equation (9), and \(\hat{\rho }(\textbf{k},t)\). \(\hat{{\textbf {A}}}_\perp ({\textbf {k}},t)\) You can find it below \(\hat{{\textbf {J}}}_\perp ({\textbf {k}},t)\) via equation (8). Based on this, we propose to find the generated electromagnetic field using the approach shown in the diagram in Figure 1, where the hard constraints (constraints that are automatically respected) are listed. Since this approach incorporates the Fourier transform, Helmholtz decomposition, and Maxwell's equations within a neural operator framework, we call it the Fourier-Helmholtz-Maxwell Neural Operator (FoHM-NO) method.
The FoHM-NO method and its built-in hard physical constraints. \(\mathscr {F}\) Represents the Fourier transform. \(\mathscr {P}_\perp \) The projection of the vector field onto the transverse component is \(\textbf{k}\)NN is a neural network, \(\mathscr {F}^{-1}\) This is the inverse Fourier transform.
In this study, we attempt to estimate the electromagnetic field. \(\rho \) and \(\textbf{J}\) We use the FoHM-NO method, specifically convolutional neural networks. When applying neural networks to physics problems, we rely on the useful quantities that physicists have already identified (i.e. \(\hat{{\textbf {A}}}_\perp ({\textbf {k}}, t)\)), those that are directly observable (i.e. \({\textbf {E}}({\textbf {r}}, t)\) and \({\textbf {B}}({\textbf {r}}, t)\)This has been demonstrated previously by using neural networks to model the Hamiltonian of a classical system.43scalar and vector potentials for estimating electromagnetic fields40 Stream function of an incompressible fluid velocity flow field44.
Finally, the FoHM-NO method exploits the fact that the electromagnetic field has only two independent degrees of freedom, as can be seen from equation (9) in the source-free case. However, at a fundamental level, this is due to the photon being a massless particle with spin 1. One advantage of this method is that it requires fewer fields than the six required to directly estimate both the electric and magnetic fields, or the four required when using the full scalar and vector potentials. Fitting more fields increases the tension between fitting the data while satisfying physical conditions. In contrast, the procedure, by construction, incorporates three of the four Maxwell equations (equations (5) and (7)) as hard constraints. This is also true for the longitudinal component of the Ampere-Maxwell law, which is derived from the continuity equation and Gauss's law for the electric field (see Appendix B). On the other hand, working in Fourier space has the advantage that spatial derivatives are exchanged for wavevector multiplications. Therefore, constraints such as those in equation (4) are easier to satisfy than in the spatial derivative case. In the case of the Fast Fourier Transform (FFT), no For the total number of spatial data points, the time complexity of the FFT is \(\mathscr {O} \left( N \log N \right) \)This, coupled with modern GPU computing, means that the move to Fourier space can be performed very quickly, even for very large data sets. We did not perform a temporal Fourier transform because the time length of the simulation varies. This variation poses challenges for the network to learn in the frequency domain. Furthermore, employing 4D CNNs brings computational challenges, significantly increasing the number of parameters in the model.
Computational EM methods conducted in configuration space have the notable advantage that they can easily and readily incorporate spatial boundary conditions that are useful for solving many real-world EM problems. We do not consider any special spatial boundary conditions here. First, the simulation was simply started with particle initial conditions. Second, neural networks have been shown to be useful for solving inverse PDE problems where the problem may be ill-posed, e.g. when boundary conditions are poor and classical methods are difficult or unavailable.45,46,47In future research, we would like to apply this approach to inverse problems.
Another advantage of FoHM-NO is \(\hat{{\textbf {A}}}_\perp ({\textbf {k}}, t)\) The gauge redundancy problem is avoided for all potential fields, but not all of them. To check this, under the gauge transformation,
$$\begin{aligned} \hat{\phi } ({\textbf {k}}, t) \rightarrow \hat{\phi } ({\textbf {k}}, t) – \frac{\partial \hat{f} ({\textbf {k}}, t) }{\partial t}, \quad \hat{{\textbf {A}}} ({\textbf {k}}, t) \rightarrow \hat{{\textbf {A}}}({\textbf {k}}, t) + i {\textbf {k}} \hat{f} ({\textbf {k}}, t), \end{aligned}$$
(Ten)
A scalar function \(\hat{f} ({\textbf {k}}, t)\)The change in vector potential is \({\textbf {k}}\) The direction is
$$\begin{aligned} \hat{{\textbf {A}}}_\parallel ({\textbf {k}}, t) \rightarrow \hat{{\textbf {A}}}_\parallel ({\textbf {k}}, t) + i {\textbf {k}} \hat{f} ({\textbf {k}}, t), \quad \hat{{\textbf {A}}}_\perp ({\textbf {k}}, t) \rightarrow \hat{{\textbf {A}}}_\perp ({\textbf {k}}, t). \end{aligned}$$
(11)
therefore, \(\hat{\textbf{A}}_\perp (\textbf{k}, t)\) There is no need to select a specific gauge. \(\hat{{\textbf {A}}}_\perp ({\textbf {k}}, t)\) He is well known for his research on the quantization of electromagnetic fields.41,48,49 Spin-orbit decomposition of gauge fields50.
However, care must be taken when changing the inertial reference frame. \(\hat{\textbf{A}}_\perp (\textbf{k}, t) \)Unlike gauge dependence, \(A^\mu = (\phi /c, \textbf{A})\)is not a Lorentzian 4-vector. This is not a serious problem since our goal is to find efficient and accurate computational methods for accelerator physics.
The remainder of this article is a proof-of-concept application of FoHM-NO to predict the electromagnetic fields generated by a relativistically charged particle beam. \(\rho \) and \(\textbf{J}\) is given, but can also be extended to include the matter density field in the predictions if an equation of state is used. FoHM-NO may not generally perform as accurately as mature state-of-the-art EM codes, but the use of neural networks here has the advantage of very fast run times. This can be useful when speed is essential, such as in accelerator control or diagnostic setups, where beam time may be quite limited. Alternatively, it can be used for exploratory analysis to find an approximate solution before deploying more accurate but time-consuming refinement methods.
