Physics-constrained machine learning for electrodynamics without gauge ambiguity based on Fourier transformed Maxwell’s equations

Maxwell’s equations in Fourier areaThe formalism that will probably be employed right here can also be used within the canonical quantization of the electromagnetic field41, the place distinguishing between the bodily levels of freedom and the gauge redundancies is vital. In the context of our research, we use it to attenuate the variety of fields wanted to be modeled, whereas additionally guaranteeing {that a} substantial variety of Maxwell’s equations are routinely glad as onerous constraints. We begin with Maxwell’s equations in place area:$$start{aligned} nabla cdot textbf{E}= frac{rho }{varepsilon _0}, quad nabla cdot textbf{B}= 0, quad nabla instances textbf{E}= -frac{partial textbf{B}}{partial t}, quad nabla instances textbf{B}= mu _0 left( textbf{J}+ varepsilon _0 frac{partial textbf{E}}{partial t} proper) , finish{aligned}$$
(1)
the place (textbf{E}(textbf{r},t)), (textbf{B}(textbf{r},t)), (textbf{J}(textbf{r},t)), and (rho (textbf{r},t)) are the time-varying electrical discipline, magnetic discipline, present density, and cost density, respectively42. We carry out a 3D spatial Fourier rework on Maxwell’s equations, which is outlined for a vector discipline (textbf{F}(textbf{r},t)) as$$start{aligned} hat{textbf{F}}(textbf{okay},t) = frac{1}{(2pi )^{3/2}}iiint textbf{F}(textbf{r},t)e^{-itextbf{r}cdot textbf{okay}}d^3textbf{r}, finish{aligned}$$
(2)
and rewrite Eq. (1) (assuming (textbf{okay}ne textbf{0}); see “Appendix A” for a dialogue of the zero case):$$start{aligned} itextbf{okay}cdot hat{textbf{E}}= frac{hat{rho }}{varepsilon _0}, quad itextbf{okay}cdot hat{textbf{B}}=0, quad itextbf{okay}instances hat{textbf{E}}= – frac{partial hat{textbf{B}}}{partial t}, quad itextbf{okay}instances hat{textbf{B}}=mu _0left( hat{textbf{J}}+varepsilon _0frac{partial hat{textbf{E}}}{partial t} proper) . finish{aligned}$$
(3)
For every wave vector (textbf{okay}), a generic vector may be separated into parts longitudinal and transverse to (textbf{okay}):$$start{aligned} hat{{textbf {F}}} (textbf{okay}, t) = hat{{textbf {F}}}_{parallel } (textbf{okay}, t) + hat{{textbf {F}}}_{perp } (textbf{okay}, t), quad textual content {such that} quad textbf{okay}cdot hat{{textbf {F}}}_{perp } (textbf{okay}, t) = 0, quad textbf{okay}instances hat{{textbf {F}}}_{parallel } (textbf{okay}, t) = {textbf {0}}. finish{aligned}$$
(4)
An inverse Fourier transformation of Eq. (4) will result in a Helmholtz decomposition of ({textbf {F}}({textbf {r}},t)). Gauss’s legislation for the electrical and magnetic fields (Eq. (3)) can then be expressed as:$$start{aligned} hat{{textbf {E}}}_{parallel } (textbf{okay}, t) = -itextbf{okay}frac{ hat{rho } (textbf{okay}, t) }{varepsilon _0 |textbf{okay}|^2}, quad hat{{textbf {B}}}_{parallel } (textbf{okay}, t) = {textbf {0}}. finish{aligned}$$
(5)
In Fourier area, the electrical and magnetic fields’ dependence on the Fourier transformed scalar and vector potential, (hat{phi } (textbf{okay}, t)) and (hat{{textbf {A}}}(textbf{okay}, t)), is given by (hat{{textbf {E}}}(textbf{okay},t) = – i textbf{okay}hat{phi }(textbf{okay},t) – frac{partial hat{{textbf {A}}}(textbf{okay},t)}{partial t}, hat{{textbf {B}}}(textbf{okay},t) = i textbf{okay}instances hat{{textbf {A}}}(textbf{okay},t)). The transverse discipline parts are:$$start{aligned} hat{{textbf {E}}}_perp (textbf{okay},t) = -frac{partial hat{{textbf {A}}}_perp (textbf{okay},t)}{partial t}, quad hat{{textbf {B}}}_perp (textbf{okay},t) = i textbf{okay}instances hat{{textbf {A}}}_perp (textbf{okay},t). finish{aligned}$$
(6)
Using Eq. (3), Faraday’s legislation trivially follows:$$start{aligned} – itextbf{okay}instances frac{partial hat{textbf{A}}_perp (textbf{okay}, t)}{partial t} = – frac{partial }{partial t} i textbf{okay}instances hat{textbf{A}}_perp (textbf{okay}, t). finish{aligned}$$
(7)
The transverse parts of the Ampère–Maxwell equation grow to be:$$start{aligned} frac{1}{c^2} frac{partial ^2 hat{{textbf {A}}}_perp (textbf{okay},t) }{partial t^2} +|textbf{okay}|^2 hat{{textbf {A}}}_perp (textbf{okay},t) = mu _0 hat{{textbf {J}}}_perp (textbf{okay},t), finish{aligned}$$
(8)
whereas utilizing Eq. (5) it may be proven that the longitudinal parts of the Ampère–Maxwell equation is equal to the continuity equation. The continuity equation additionally relates (hat{textbf{J}}_parallel ) to (hat{rho }), thus solely (hat{rho }) and (hat{textbf{J}}_perp ) want be used.From Eqs. (5) and (6), the EM fields can then be decided by:$$start{aligned} hat{textbf{E}} (textbf{okay}, t) = -itextbf{okay}frac{ hat{rho } (textbf{okay}, t) }{varepsilon _0 |textbf{okay}|^2} – frac{partial hat{{textbf {A}}}_perp (textbf{okay},t)}{partial t}, quad hat{textbf{B}} (textbf{okay}, t) = i textbf{okay}instances hat{{textbf {A}}}_perp (textbf{okay},t), finish{aligned}$$
(9)
adopted by inverse Fourier transforms.Fourier–Helmholtz–Maxwell neural operatorGiven the cost and present density distributions, the EM Fields may be by decided by (hat{{textbf {A}}}_perp ({textbf {okay}},t)), as per Eq. (9), and (hat{rho }(textbf{okay},t)). (hat{{textbf {A}}}_perp ({textbf {okay}},t)) may be discovered from (hat{{textbf {J}}}_perp ({textbf {okay}},t)) by way of Eq. (8). From this, we suggest discovering the generated EM fields utilizing the strategy depicted within the diagram of Fig. 1, the place the onerous constraints (ones routinely obeyed) are listed. Since this strategy incorporates Fourier transformations, Helmholtz decompositions, and Maxwell’s equations inside a neural operator framework, we discuss with it because the Fourier–Helmholtz–Maxwell neural operator (FoHM-NO) methodology.Figure 1The FoHM-NO methodology and its built-in onerous physics constraints. (mathscr {F}) denotes Fourier transformation, (mathscr {P}_perp ) is the projection of the vector discipline to parts transverse to (textbf{okay}), NN a neural community, and (mathscr {F}^{-1}) is inverse Fourier transformation.In this work, we try to estimate the EM fields from (rho ) and (textbf{J}) utilizing the FoHM-NO methodology, particularly with convolutional neural networks. When making use of neural networks to physics issues, prior theoretical data may be leveraged to mannequin helpful portions already recognized by physicists (i.e., (hat{{textbf {A}}}_perp ({textbf {okay}}, t))), reasonably than the direct observables (i.e., ({textbf {E}}({textbf {r}}, t)) and ({textbf {B}}({textbf {r}}, t))). This has been demonstrated beforehand by way of using neural networks to mannequin the Hamiltonian of classical systems43, the scalar and vector potentials to estimate EM fields40 and the stream operate of an incompressible fluid’s velocity move field44.Ultimately, the FoHM-NO methodology exploits that the EM fields have solely 2 impartial levels of freedom, as may be seen from Eq. (9) within the source-less case. On a elementary stage although, this stems from the photon being a spin-1 massless particle. One advantage of the strategy is that it require fewer fields than the 6 that will be wanted for straight estimating each the electrical and magnetic discipline, or the 4 needed through the use of the scalar potential and all the vector potential. With extra fields to suit, there comes growing stress between becoming the info whereas additionally satisfying bodily circumstances. In distinction, with this process by building three of the 4 Maxwell’s equations (Eqs. (5) and (7)) are built-in as onerous constraints. This can also be the case for the longitudinal part of Ampère–Maxwell’s Law, which follows from the continuity equation and Gauss’s legislation for the electrical discipline (see “Appendix B”). Working in Fourier area, in the meantime, has the benefit that spatial derivatives are traded for multiplication by wave vectors. Thus, constraints like Eq. (4) may be simpler to fulfill than their spatial by-product counterparts. For a quick Fourier rework (FFT), if N is the full variety of spatial information factors, then the FFT has a time complexity of (mathscr {O} left( N log N proper) ). This, together with fashionable GPU computing, signifies that transferring back and forth from Fourier area may be carried out extraordinarily shortly, even for very massive information units. We didn’t carry out a temporal Fourier rework because the time size of the simulations assorted. This variation would pose challenges to the networks to be taught within the frequency area. Additionally, the adoption of a 4D CNN would current computational challenges and considerably will increase the variety of parameters in fashions.Computational EM strategies carried out in configuration area have a notable benefit: a straightforward, straight-forward incorporation of spatial boundary circumstances, which may be helpful in fixing many real-world EM issues. Here, we don’t contemplate any particular spatial boundary circumstances. For one, the simulation was simply began with the preliminary circumstances of the particles. Secondly, neural networks have been discovered to be helpful in fixing inverse PDE issues, the place the issue could also be ill-posed. For instance, inadequate boundary circumstances, a case the place classical strategies battle or can’t be used45,46,47. We want to apply our strategy to inverse issues in future works.An further advantage of FoHM-NO is that by focusing on simply (hat{{textbf {A}}}_perp ({textbf {okay}}, t)) reasonably than all of the potential fields, the problem of gauge redundancy is bypassed. To see this, observe that below a gauge transformation,$$start{aligned} hat{phi } ({textbf {okay}}, t) rightarrow hat{phi } ({textbf {okay}}, t) – frac{partial hat{f} ({textbf {okay}}, t) }{partial t}, quad hat{{textbf {A}}} ({textbf {okay}}, t) rightarrow hat{{textbf {A}}}({textbf {okay}}, t) + i {textbf {okay}} hat{f} ({textbf {okay}}, t), finish{aligned}$$
(10)
for some scalar operate (hat{f} ({textbf {okay}}, t)). Since the vector potential change happens within the ({textbf {okay}}) path then,$$start{aligned} hat{{textbf {A}}}_parallel ({textbf {okay}}, t) rightarrow hat{{textbf {A}}}_parallel ({textbf {okay}}, t) + i {textbf {okay}} hat{f} ({textbf {okay}}, t), quad hat{{textbf {A}}}_perp ({textbf {okay}}, t) rightarrow hat{{textbf {A}}}_perp ({textbf {okay}}, t). finish{aligned}$$
(11)
Hence, through the use of simply (hat{textbf{A}}_perp (textbf{okay}, t)) there isn’t any want to decide on a specific gauge. The gauge invariance of (hat{{textbf {A}}}_perp ({textbf {okay}}, t)) is well-known in works involving the quantization of the EM field41,48,49 and the spin-orbit decomposition of gauge fields50.Care have to be taken, although, if one have been to vary inertial reference frames as (hat{textbf{A}}_perp (textbf{okay}, t) ), in contrast to the gauge-dependent (A^mu = (phi /c, textbf{A})), just isn’t a Lorentz 4-vector. For our functions, which is discovering environment friendly and correct computational strategies for accelerator physics, this isn’t a severe subject.The remainder of this text is a proof-of-concept software of FoHM-NO for predicting the EM fields generated by relativistically charged particle beams. However, it’s relevant to any EM setting the place (rho ) and (textbf{J}) are given, though it may also be prolonged to incorporate the matter density fields within the predictions if the equations of state are used. While FoHM-NO could usually not carry out as precisely as mature, state-of-the-art EM codes, using neural networks right here offers it the benefit of getting a really quick runtime. This may be helpful in circumstances the place velocity is important, equivalent to accelerator management and diagnostics settings, the place beam time may be fairly restricted. Alternatively, it may be used for exploratory analyses to search out approximate options earlier than bringing in additional correct, however slower, strategies for refinement.

https://www.nature.com/articles/s41598-024-65650-9