Neural Networks as Effective Surrogate Models of Radio-Frequency Quadrupole Particle Accelerator Simulations

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Neural Networks as Effective Surrogate Models of

Radio-Frequency Quadrupole Particle Accelerator

Simulations

Joshua Villarreal1, Daniel Winklehner1, Daniel Koser1Janet

M. Conrad1

1Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA

02139, USA

E-mail: villaj@mit.edu

Abstract. Radio-Frequency Quadrupoles (RFQs) are multi-purpose linear particle

accelerators that simultaneously bunch and accelerate charged particle beams. They

are ubiquitous in accelerator physics, especially as injectors to higher-energy machines,

owing to their impressive efﬁciency. The design and optimization of these devices can

be lengthy due to the need to repeatedly perform high-ﬁdelity simulations. Several

recent papers have demonstrated that machine learning can be used to build surrogate

models (fast-executing replacements of computationally costly beam simulations) for

order-of-magnitude computing time speedups. However, while these pilot studies

are encouraging, there is room to improve their predictive accuracy. Particularly,

beam summary statistics such as emittances (an important ﬁgure of merit in particle

accelerator physics) have historically been challenging to predict. For the ﬁrst time,

we present a surrogate model trained on 200,000 samples that yields <6 % mean

average percent error for the predictions of all relevant beam output parameters from

deﬁning RFQ design parameters, solving the problem of poor emittance predictions by

identifying and including hidden variables which were not accounted for previously.

These surrogate models were made possible by using the Julia language and GPU

computing; we brieﬂy discuss both. We demonstrate the utility of surrogate modeling

by performing a multi-objective optimization using our best model as a callback in

the objective function to select an optimal RFQ design. We consider trade-offs in RFQ

performance for various choices of Pareto-optimal design variables—common issues

for any multi-objective optimization scheme. Lastly, we make recommendations for

input data preparation, selection, and neural network architectures that pave the way

for future development of production-capable surrogate models for RFQs and other

particle accelerators.

arXiv:2210.11451v2 [physics.comp-ph] 15 Mar 2024

Surrogate Modeling of Radio-Frequency Quadrupole Particle Accelerators 2

1. Introduction

The replacement of highly accurate, but computationally costly, particle-in-cell

simulations with surrogate models (sometimes called virtual accelerators) is a topical

ﬁeld of increased interest [1, 2, 3, 4, 5, 6]. Surrogate models use machine learning

(ML) to create fast-executing virtual representations of a complex system like a

particle accelerator. We can then use this surrogate model to, for instance, speed

up (multi-objective) design optimization, or obtain real-time feedback during the

commissioning, tuning, and running of the particle accelerator. The surrogate model is

typically built from a neural network (NN) or some other statistical learning technique

(like polynomial chaos expansion [2]). In the case of using the surrogate model as an

autonomous tuning tool, training data can be obtained not just from simulations, but

also by measurements from existing hardware [7].

The design of the IsoDAR (isotope decay-at-rest) project [8, 9, 10], a planned

experiment in neutrino physics, is the primary motivation for this work. In IsoDAR,

a compact particle accelerator produces a 10 mA proton beam that impinges with

an energy of 60 MeV/amu on a beryllium target surrounded by lithium-7, producing

electron antineutrinos (¯νe) with a well-understood energy distribution through isotope

decay-at-rest [10] (as opposed to other experiments using decay-in-ﬂight). The ¯νe

can then be measured in a nearby liquid scintillator detector via inverse beta decay

(IBD). This conﬁguration yields unprecedented sensitivity to so-called sterile neutrinos,

hypothesized new particles thought to resolve ¯νedeﬁcits observed at experiments

worldwide [11, 12, 13, 14]. The requirements for IsoDAR are 10 mA of protons on

target in a continuous wave beam at 80% duty factor to produce about 1.15 ·1023 ¯νe

over the course of 5 years. Paired with the planned 2.3 kiloton liquid scintillator

detector (LSC) [15] in Korea, this will yield 1.67 million IBD events in the detector.

The IsoDAR particle accelerator comprises an ion source, radio-frequency

quadrupole (RFQ), and a cyclotron [16, 17]. Surrogate modeling has proven

invaluable to IsoDAR’s development, allowing us to demonstrate the robustness of

IsoDAR’s cyclotron design [17] through uncertainty quantiﬁcation [2], and to perform

a small pilot study to investigate the use of surrogate models for RFQs [18].

In this paper, we expand upon work presented in Ref. [18] to build a neural

network-based surrogate model of an RFQ, but rethink the RFQ parametrization to

account for collinear effects in the feature space, physical RFQ design constraints, and

incorporate variables previously hidden from trained surrogate models. We use these

insights to generate an accurate surrogate model for a 32.8 MHz RFQ covering a wide

design parameter space (subject to physical design constraints). We use the highly

efﬁcient Julia programming language [19] to train our NNs with a widely cast net for

hyperparameter tuning and an unusually large batch size.

The working principle of an RFQ and the generation of training data for the

surrogate model were discussed in detail in Ref. [18]. We brieﬂy summarize them in

Sec. 1.1, and Sec. 2.1, respectively. In Sec. 2, we discuss our methods, including data

Surrogate Modeling of Radio-Frequency Quadrupole Particle Accelerators 3

Figure 1. A single RFQ cell with electric ﬁelds in quasi-static approximation. Left:

Front view (beam and z-axis point into the paper). The green arrows indicate the

focusing of the charged particle beam and the red arrows the defocusing. Right: Side

view (beam propagates from left to right). Cell geometry parameters are shown, and

the electric ﬁeld is indicated by arrows. From Ref. [18].

preparation and how we enhance the predictive accuracy of beam summary statistics

outputs like the beam emittance (an important ﬁgure of merit for beam quality),

followed by our results for training of several NNs and optimization of the RFQ in

Sec. 3. Finally, we discuss our results, general observations, and recommendations for

future development of NN-powered surrogate models in Sec. 4- 5. Code relevant to

this project is available on GitHub [20].

1.1. The Radio-Frequency Quadrupole

An RFQ is a multi-purpose linear particle accelerator element able to bunch (compress

along the direction of movement) and accelerate a high-current ion beam, while

keeping the beam tightly focused in the transverse direction [21, 22, 23]. In an RFQ,

an oscillating electric ﬁeld is generated between four vanes (or rods). The arrangement

for one cell can be seen in Fig. 1, where the length is β·λ/2, with βthe ratio of

the beam velocity to the speed of light, and λthe wavelength corresponding to the

electromagnetic wave driving the RFQ. Typical RFQs comprise tens to hundreds of

such cells, each deﬁned by three main parameters (the focusing strength B, the phase

Φ, and the modulation m). The IsoDAR RFQ is discussed in detail elsewhere [24, 25].

To optimize an RFQ, the parameters of each cell have to be ﬁne-tuned to yield

desirable beam output qualities like a high beam transmission. In this paper, we

call the RFQ parameters design variables (DVARs) and the beam output parameters

objectives (OBJs).

1.2. The Julia Language

Julia is an open-source dynamically typed programming language with signiﬁcant

performance improvements over other languages common in scientiﬁc computing like

Python and Matlab [19]. Julia’s computational efﬁciency is especially apparent when

performing costly calculations like neural network training, motivating our use of the

Surrogate Modeling of Radio-Frequency Quadrupole Particle Accelerators 4

Figure 2. The three main cell parameters (focusing function B, phase Φ, and

modulation m), parametrized along the length of the RFQ by fourteen DVARs

(DVAR14, not shown, is the design energy determining the overall length). From

Ref. [18].

language throughout this analysis and allowing for straightforward implementation of

multi-threading and distributed computing to facilitate the completion of expensive

computational tasks. In addition, Julia has built-in support for GPU programming

using CUDA.jl [26, 27], making the language an obvious choice for building and

training series of NNs on both local (CPU) and remote (GPU) machines.

2. Methodology

2.1. Data Generation

In this study, we reuse the dataset from Ref. [18], consisting of 217,292 samples,

each representing a randomly conﬁgured RFQ with corresponding output beam

parameters obtained from beam dynamics simulations using the well-established code

PARMTEQM [28]. The beam dynamics characteristics of an RFQ are fully described

by a set of three main cell parameters (Bn,Φnand mn) for each RFQ cell n, resulting

in a total of 3ndesign variables. While historically the LANL Four-Section Procedure

(FSP) has been used as a design strategy since the very ﬁrst proof-of-principle RFQs

at LANL at the end of the 1970s [22], over the years, modiﬁed approaches have

evolved to improve RFQ performance for speciﬁc applications. In the FSP, the RFQ

is divided into four sections with dedicated functions: a radial matcher, a shaper, a

gentle buncher and an accelerator, with the overall aim of smoothly matching the

beam to the structure, bunching it, and accelerating it with as little losses as possible.

As a ﬁrst approach for the IsoDAR RFQ, a baseline design was created via an

adaptation of the FPS framework, but with special characteristics: (1) in addition to

the typical linear increase of the synchronous phase Φand the modulation min the

shaper (called the “linear shaper”), a section of exponential increase is introduced

(called the “exponential shaper”); (2) in the gentle buncher, not only the modulation

but also the synchronous phase is ramped up exponentially (called the “exponential

Surrogate Modeling of Radio-Frequency Quadrupole Particle Accelerators 5

Label Variable Lower Bound*Upper Bound Physical Meaning

DVAR1 Bmax [1] 8.5 12.0(Constant) focusing strength

DVAR2 mX1 [cm] 5 140 End position of linear shaper

DVAR3 mX2 [cm] DVAR2 + 10 160 End position of exponential shaper

DVAR4 mY1 [1] 1.005 1.7Modulation at end of linear shaper

DVAR5 mY2 [1] DVAR4 + 0.05 1.85 Modulation at end of exponential shaper

DVAR6 mtau1 [cm] 1 500 Exp. param. for modulation in exp. shaper

DVAR7 mtau2 [cm] 1 500 Exp. param. for modulation in exp. buncher

DVAR8 PhiY1 [deg] −89.95 −30 Synchronous phase at end of lin. shaper

DVAR9 PhiY2 [deg] DVAR8 + 2.5−25 Synchronous phase at end of exp. shaper

DVAR10 Phitau1 [cm] 1 500 Exp. param. for sync. phase in exp. shaper

DVAR11 Phitau2 [cm] 1 500 Exp. param. for sync. phase in exp. buncher

DVAR12 mY3ref [1] DVAR5 + 0.05 2.0Modulation at end of exp. buncher

DVAR13 PhiY3ref [deg] DVAR9 + 2.5−20 Synchronous phase at end of exp. buncher

DVAR14 Eref [MeV] 0.055 0.075 Design energy (at exit)

OBJ1 Transmission [%]

OBJ2 Output energy [MeV] (Eout)

OBJ3 RFQ length [cm]

OBJ4 Longitudinal emittance [MeV deg] (ϵlong)

OBJ5 x-emittance [cm mrad] (ϵx)

OBJ6 y-emittance [cm mrad] (ϵy)

Table 1. Summary of design variables (DVARs) and objectives (OBJs) used for

surrogate model training. *In uniformly drawing random RFQ design variables, certain

relationships between these parameters must be satisﬁed to ensure the RFQ is physical.

This is further discussed in Sec. 2.2.

buncher”); and (3) the accelerator section is omitted due to the IsoDAR RFQ being

intended for use as a dedicated pre-buncher.

For a machine learning based design optimization, a reduction of the number of

the initially 3ndesign variables was pursued and thus a parametrization according

to Fig. 2 was performed. This allowed to capture all crucial characteristics of the

design functions while reducing the number of design variables to 14. While DVAR1

corresponds to the value of the (constant) focusing strength and DVAR2 and DVAR3 set

the lengths of the linear and exponential shaper, the total slope and smoothness of

the shaping/bunching effect are characterized by DVAR4–DVAR13.DVAR14 speciﬁes the

design energy at the RFQ exit, which determines the length of the RFQ.

Besides the 14 design variables (DVARs), the dataset from Ref. [18] contains

6 objectives (OBJs): beam transmission, output energy, RFQ length, and three

beam emittances (one longitudinal, two transverse to the beamline). This data is

summarized in Tab. 1.

2.2. Data Preprocessing

Unlike Ref. [18], we perform transformations on the data set, as some variables have

values that directly affect values of others. In particular, for some upper bounds ui

and buffers δi,i∈ {3,5,9,12,13}, after a subset of the 14 DVARs for the jth sample are

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NeuralNetworksasEffectiveSurrogateModelsofRadio-FrequencyQuadrupoleParticleAcceleratorSimulationsJoshuaVillarreal1,DanielWinklehner1,DanielKoser1JanetM.Conrad11MassachusettsInstituteofTechnology,77MassachusettsAve,Cambridge,MA02139,USAE-mail:villaj@mit.eduAbstract.Radio-FrequencyQuadrupoles(RFQs)are...

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Neural Networks as Effective Surrogate Models of Radio-Frequency Quadrupole Particle Accelerator Simulations

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