Multi-objective and multi-delity Bayesian optimization of laser-plasma acceleration F. Irshad1S. Karsch1 2and A. D opp1 2 1Ludwig-Maximilian-Universit at M unchen Am Coulombwall 1 85748 Garching Germany

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Multi-objective and multi-ﬁdelity Bayesian optimization of laser-plasma acceleration

F. Irshad,1S. Karsch,1, 2 and A. D¨opp1, 2

1Ludwig-Maximilian-Universit¨at M¨unchen, Am Coulombwall 1, 85748 Garching, Germany

2Max Planck Institut f¨ur Quantenoptik, Hans-Kopfermann-Strasse 1, Garching 85748, Germany

Beam parameter optimization in accelerators involves multiple, sometimes competing objectives.

Condensing these individual objectives into a single ﬁgure of merit unavoidably results in a bias

towards particular outcomes, in absence of prior knowledge often in a non-desired way. Finding an

optimal objective deﬁnition then requires operators to iterate over many possible objective weights

and deﬁnitions, a process that can take many times longer than the optimization itself. A more

versatile approach is multi-objective optimization, which establishes the trade-oﬀ curve or Pareto

front between objectives. Here we present the ﬁrst results on multi-objective Bayesian optimiza-

tion of a simulated laser-plasma accelerator. We ﬁnd that multi-objective optimization reaches

comparable performance to its single-objective counterparts while allowing for instant evaluation of

entirely new objectives. This dramatically reduces the time required to ﬁnd appropriate objective

deﬁnitions for new problems. Additionally, our multi-objective, multi-ﬁdelity method reduces the

time required for an optimization run by an order of magnitude. It does so by dynamically choos-

ing simulation resolution and box size, requiring fewer slow and expensive simulations as it learns

about the Pareto-optimal solutions from fast low-resolution runs. The techniques demonstrated in

this paper can easily be translated into many diﬀerent computational and experimental use cases

beyond accelerator optimization.

I. INTRODUCTION

Laser-plasma interaction [1,2] and in particular its

sub-ﬁeld of laser-plasma acceleration [3,4] are highly

researched areas with prospects for numerous scientiﬁc

and societal applications [5,6]. Until the past decade,

both experimental and numerical investigations in these

ﬁelds were often based on single or a few laser shots

and particle-in-cell simulations [7], respectively. Since

then, improvements in laser technology as well as com-

puting hard- and software have made it possible to gather

data for hundreds or thousands of diﬀerent conﬁgura-

tions in both experiments and simulations [8–10]. This

has sparked interest in using advanced techniques from

computer science, particularly machine learning meth-

ods, which can deal more eﬃciently with large multi-

dimensional data sets than human operators [11].

Early examples include the use of genetic algorithms

[12,13] and, more recently, the ﬁrst measurements using

surrogate models have been presented [14,15]. The latter

are intermediate models that are generated based on ex-

isting data during optimization and that can be quickly

explored numerically. Studies involving this Bayesian

optimization have demonstrated clear optimization of a

carefully chosen optimization goal. Importantly, this goal

has to be encoded in form of a so-called objective func-

tion, which acts on the measurement and gives a scalar

output. In the case of a particle accelerator, the beam can

generally be described by the charge distribution ρ(~x, ~p)

in the six-dimensional phase space, and an objective func-

tion that optimizes beam parameters will act on this dis-

tribution or a subset of it. One of the simplest examples

of an objective function is the charge objective function

gQ(ρ(~x, ~p)) = Zρ(~x, ~p)d~xd~p.

While this function can in principle be used as an ob-

jective function in a particle accelerator, it will usually

not yield a useful optimization result. This is because

it optimizes solely the charge and all other beam pa-

rameters such as divergence and energy are lost in the

integration process. In fact, due to energy conservation,

this optimizer tends to reduce the beam energy, which

is an unintended consequence in almost all conceivable

applications of particle accelerators.

In practice, one usually uses a combination of objec-

tives, e.g. reaching a certain charge above a certain en-

ergy or the total beam energy. The design of objective

functions for these problems is even more diﬃcult be-

cause they need to give some constraints or limits to

the single objectives. Many multi-objective scalarizations

take the form of a weighted product g=Qgαi

ior sum

g=Pαigiof the individual objectives giwith the hyper-

parameter αidescribing its weight. For instance, Jalas

et al. [15] optimized the spectrum of a laser-accelerated

beam using an objective function that combines the beam

charge Q, the median energy ˜

Eand the median absolute

deviation ∆ ˜

E. Their proposed objective function to be

maximized is √Q˜

E/∆˜

E, i.e. the exponential weights are

α1= 0.5, α2= 1 and α3=−1. Here, the use of median-

based metrics will result in less sensitivity to outliers in

the spectrum, while the weight parameter α1= 0.5 ex-

plicitly reduces the relevance of charge compared to beam

energy and spread.

The choice of particular weights is, however, entirely

empirical and usually the result of trial and error. An ob-

jective function is thus not necessarily aligned with the

actual optimization goal, and one often needs to manu-

ally adjust the parameters of the objective function over

multiple optimization runs. In essence, instead of scan-

ning the input parameters of an experiment or simula-

tion, the human operator will be scanning hyperparame-

arXiv:2210.03484v2 [physics.acc-ph] 23 Dec 2022

ters of the objective function many times over. The less

prior knowledge about the system is known, the longer

this process may take.

The underlying problem is essentially one of compres-

sion, i.e. that the objective function needs to reduce a

complex distribution function to a single number char-

acterizing said distribution. It is impossible to do this

without information loss for an unknown distribution

function. In fact, even if we knew the distribution, e.g.

a normal distribution, one would still need both mean

and variance to describe it without ambiguities. In the

case of an unknown one-dimensional distribution func-

tion, we can use multiple statistical descriptions to cap-

ture essential features of the distribution such as the cen-

tral tendency (weighted arithmetic or truncated mean,

the median, mode, percentiles, etc.) and the statistical

dispersion of the distribution (full width at half max-

imum, median absolute deviation, standard deviation,

maximum deviation, etc.). These measures weigh dif-

ferent features in the distribution diﬀerently. One may

also include higher-order features such as the skewness,

which occurs for instance as a sign of beam loading in en-

ergy spectra of laser-plasma accelerators [8], or coupling

terms between the diﬀerent parameters. Last, the am-

plitude or integral of the distribution function are often

parameters of interest [16].

In the following, we will discuss optimizations of elec-

tron energy spectra according to diﬀerent objective def-

initions and then present a more general multi-objective

optimization.

The paper is structured as follows: First, we are going

to discuss details of the simulated laser-plasma accelera-

tor used for our numerical experiments (Section II) and

introduce Bayesian optimization (Section III). Then we

present results from optimization runs using diﬀerent def-

initions of scalarized objectives that aim for beams with

high charge and low energy spread at a certain target en-

ergy (Section IV). We then compare these results with an

optimization using eﬀective hypervolume optimization of

all objectives (Section V). In Section VI we discuss some

of the physics that the optimizer ’discovers’ during opti-

mization and in the last section, we summarize our results

and outline perspectives for future research (Section VII).

II. LASER-PLASMA ACCELERATOR

As a test system for optimization, we use an example

from the realm of plasma-based acceleration, i.e. a laser

wakeﬁeld accelerator with electron injection in a sharp

density downramp [8,17]. The basic scenario here is that

electrons get trapped in a laser-driven plasma wave due

to a local reduction in the plasma density, which is of-

ten realized experimentally as a transition from one side

to the other of a hydrodynamic shock, hence the often-

used name ”shock injection”. The number of electrons

injected at this density transition strongly depends on

the laser parameters at the moment of injection, but also

°1 0 1 2 3

Position z[mm]

0.0

0.5

1.0

1.5

Plasma density [ne]

ldown

lup

FIG. 1. Illustration of the four variable input parame-

ters from Table I, namely the upramp length lup, the down-

ramp length ldown, the plateau density neand the focus po-

sition z0.

on the plasma density itself. Both parameters also aﬀect

the ﬁnal energy spectrum the electrons exhibit at the end

of the acceleration process. Here we will use simulations

to investigate this system, the primary reason being that

they are perfectly reproducible and do not require addi-

tional handling of jitter, drifts, and noise. However, the

methods outlined in this paper are equally relevant to

experiments. The input space consists of four variable

parameters, namely the plateau plasma density, the po-

sition of laser focus, as well as the lengths of the up- and

downramps of the plasma density close to the density

transition.

While the shock injection scenario is suﬃciently com-

plex to require particle-in-cell codes, we use the code

FBPIC by Lehe et al. [18] in conjunction with various

optimizations to achieve an hour-scale run-time. On the

hardware side, the code is optimized to run on NVIDIA

GPUs (here we used Tesla V100 or RTX3090), while the

physical model includes optimizations such as the usage

of a cylindrical geometry with Fourier decomposition in

the angular direction and boosted-frame moving windows

[19]. Additionally, we can take advantage of the very lo-

calized injection to locally increase the macro-particles

density in the injection area [8]. Similarly, the linear

wakeﬁelds forming in regions of lower laser intensity re-

sult in a nearly laminar ﬂow of particles, meaning that

we can decrease the macro-particle density far away from

the laser axis [20].

One particular challenge that arises in simulations over

a large range of parameters is that diﬀerent input param-

eters may result in diﬀerent computational requirements.

For instance, the transverse box size needs to be several

times larger than the beam waist to assure that the en-

ergy of a focusing beam is not lost. Hence, a laser that is

initialized out of focus requires a larger box size than a

beam initialized in focus. We address this by scaling the

transverse box size lras a function of the laser waist w(z)

at the beginning of the simulation. Similarly, the size of

the wakeﬁeld depends on the plasma density, and accord-

ingly, we scale the longitudinal size lzof the box with the

estimated wakeﬁeld size. By using these adapted simu-

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Multi-objectiveandmulti-delityBayesianoptimizationoflaser-plasmaaccelerationF.Irshad,1S.Karsch,1,2andA.Dopp1,21Ludwig-Maximilian-UniversitatMunchen,AmCoulombwall1,85748Garching,Germany2MaxPlanckInstitutfurQuantenoptik,Hans-Kopfermann-Strasse1,Garching85748,GermanyBeamparameteroptimizationinacce...

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Multi-objective and multi-delity Bayesian optimization of laser-plasma acceleration F. Irshad1S. Karsch1 2and A. D opp1 2 1Ludwig-Maximilian-Universit at M unchen Am Coulombwall 1 85748 Garching Germany

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