Vertical Fixed-Field Alternating Gradient (vFFA) accelerators exhibit particle orbits which move vertically during acceleration. This recently rediscovered circular accelerator type has several advantages over conventional ring accelerators, such as zero momentum compaction factor. At the same time, inherently non-planar orbits and a unique transverse coupling make controlling the beam dynamics a complex task. In general, betatron tune adjustment is crucial to avoid resonances, particularly when space charge effects are present. Due to highly nonlinear magnetic fields in the vFFA, it remains a challenging task to determine an optimal lattice design in terms of maximising the dynamic aperture.
					This contribution describes a deep learning based algorithm which strongly improves on regular grid scans and random search to find an optimal lattice: a surrogate model is built iteratively from simulations with varying lattice parameters to predict the dynamic aperture. The training of the model follows an active learning paradigm, which thus considerably reduces the number of samples needed from the computationally expensive simulations

Andrian, Ivan

Assmann, Ralph

Bisoffi, Giovanni

Fabris, Alessandro

Hirlaender, Simon

Lagrange, Jean-Baptiste

McIntosh, Peter

Oeftiger, Adrian

Santamaria Garcia, Andrea

Vinicola, Giulia

English

KITopen

ACTIVE DEEP LEARNING FOR NONLINEAR OPTICS DESIGN OF AVERTICAL FFA ACCELERATORA. Oeftiger∗, GSI Helmholtz Centre, Darmstadt, GermanyA. Santamaria Garcia, Karlsruhe Institute of Technology (KIT), Karlsruhe, GermanyJ.-B. Lagrange, ISIS Department, RAL, STFC, UKS. Hirlaender, Paris Lodron Universität Salzburg, Salzburg, AustriaAbstractVertical Fixed-Field Alternating Gradient (vFFA) accel-erators exhibit particle orbits which move vertically duringacceleration. This recently rediscovered circular accelera-tor type has several advantages over conventional ring ac-celerators, such as zero momentum compaction factor. Atthe same time, inherently non-planar orbits and a uniquetransverse coupling make controlling the beam dynamics acomplex task. In general, betatron tune adjustment is cru-cial to avoid resonances, particularly when space chargeeffects are present. Due to highly nonlinear magnetic fieldsin the vFFA, it remains a challenging task to determine anoptimal lattice design in terms of maximising the dynamicaperture. This contribution describes a deep learning basedalgorithm which strongly improves on regular grid scans andrandom search to find an optimal lattice: a surrogate modelis built iteratively from simulations with varying lattice pa-rameters to predict the dynamic aperture. The training ofthe model follows an active learning paradigm, which thusconsiderably reduces the number of samples needed fromthe computationally expensive simulations.OVERVIEWThe concept of a vertical FFA (vertical Fixed Field al-ternating gradient Accelerator) [1] has recently gained pop-ularity [2]. In this type of circular accelerator, the beamsmove vertically when accelerated as can be seen in Fig. 1. Incontrast, orbits in the original FFA remain in the horizontalplane. Vertical FFAs are characterised by strong transversecoupling and highly nonlinear magnetic fields. The designof such lattices was so far carried out in a trial-and-errorfashion [3], and it remains challenging to optimise them.The goal of this study is to efficiently explore the latticeparameter space in order to identify a lattice design with max-imum dynamic aperture (DA). In such higher-dimensionalproblems, an exhaustive grid scan can easily become imprac-tical and a mere random selection too sparse to resolve the im-portant regions. Data-driven approaches lend themselves ascomputationally economical and rewarding solutions. Thispaper guides the exploration by iterative supervised learning,which is also referred to as active learning [4]. Since manylattice configurations do not even yield a closed orbit (CO)around the accelerator, the domain of valid CO needs to beidentified before searching for maximum DA.∗ a.oeftiger@gsi.deThe present study was staged in three steps. First, thelattice parameter space was randomly sampled to obtainan initial set of lattices which possess a CO. In a secondstep, a classification algorithm was trained on predicting COexistence for a given lattice configuration. Rejection sam-pling based on the predicted probability then significantlyimproved the efficiency of gathering more samples with avalid CO. The third and last stage qualifies the DA across thevalid CO domain: an uncertainty-aware surrogate model wasinitially trained to predict the DA, and was later re-trainedwith additional simulations where predictions were mostuncertain. With this approach, several interesting parameterregions could be identified where particular lattices yield alarger DA compared to previous studies based on grid scans.MACHINE AND SIMULATION MODELThe vertical beam excursion of a vFFA has several advan-tages. The momentum compaction factor is zero becauseof the vanishing radial dispersion function, which would beadvantageous e.g. for acceleration at ultra-relativistic ener-gies in a muon collider, a neutrino factory, or a relativisticcyclotron. It would also vertically separate the light raysof different radiation spectra in a light source. However,these benefits come at the cost of coupling the particle mo-tion between the two transverse planes [5]. The verticalfield gradient also implies a transversely oscillating closedorbit, making it challenging to derive essential acceleratorparameters and optics other than by numerical modelling.In addition to this, the scaling condition requires magneticfields that increase exponentially in the vertical direction𝑧 as 𝐵𝑥,𝑦,𝑧 ∝ 𝐵0 exp(𝑚𝑧) to obtain zero-chromatic opera-tion, leading to highly nonlinear magnetic fields. The DAis a critical parameter in high-power machines to avoid un-Figure 1: Closed orbit in a reference vFFA triplet lattice.controlled beam loss, and numerical simulations have beenconducted to investigate it in vFFAs in Ref. [3]. The layoutof the vFFA lattice is similar in our study with a ring madeof 10 FDF triplet cells with an average radius of 4.45 m. Therectangular F and D magnets have identical lengths of 50 cm.Preliminary hardware studies have shown that tanh fringefields used previously [3] do not properly fit first magnetprototypes, so the fringe field model here is of the form𝑓 (𝑦) = 1𝜋 (arctan (𝑦 − 𝑦ent𝜆 ) − arctan (𝑦 − 𝑦ex𝜆 )) , (1)where 𝑦 is the longitudinal coordinate, 𝑦en and 𝑦ex the effec-tive entrance and exit boundaries of the magnet, respectively,and 𝜆 the characteristic fringe field extent.Analogous to the previous study in Ref. [3], the following5 lattice parameters are varied in this study: the field atthe centre of the F magnet (𝐵0𝑓) and D magnet (𝐵0𝑑) at thereference altitude 𝑧 = 0, the normalised field gradient 𝑚,the radial displacement between F and D magnet 𝑥𝑠, andthe tilt angle 𝑡𝑓 of F magnets with respect to the D magnet(cf. the sketch on the right of Fig. 1). The characteristicfringe field extent 𝜆 is fixed at 15 cm, and the extrapolationoff the magnetic median plane is truncated at the 10th order.The neighbouring triplet cells are included in the simulationdue to overlapping fringe fields in the long straight section.Simulations were carried out with the particle tracking codeFixField [6].DOMAIN OF VALID CLOSED ORBITSSteps 1 and 2 of the study identify the domain ofvalid COs in the 5-dimensional parameter space. Due tothe nonlinear potential, a lattice described by the 5-tuple(𝐵0𝑓, 𝐵0𝑑, 𝑚, 𝑥𝑠, 𝑡𝑓) might not feature a CO. In this case thetracked particles either turn backwards, get lost in the far-located collimators1, or even get quasi-trapped2. Figure 1shows CO projections for various particle momenta in a refer-ence vFFA triplet lattice (−1 T, 1.15 T, 1.31 m−1, 0, 0). Theparticle used in the present study is a 3 MeV proton. At thecentre of the long straight section (the beginning of a tripletcell), the CO of the reference lattice is located radially (hor-izontally) at 𝑥𝑐𝑜 = 4.36 m and vertically at 𝑧𝑐𝑜 = −0.73 m.Owing to the symmetry, the corresponding transverse mo-menta vanish.In step 1, an initial data set is generated via uniform ran-dom sampling of the 5D parameter space to generate 20 000lattice configurations. Simulations computed valid COs for429 lattices i.e., 2% success rate for random sampling. Step2 involves a classification algorithm which is trained to pre-dict the probability for a new random lattice configurationto possess a valid CO. Samples from the uniform randomdistribution can then be rejected based on probability limits.A new set of 10 000 lattices is readily generated using therejection sampling, and simulations are run to determinethe CO. With each iteration of 10 000 lattices, the classifier1 limiting to radial 2 m < 𝑥 < 6 m and vertical −10 m < 𝑧 < 10 m.2 i.e. reaching a maximum number of integration steps ≫ than for the CO.improves in prediction quality as the data set increases. Theemployed rejection strategy first gathered more data by ex-cluding lattices below a 30% probability of having a CO,then resolved the domain boundary in finer detail by accept-ing lattices between 30% and 70%, and eventually focusedon gathering more CO by accepting only lattices above a90% probability. Two initial iterations used Gaussian NaiveBayes classification [7], resulting in ≈ 10% valid COs. How-ever, the emerging rather complex domain shape in the 5Dspace was not well captured by the classifier. Changing torandom forest classification [7] resulted in reproducing thedomain shape in much more detail, which in turn tremen-dously improved the fraction of valid COs. Approximately85% of valid COs were obtained when rejecting latticesbelow 90% of predicted probability.Proceeding with the rejection sampling stage, a total of170 000 lattices was simulated, with more than a quarterpossessing a valid CO. Figure 2 presents an overview of thedomain of valid CO (after step 3) along with a histogram ofall simulated lattices in 2D projections of the 5D parameterspace. The clear overlap of the highest number of COs withthe highest number of launched simulations show that theclassifier-based rejection sampling successfully guided theparameter space exploration to the most interesting regions.2.0 1.5 1.0 0.5 0.0B0f [T]0.00.51.01.52.0B 0d [T] 110002000#lattices2.0 1.5 1.0 0.5 0.0B0d/B0f0.81.01.21.41.6m [1/m]110002000#lattices2.0 1.5 1.0 0.5 0.0B0d/B0f5.02.50.02.55.0x s [cm]125005000#lattices2.0 1.5 1.0 0.5 0.0B0d/B0f505t f [deg]15001000#latticesFigure 2: Distribution of identified closed orbits in 5D lat-tice parameter space. The number of lattices simulated perbin is shown in grey, where non-white bins contain at leastone simulation. The red contours show the kernel densityestimation for the distribution of valid closed orbits.DYNAMIC APERTURE QUALIFICATIONStep 3 continues the exploration of lattice space by mov-ing from fast CO simulations ( ≈ 5 min) to slow DA simula-tions (≈ 2 × 3 h). Ref. [3] establishes a useful approach todetermine the dynamic aperture (DA) of a vFFA lattice in de-coupled transverse space, (𝑢, 𝑝𝑢, 𝑣, 𝑝𝑣): particles are trackedfor 1000 lattice cells starting from an initial amplitude, onceby scanning an initial 𝑢𝑖 as (𝑢𝑖, 0, 0, 0) and likewise for 𝑣𝑖as (0, 0, 𝑣𝑖, 0). The amplitude is considered unstable if theparticle gets lost, turns backwards, or is quasi-trapped dur-ing the tracking. More than 1000 lattice periods are notrequired as the DA estimate remained almost identical. Thepresent paper builds on this approach. For a given latticeconfiguration, a DA amplitude is identified using the divide-and-conquer algorithm separately for both decoupled planes,𝑢𝑖 and 𝑣𝑖. Thirteen recursions of interval bisection yield a2−13 ≈ 10−4 accuracy. This estimate is then confirmed byprobing the interval from zero in ten equidistant steps toexclude the existence of smaller unstable amplitudes.An initial DA data set is evaluated for the previously iden-tified lattices with a valid CO. To continue exploring thelattice parameter space, a surrogate model is built to predictthe two transverse DA values for a given 5D lattice configura-tion. The uncertainty-aware surrogate model class employedhere is based on the ensembling of deep neural networks [8],which is well suited for large and growing data sets (wheree.g. Gaussian processes do not scale well). In our case, fivestructurally equivalent pyramid-shaped networks with threehidden layers of 2048-1024-512 neurons were initialisedwith different random seeds and then trained independentlyon the data. The Adam algorithm [9] was used as optimiserand the mean squared error (MSE) as loss function. Thestandard deviation of the prediction for a given sample istaken as estimate for the epistemic uncertainty [4], whichis used to guide the exploration. The devised algorithmiteratively follows these steps:1. Train the model on the existing DA data;2. Draw new rejection-sampled lattice configurationsbased on the trained classifier to ensure a valid CO;3. Predict DA and epistemic uncertainty for new samples;4. Rank new samples w.r.t. their epistemic uncertainty;5. Run DA simulations for selected new samples.Step 3 added three more iterations of 10 000 lattice sam-ples each, which were selected as the top 25% quantile fromthe ranked samples. The RMS prediction error from eachiteration is shown in Fig. 3 against the DA amplitude in de-coupled 𝑢 and 𝑣 spaces. The underlying inference samplesare indicated by the grey histogram. Only a fraction of 0.5%lattices does not feature a CO in this last iteration. Fromthe total of 200 000 simulated lattices, 25% possess a validCO. Figure 4 displays a maximum DA figure3 in the 5Dlattice parameter space projections. With a DA of 5.5 cm thereference lattice lies in the light green area, red indicates thelocation of the ten lattices with the largest DA up to 6.5 cm.CONCLUSIVE REMARKSThe employed active learning approach tremendously im-proved the efficiency of finding closed orbits in the 5D latticeparameter space: from an initial 2% with random samplingto 85% using classifier-based rejection sampling, and finally3 sum of 𝑢 + 𝑣/2 since the 𝛽-function in 𝑣 is about twice as large as in 𝑢.0.00 0.01 0.02 0.03u [m]0.00.20.40.60.8MSEinference sampleRMSE iteration 1RMSE iteration 2RMSE iteration 30.00 0.02 0.04 0.06 0.08v [m]Figure 3: Training results after each active-learning iteration.2.0 1.5 1.0 0.5B0f [T]0.51.01.52.0B 0d [T]2.0 1.5 1.0 0.5B0d/B0f1.01.21.41.6m [1/m]2.0 1.5 1.0 0.5B0d/B0f5.02.50.02.55.0x s [cm]2.0 1.5 1.0 0.5B0d/B0f505t f [deg]0.00 0.01 0.02 0.03 0.04 0.05 0.06Maximum u + v/2 [m] in projected binFigure 4: Simulation results for dynamic aperture (DA) in5D lattice parameter space. Each panel plots the maximumDA per projected bin for the nearly 50 000 valid closed orbits.Red bins contain the top 10 lattices (overall maximum DA).0.00 0.05 0.10 0.15 0.20 0.25|qu|0.000.050.100.150.200.25|qv|150100150200250300350#latticesFigure 5: Histogram of stable lattices in tune space.99.5% via ranking using the deep neural network ensemble.The gathered data set and established surrogate models linklattice parameter space to dynamic aperture as well as tunes(see Fig. 5). This result now allows to guide vFFA latticedesign by choosing a target tune and then obtaining a latticeconfiguration with maximum dynamic aperture: a helpfultool to gain space toward resonances located close by in(decoupled) tune space, in particular in view of intensitylimitations due to space-charge detuning.ACKNOWLEDGEMENTShinji Machida provided inputs and valuable feedback.REFERENCES[1] J. Teichmann, “Accelerators with a vertically increasing field,”J. Nucl. Energy Part C, vol. 5, no. 1, p. 57, 1963.doi:10.1088/0368-3281/5/1/312[2] S. Brooks, “Vertical orbit excursion fixed field alternatinggradient accelerators,” Phys. Rev. ST Accel. Beams, vol. 16,no. 8, p. 084 001, 2013.doi:10.1103/PhysRevSTAB.16.084001[3] S. Machida, D. J. Kelliher, J.-B. Lagrange, and C. T. Rogers,“Optics design of vertical excursion fixed-field alternatinggradient accelerators,” Phys. Rev. Accel. Beams, vol. 24,p. 021 601, 2021.doi:10.1103/PhysRevAccelBeams.24.021601[4] K. P. Murphy, Probabilistic Machine Learning: Advanced Top-ics. MIT Press, 2023. http://probml.github.io/book2[5] M. Vanwelde, C. Hernalsteens, S. A. Bogacz, S. Machida, andN. Pauly, “Review of coupled betatron motion parametriza-tions and applications to strongly coupled lattices,” arXivpreprint, 2022. doi:10.48550/arXiv.2210.11866[6] J.-B. Lagrange, “The Particle Tracking Code Fixfield,” in Proc.IPAC’21, Campinas, Brazil, May 2021, pp. 1905–1906.doi:10.18429/JACoW-IPAC2021-TUPAB209[7] L. Buitinck et al., “API design for machine learning software:Experiences from the scikit-learn project,” in ECML PKDDWorkshop: Languages for Data Mining and Machine Learning,2013, pp. 108–122. doi:10.48550/arXiv.1309.0238[8] B. Lakshminarayanan, A. Pritzel, and C. Blundell, “Simpleand scalable predictive uncertainty estimation using deep en-sembles,” Advances in neural information processing systems,vol. 30, 2017. doi:10.48550/arXiv.1612.01474[9] D. P. Kingma and J. Ba, “Adam: A method for stochastic opti-mization,” arXiv preprint, 2014.doi:10.48550/arXiv.1412.6980

Active deep learning for nonlinear optics design of a vertical FFA accelerator

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Active deep learning for nonlinear optics design of a vertical FFA accelerator

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