Genetic algorithm enhanced by machine learning in dynamic aperture optimization

With the aid of machine learning techniques, the genetic algorithm has been enhanced and applied to the multi-objective optimization problem presented by the dynamic aperture of the National Synchrotron Light Source II (NSLS-II) Storage Ring. During the evolution processes employed by the genetic algorithm, the population is classified into different clusters in the search space. The clusters with top average fitness are given “elite” status. Intervention on the population is implemented by repopulating some potentially competitive candidates based on the experience learned from the accumulated data. These candidates replace randomly selected candidates among the original data pool. The average fitness of the population is therefore improved while diversity is not lost. Maintaining diversity ensures that the optimization is global rather than local. The quality of the population increases and produces more competitive descendants accelerating the evolution process significantly. When identifying the distribution of optimal candidates, they appear to be located in isolated islands within the search space. Some of these optimal candidates have been experimentally confirmed at the NSLS-II storage ring. The machine learning techniques that exploit the genetic algorithm can also be used in other population-based optimization problems such as particle swarm algorithm.

Figure 1

Schematic illustration of intervention using machine learning techniques. Here, a search space of two free variables is assumed. The distribution of the original population is shown in the subplot (a). The candidates are classified into three colored clusters with the K-means algorithm in the subplot (b). The average fitness of each cluster has been evaluated, sorted, and given a status labeled with “Best (elite),” “Good,” and “Poor” respectively in the subplot (c). In the subplot (d), some potential competitive candidates (marked as the magenta dots) are repopulated inside the range of the “Best (elite)” cluster and then are used to replace the same amount of candidates from the original data pool. After the replacement, the post-population densities of the “Good” and the “Poor” cluster become low. In reality, there may not exist obvious boundaries to separate each cluster and cluster classification is not unique either.

Figure 2

The root means squared (rms) spread of action from a constant is used as an optimization objective. The dashed circle represents a constant linear action at different angles. The dots are the normalized turn-by-turn coordinates.

Figure 3

Five initial particle coordinates in the $x-y$ plane with their conjugate momenta $p_{x,y}=0$ used for tracking. The turn-by-turn data are used to evaluate the spread of their linear actions. The dashed line is the size of the desired dynamic aperture. The 5th particle is chosen beyond the desired dynamic aperture in order to obtain a safe margin. The choice of the initial coordinates is not arbitrary. It may depend on the local optics functions, and physical aperture, etc.

Figure 4

Comparison of the evolution of average fitness with and without machine learning for 135 generations. Without machine learning, the evolution process can sometimes stop, or even regress. On the other hand, the fitness convergence becomes faster and steadier with the introduction of machine learning.

Figure 5

Variation of the ranges for generating new elite population in six-dimensional search space (sextupole’s $K_2$) along the evolution. The elite ranges fluctuate, but gradually converge toward much narrower ranges.

Figure 6

Relative distances of six neighbors from one elite candidate, which is used as the reference here in the search space. The reference sextupole settings are given as $K_{2,SH1}=26.20891{\mathrm{m}}^{-3}$, $K_{2,SH3}=-17.87664{\mathrm{m}}^{-3}$, $K_{2,SH4}=-6.39466{\mathrm{m}}^{-3}$, $K_{2,SL3}=-22.42607{\mathrm{m}}^{-3}$, $K_{2,SL2}=28.54735{\mathrm{m}}^{-3}$, $K_{2,SL1}=-0.22496{\mathrm{m}}^{-3}$.

Figure 7

Simulated multiturn trajectories in the phase space (left: the horizontal plane, right: the vertical plane) for five different initial conditions. The maximum spread for the initial condition $x=20\mathrm{mm}$, $p_x=0\mathrm{mrad}$and $y=3\mathrm{mm}$, $p_y=0\mathrm{mrad}$ is around 3%.

Figure 8

Dynamic aperture for on-momentum particles. The color represents the tune diffusion obtained by turn-by-turn tracking simulation. Diffusion [37] is defined as the difference of tunes $\mathrm{\Delta }\nu$ extracted from the different time durations $\mathrm{Diff}={\log }_{10}\sqrt{\mathrm{\Delta }{\nu }_x^2+\mathrm{\Delta }{\nu }_y^2}$. A cool color means the motion is less chaotic and vice versa.

Figure 9

Frequency map corresponding to the on-momentum dynamic aperture in the $x–y$ planes. A large tune-shift-with-amplitude is observed in this lattice. The third order resonance line can be crossed stably (without obvious diffusion).

Figure 10

Frequency map of energy (momentum) acceptance in the $x-\delta$ planes. Here $\delta =\frac{\mathrm{\Delta }p}{p}$ is the particles’ relative momentum deviation.

Figure 11

Simulated turn-by-turn trajectories in phase space. A large tune-shift-with-amplitude coefficient shifts the tune away from the third order resonance quickly when the betatron amplitude is slightly off. The stop-band width for this lattice is narrow, which means particle motion is stable even if its tune sits on the resonance.

Figure 12

Left: measured beam turn-by-turn trajectories in the phase space with two neighboring beam position monitors (BPM). Three isolated islands are observed with a phase advance of $\frac{2\pi }{3}$ in-between as expected. Right: the FFT spectrum confirms that the beam remains stable on the third order resonance.

Figure 13

Measured closed orbit at ${\nu }_x=1/3$ with BPM turn-by-turn data. The periodicity of the closed orbit became $1/3$ rather than 1. In other words, the closed orbit repeated itself every 3 turns.