Bayesian optimization experiment for trajectory alignment at the low energy RHIC electron cooling system

The low energy RHIC electron cooling (LEReC) system is the world’s first electron cooler utilizing radio frequency (rf) accelerated electron bunches, and a nonmagnetized electron beam. It is also the first electron cooler applied directly to colliding hadron beams. The unique approach to cooling makes beam dynamics in LEReC very different from the conventional electron coolers. Numerous LEReC parameters can affect the cooling rate. One of the most critical factors is the alignment of the electron and ion trajectories in the cooling section. In this work, we apply Bayesian optimization to check and if needed to optimize the trajectories’ alignment. Experimental results are presented and it is demonstrated that machine learning (ML) methods can be applied to perform the control tasks effectively in the RHIC controls system.

Figure 1

LEReC system layout.

Figure 2

Bayesian optimization process.

Figure 3

The simulation considers a real-world scenario where electrons travel through the cooling sections and are monitored by 8 BPMs.

Figure 4

Comparisons of rms and std values of BPM readings from the random samples (blue) and Bayesian samples (red). The Bayesian samples clearly have both rms and std values closer to 0 compared with the random samples. It indicates that the algorithm learns a way to optimize the cooling rate by reducing both rms and std values of electron trajectories, which matches our expectations.

Figure 5

The statistical distribution shows that the Bayesian samples (red) have a larger percentage of populations that possess a higher cooling rate compared with the random samples (blue).

Figure 6

40 training points are sampled using Algorithm lg2. The inputs step through the entire range (top) which helps to reveal a clear pattern of the outputs (bottom). As we can see, input positions around 0 potentially have higher objectives.

Figure 7

An example period of transverse ion beam size data are fetched from the real system during the experiment. It shows the data are noisy.

Figure 8

Demonstration of the volatility of the original objective function. Due to the use of point $\delta$ values in $\lambda =(1/\delta )(d\delta /dt)$ and the noise in $\delta$, the original objective varies too fast (bottom) even when the algorithm converges to an optimum strategy (top, e.g., from step 7 to 11 or 13 to 17). This eventually makes the algorithm diverge (after step 17) as the algorithm does not have enough time to learn properly the correlation between the inputs and outputs.

Figure 9

Effects of different radius values on the algorithm’s behaviors. For a small radius $r=10$ (left column), the objective function is still sensitive to changes (e.g., sample 10 to 14). But compared with the original objective (see Fig. 8), the process has become more stable (e.g., sample 7 to 10). For a large radius $r=30$ (right column), the objective function becomes too insensitive. It is unable to change properly according to the inputs. As the inputs step through the entire range (top right), the objective values do not change with a proper scale (bottom right). Both cases should be avoided as they do not provide correct system information for the algorithm to learn.

Figure 10

The final experiment is performed by using a radius of 15. The outputs is not too sensitive to the noise but also sensitive enough to respond to the input changes properly. The first 40 points are the training samples. After training, the algorithm converges very quickly (already reaches a close neighborhood after the first 3 steps, top plot) to an optimum solution, which corresponds to the center position of $(x=0,y=0)$.

Figure 11

After training, the algorithm is used to control the electron beam trajectories. As we can see, the algorithm tunes the electrons back to the center and maintains it there, which in turns verifies the correctness of the beam-based alignment method.

Figure 12

20 training points are sampled using Algorithm lg2. Both the data-informed (DI) method and the physics model-informed (PMI) method use the same training data.

Figure 13

Comparison of results from the optimization performed by the data-informed (DI, top left) and by the physics model-informed BO (PMI, top right). Both methods converge to the optimum solution very quickly. Moreover, the PMI method produces a more steady solution (top row), and hence a higher level of objective values (bottom row).

Figure 14

20 training samples with a sinusoidal context (purple line) added. The objective value is now dependent on both the inputs and the context. Compared with Fig. 12, the objective’s pattern is more volatile.

Figure 15

Physics model-informed BO is used to optimize the contextual objective function. Without the CGP feature (top left), the BO is unable to converge as it cannot correctly model the relationship between the inputs and outputs. Whereas with the CGP modeling the context (top right), the BO successfully handles the dynamic environmental factors and is able to find the optimum solution.