Applying Bayesian Optimization to Achieve Optimum Cooling at the Low Energy RHIC Electron Cooling System

11/09/2021

Yuan Gao, Weijian (Lucy) Lin, Kevin Brown, Xiaofeng Gu,

Georg Hoffstaetter, John Morris, Sergei Seletskiy

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Relativistic Heavy Ion Collider

• Two 3.8 km counter-rotating supper-conducting rings;
• 6 Interaction Regions (IR);

LEReC

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• LEReC is used to increase the luminosity, it was successfully improved the luminosity multifold in 2020 and 2021 runs.

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• 704 MHz e-bunches (grouped into 9 MHz macro-bunches) are produced from the photocathode and accelerated in the SRF cavity to the designed energy (1.6 MeV, 2 MeV).

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• Those e-bunches are delivered to the cooling sections (20 meter), where they co-travel with ion bunches.

LEReC System Overview

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• BPM Measurement errors;
• An independent way to optimize the cooling performance.

Method

• Bayesian Optimization (BO): a powerful tool for finding the extrema of objective functions that are expensive to evaluate;
• It is called Bayesian because it uses the famous “Bayes’ theorem”

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Motivations

Expensive function

Surrogate model

Acquisition function sampling

Update

Output

Criterion met?

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Simulation Settings

• A LEReC simulator is used to output the transverse cooling rate, taking the BPMs as the inputs.

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• Ions are assumed at the center position (x=0, y=0).

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• To get a quick idea of the optimization, BO takes only 2 parameters as the inputs, rms and std of the electron positions.

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Simulation Results

• An electron trajectory that has more overlapping with the ion beam produce faster cooling rate.

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Experiment Settings

Goal

correctors

BPMs

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• Input: 4 BPMs, each has a range of [-3, 3] mm;
• Objective: cooling rate;
• 40 initial samples, go through the entire range randomly;
• The objective exhibits a pattern, it favors input positions around 0.

Initial Sampling

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Optimization Strategy in the Presence of Noise

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Smoothing by Moving Average

Number

Number

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• Results are generated using an average window of 15 points;

Results

Number

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• The algorithm can tune electrons from the farthest positions to the center and maintain the trajectories.

Electron Positions Controlled by BO

• Only picked first 4 BPMs as input parameters due to limited beam time

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• Real system: 16 BPMs in two cooling sections, other possible system parameters that effect cooling rate

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• algorithm that converges faster and consider more information/parameters of the real system
• Physics-informed Gaussian Process Optimization
• Contextual Gaussian Process Optimization

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Future Work

Goal

• A probability distribution over possible functions that fit a set of points
• Mean function + Covariance function

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Gaussian Process

Kernel

[Brochu et al, 2010]

• Guide how input space should be explored during optimization
• Combine predicted mean and variance from Gaussian Process model
• Probability Improvement (PI)
• Expected Improvement (EI)
• Upper Confidence Bound (UCB)

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Acquisition Function

[Brochu et al, 2010]

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Physics-informed Gaussian Process Optimization

[Hanuka et al, 2020]

Physics-informed Gaussian Process Optimization

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• Test function: 4-dimensional Gaussian-like function centered at the origin
• Both data-informed GP and physics-informed GP converges
• Physics-informed is faster and more stable

Result comparison: Physics-informed GP

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Contextual Gaussian Process (CGP) Optimization

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[Krause & Ong, 2021]

• Test function: sum of 4-dimensional Gaussian-like function centered at zero and periodic sinusoidal function
• 20 initial samples are used to train the algorithms
• Without CGP: GP model has no knowledge of the sinusoidal function, takes 4-dimensional inputs
• With CGP: GP model takes 5-dimensional inputs, kernel is the sum of input kernel and context kernel

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Test CGP: objective function with context

• Without CGP: algorithm unable to converge due to varying context
• With CGP: algorithm converges in 7 steps with 20 initial training samples

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Result comparison: Contextual GP

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• The BO method is very effective in optimizing the cooling performance at LEReC

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• It also verifies the correctness of the traditional orbit correction program and the BPM calibrations

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• It opens many possibilities of trying different machine learning methods on optimizing performance for control tasks in the RHIC complex, as well as the future EIC
• i.e. Coherent electron Cooling (CeC) experiment at RHIC

Conclusion & Outlook

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Conclusion & Outlook

• Machine learning for improving CeC operations

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• [1] Y. Gao, W. Lin, et al., Applying Bayesian Optimization to Achieve Optimum Cooling at the Low Energy RHIC Electron Cooling System, Manuscript submitted to Phys. Rev. Accel. Beams, Sept. 2021.
• [2] E. Brochu, V. M. Cora, and N. de Freitas, A Tutorial on Bayesian Optimization of Expensive Cost Functions, with Application to Active User Modeling and Hierarchical Reinforcement Learning (2010), arXiv:1012.2599.
• [3] A. Hanuka, X. Huang, J. Shtalenkova, et al., Physics model-informed gaussian process for online optimization of particle accelerators, Phys. Rev. Accel. Beams 24, 072802 (2021).
• [4] A. Krause and C. Ong, Contextual gaussian process bandit optimization, in Advances in Neural Information Processing Systems (NIPS), Vol. 24, edited by J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and K. Q.Weinberger (Curran Associates, Inc., 2011).

References