# Automated Machine Learning Strategies for Multi-Parameter Optimisation of a Caesium-Based Portable Zero-Field Magnetometer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Set-Up

#### 2.2. Hanle Resonance

#### Machine Learning

#### 2.3. Optimisation Techniques

#### 2.3.1. Evolutionary Algorithms

#### 2.3.2. Gradient Ascent

#### 2.3.3. Gaussian Process Regression

#### 2.4. Parameters

## 3. Results

- Scheme 1.
- Cost = ${C}_{1}$, $M=3$
- Scheme 2.
- Cost = ${C}_{1}$, $M=5$
- Scheme 3.
- Cost = ${C}_{2}$, $M=3$
- Scheme 4.
- Cost = ${C}_{2}$, $M=5$

- Scheme 1.
- ${C}_{1}$. All MLAs converged at $2.5\pm 1$ mV/nT, equating to a measured sensitivity of 163 $\pm \phantom{\rule{3.33333pt}{0ex}}20\phantom{\rule{3.33333pt}{0ex}}\mathrm{fT}/\sqrt{\mathrm{Hz}}$.
- Scheme 2.
- ${C}_{1}$. All MLAs converged at $4.4\pm 0.4$ mV/nT, equating to a measured sensitivity of 147 $\pm \phantom{\rule{3.33333pt}{0ex}}11\phantom{\rule{3.33333pt}{0ex}}\mathrm{fT}/\sqrt{\mathrm{Hz}}$.
- Scheme 3.
- ${C}_{2}$. All MLAs converged at a measured sensitivity of 163 $\pm \phantom{\rule{3.33333pt}{0ex}}15\phantom{\rule{3.33333pt}{0ex}}\mathrm{fT}/\sqrt{\mathrm{Hz}}$, equating to a demodulated gradient of $2.2\pm 0.15$ mV/nT.
- Scheme 4.
- ${C}_{2}$. All MLAs converged at a measured sensitivity of 132 $\pm \phantom{\rule{3.33333pt}{0ex}}23\phantom{\rule{3.33333pt}{0ex}}\mathrm{fT}/\sqrt{\mathrm{Hz}}$, equating to a demodulated gradient of $2.8\pm 0.9$ mV/nT.

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Experimental setup. Elliptically polarised light from a distributed Bragg reflector (DBR) laser close to the $F=4\to {F}^{\prime}=3$ hyperfine transition of the Cs ${D}_{1}$ line is fibre-coupled to pass through a micro-fabricated atomic vapour cell [25,26] filled with a saturated vapour of Cs and 211 Torr of nitrogen buffer gas. The cell is heated through resistive heating by square-wave modulated current provided by a custom high efficiency heater driver. Three pairs of biplanar coils, ${B}_{x}$, ${B}_{y}$, ${B}_{z}$, control the static magnetic field along each axis, and an additional modulation coil, ${B}_{RF}$, allows the application of an oscillating field along the y-axis. The static field coils are driven using a custom low-noise current driver [27]. The photodetector (PD) measures light transmitted through the vapour cell. A low nT-level magnetic field environment is provided by a 5-layer $\mu $-metal shield. $\lambda /4$, quarter waveplate; Cs, caesium vapour cell; ADC, analog-to-digital converter; DAC, digital-to-analog converter.

**Figure 2.**(

**a**) (top), Hanle resonances, showing transmission, A, across the transverse axis as a function of longitudinal magnetic field; far detuned from zero-field (light line) to zero field, ${B}_{{z}_{0}}$ (darkest line). (

**a**) (bottom), Hanle resonances, showing transmission, A, across the longitudinal axis as a function of the transverse magnetic field; far detuned from zero-field (light line) to zero field, ${B}_{{x}_{0}}$ or ${B}_{{y}_{0}}$ (darkest line). (

**b**), Hanle resonances across two axes. The transverse and longitudinal magnetic fields, ${B}_{x}\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}{B}_{z}$, are swept across the x- and z-axes to generate a 2D landscape of the Hanle resonance. Colour indicates the measured light transmission amplitude (A) on the photodetector, normalised with respect to the maximum (1) and minimum (0) transmission. (

**c**), modulation of the magnetic field is applied across the y-axis as the transverse field, ${B}_{y}$, is swept from ${B}_{{y}_{MIN}}$ to ${B}_{{y}_{MAX}}$. The resultant photodiode signal is demodulated and the demodulated amplitude with respect to (${B}_{y}$) is shown by the black solid line, the linear sensing region is shown by the red dashed line.

**Figure 3.**Two evolutionary algorithm processes. (

**a**,

**b**) share evolutionary elements of initial population formation, selection, crossover and mutation. For both algorithms, the initial population $X\left(t\right)$ contains a population of N sets of parameter settings. The colour indicates each set of parameter settings. t, generation or loop number; $t=t+1$, the next generation; and $C\left(t\right)$, measured cost. Both algorithms repeat until the end condition is met, where the number of sets of parameters tested N is equal to 250 (${N}_{end}$). (

**a**) Genetic algorithm (GA) process. The initial population is generated and evaluated for cost, with individual costs denoted as ${C}_{i}$. $\frac{N}{2}$ parameter sets are selected for the next generation based on ranked cost. The best performing $\frac{N}{2}$ are used as “parents” to produce “children” sets during crossover with respect to the crossover point. Mutation of individual parameter values randomly occurs in the new population. (

**b**) Differential evolution (DE) process. The initial population is generated and evaluated for cost where three random sets ${X}_{a}$, ${X}_{b}$ & ${X}_{c}$ and a target set ${X}_{T}$ are selected. A new set V is created during mutation from the randomly selected sets, and used in a crossover with the target set to make a new set Q. ${C}_{Q}$, the cost of Q, is evaluated and measured against ${C}_{T}$, the cost of the target set. The target set is replaced in a new generation if ${C}_{Q}>{C}_{T}$ (for ${C}_{1}$) or ${C}_{Q}<{C}_{T}$ (for ${C}_{2}$).

**Figure 4.**Gradient ascent algorithm process. $x\left(i\right)$, a vector value for a single parameter ${x}_{i}$ ranging from minimum ${x}_{i}^{min}$ to maximum ${x}_{i}^{max}$ as defined by parameter space range. i, the individual parameter selected. Initially, the first parameter is selected for the first batch $i=1$. All other parameters are kept constant. The batch is evaluated based on cost, indicated in green, to find where the gradient tends to zero, $\frac{\partial C\left(\rho \right)}{\partial x}\to 0$ indicated in red. The corresponding parameter value ${x}^{opt}$ is then set for this parameter for the next batch, $i=i+1$. This continues until all parameters are used as batches, for a total number of parameters M. The segmented graph shows this process as a function of the run number. This process in turn repeats until the end condition is met, where the number of sets of parameters tested N is equal to 250 (${N}_{end}$).

**Figure 5.**All figure parts contain the following optimisation techniques, gradient descent algorithm in green, genetic algorithm in blue and Gaussian process regression model in pink. M, the number of parameters optimised. Row 1 & 3, (

**a**–

**c**,

**g**–

**i**), optimisation of 3 parameters ($M=3$). Row 2 & 4, (

**d**–

**f**,

**j**–

**l**) optimisation of 5 parameters ($M=5$). Row 1 & 2, (

**a**–

**f**), optimise for maximising cost function ${C}_{1}$ the demodulated line shape gradient (mV/nT). Row 3 & 4, (

**g**–

**l**), optimise for minimising cost function ${C}_{2}$, calculated sensitivity ($\mathrm{T}/\sqrt{\mathrm{Hz}}$). Column 1 “Optimisation”, (

**a**,

**d**,

**g**,

**j**), show Cost function as a function of run number. The solid line indicates the moving maximum per optimisation technique. Column 2 “Sensitivity”, (

**b**,

**e**,

**h**,

**k**), shows corresponding FFT for the optimal parameters found per optimisation technique. Sensitivity is shown as a function frequency (Hz), raw data are shown by solid lines. The frequency band of interest (5 to 20 Hz) is highlighted in grey. Averaged sensitivity in this band is shown by the dashed line (value represented in the key). Column 3, “Demodulation”, (

**c**,

**f**,

**i**,

**l**), shows a corresponding demodulated line shape for the optimal parameters found per optimisation technique.

**Figure 6.**Data and models resulting from the Gaussian process regression model MLA, from a 5 parameter optimisation scheme ($M=5$). The 5 parameters optimised are cell temperature (T), laser power (LP), laser detuning (LD), modulation amplitude factor (${A}_{Mod}$) and modulation frequency factor (${F}_{Mod}$). Each part shows a parameter as a function of the cost. Row 1, (

**a**–

**e**), shows optimisation for cost function ${C}_{1}$, the demodulated line shape gradient (mV/nT). Row 2, (

**f**–

**j**) optimisation for cost function ${C}_{2}$, calculated sensitivity ($\mathrm{T}/\sqrt{\mathrm{Hz}}$). Marks indicate measured values from optimisation, solid line indicates the Gaussian process predicted cost-landscape and shaded region indicates the model provided $95\%$ confidence interval of the cost-landscape.

**Table 1.**Definition of all controlled parameters (p) used for optimisation, with corresponding units. Min (p), the minimum value for each parameter. Max (p), the maximum value for each parameter. Default (p), chosen default value if parameter is not directly optimised during optimisation.

Parameter | Min (p) | Max (p) | Default (p) | Unit |
---|---|---|---|---|

Temperature | 115 | 140 | - | ${}^{\circ}$C |

Laser Power | 0.5 | 6 | - | mW |

Laser Detuning | −20 | 20 | - | GHz |

${A}_{Mod}$ | 0.2 | 1.5 | 0.5 | dimensionless |

${F}_{Mod}$ | 0.2 | 1.5 | 1 | dimensionless |

**Table 2.**Optimal parameters found for the following optimisation techniques, Genetic Algorithm (GA), Gradient Descent algorithm (GD) and Gaussian process (GP). The number of parameters tested, M, is specified for each optimisation run. T, cell temperature (${}^{\circ}\mathrm{C}$). LP, laser power (mW). LD, laser detuning (GHz). ${A}_{Mod}$, modulation amplitude factor (dimensionless). ${F}_{Mod}$, modulation frequency factor (dimensionless). ${m}_{i}$, modulation index (dimensionless). $C\left(\rho \right)$ defines the cost function implemented. ${C}_{1}$ is the demodulated lineshape gradient (mV/nT), with uncertainty taken as the geometric standard deviation across the frequency band of interest. ${C}_{2}$ is the calculated sensitivity ($\mathrm{fT}/\sqrt{\mathrm{Hz}}$), with uncertainty taken as the linear fitting error across demodulated linear region. $\mathrm{\Gamma}$ is the full-width at half-maximum (FWHM) of the magnetic resonance (nT), with uncertainty taken as the fit error to Equation (2). Values in grey indicate parameters that were not optimised during operation.

MLA | M | $\mathit{C}\left(\mathit{\rho}\right)$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | $\mathrm{\Gamma}$ | T | LD | LP | ${\mathit{A}}_{\mathit{Mod}}$ | ${\mathit{F}}_{\mathit{Mod}}$ | ${\mathit{m}}_{\mathit{i}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

GD | 3 | ${C}_{1}$ | 2.82 ± 0.03 | 158.62 ± 1.3 | 132.51 ± 1.5 | 119.41 | 8.24 | 6.00 | 0.50 | 1.00 | 0.55 |

GA | 3 | ${C}_{1}$ | 2.59 ± 0.02 | 182.39 ± 1.4 | 183.27 ± 2.1 | 115.00 | 3.00 | 5.35 | 0.50 | 1.00 | 0.55 |

GP | 3 | ${C}_{1}$ | 3.50 ± 0.03 | 143.40 ± 1.2 | 168.83 ± 1.6 | 115.00 | 8.00 | 6.00 | 0.50 | 1.00 | 0.55 |

GD | 5 | ${C}_{1}$ | 4.04 ± 0.02 | 150.24 ± 1.5 | 130.06 ± 2.1 | 118.85 | 10.77 | 5.58 | 1.50 | 0.30 | 5.51 |

GA | 5 | ${C}_{1}$ | 4.23 ± 0.02 | 157.62 ± 1.3 | 98.81 ± 2.0 | 123.00 | 7.00 | 5.32 | 1.48 | 0.39 | 4.21 |

GP | 5 | ${C}_{1}$ | 4.75 ± 0.03 | 136.30 ± 1.2 | 147.36 ± 1.2 | 120.13 | 6.22 | 6.00 | 1.50 | 0.21 | 7.82 |

GD | 3 | ${C}_{2}$ | 2.10 ± 0.02 | 148.28 ± 1.3 | 143.36 ± 2.5 | 117.94 | 5.88 | 5.35 | 0.50 | 1.00 | 0.55 |

GA | 3 | ${C}_{2}$ | 2.35 ± 0.02 | 152.30 ± 1.3 | 136.66 ± 1.3 | 119.00 | 4.00 | 4.66 | 0.50 | 1.00 | 0.55 |

GP | 3 | ${C}_{2}$ | 2.31 ± 0.02 | 177.40 ± 1.3 | 192.81 ± 1.6 | 115.01 | 3.49 | 5.57 | 0.50 | 1.00 | 0.55 |

GD | 5 | ${C}_{2}$ | 2.22 ± 0.03 | 109.59 ± 1.3 | 137.70 ± 1.6 | 118.85 | 7.69 | 5.58 | 0.70 | 0.80 | 0.96 |

GA | 5 | ${C}_{2}$ | 1.95 ± 0.02 | 119.76 ± 1.2 | 111.05 ± 2.1 | 121.00 | 7.00 | 5.24 | 0.97 | 1.15 | 0.93 |

GP | 5 | ${C}_{2}$ | 3.65 ± 0.02 | 154.81 ± 1.2 | 203.05 ± 1.6 | 115.00 | 3.00 | 5.50 | 1.09 | 1.00 | 1.20 |

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## Share and Cite

**MDPI and ACS Style**

Dawson, R.; O’Dwyer, C.; Irwin, E.; Mrozowski, M.S.; Hunter, D.; Ingleby, S.; Riis, E.; Griffin, P.F.
Automated Machine Learning Strategies for Multi-Parameter Optimisation of a Caesium-Based Portable Zero-Field Magnetometer. *Sensors* **2023**, *23*, 4007.
https://doi.org/10.3390/s23084007

**AMA Style**

Dawson R, O’Dwyer C, Irwin E, Mrozowski MS, Hunter D, Ingleby S, Riis E, Griffin PF.
Automated Machine Learning Strategies for Multi-Parameter Optimisation of a Caesium-Based Portable Zero-Field Magnetometer. *Sensors*. 2023; 23(8):4007.
https://doi.org/10.3390/s23084007

**Chicago/Turabian Style**

Dawson, Rach, Carolyn O’Dwyer, Edward Irwin, Marcin S. Mrozowski, Dominic Hunter, Stuart Ingleby, Erling Riis, and Paul F. Griffin.
2023. "Automated Machine Learning Strategies for Multi-Parameter Optimisation of a Caesium-Based Portable Zero-Field Magnetometer" *Sensors* 23, no. 8: 4007.
https://doi.org/10.3390/s23084007