# Applications of Machine Learning and Neural Networks for FT-ICR Mass Measurements with SIPT

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{85}Rb

^{+}dataset is presented, suggesting that SIPT is sensitive to single-ion signals. Lastly, the implications for future experiments are discussed.

## 1. Introduction

^{100}Sn and

^{78}Ni which are of current interest.

## 2. Background

#### 2.1. Ion Motion in a Penning Trap

#### 2.2. Fourier Transform Ion Cyclotron Resonance Technique

## 3. Numerical Simulation

#### 3.1. Noise Floor Determination

#### 3.2. Simulated Signal-to-Noise Ratio

^{100}Sn

^{+}and

^{78}Ni

^{+}, SIPT is expected to achieve SNR’s of roughly $5.5/\sqrt{\Delta \nu}{\mathrm{Hz}}^{1/2}$. The spectral bandwidth is left undetermined as it is limited by the measurement sample rate and acquisition time. A more precise measurement will yield a narrower resonance for the eigenfrequency being probed, which improves the SNR as a narrower spectral bandwidth can be used.

#### 3.3. Methods for Simulating FT-ICR Signals

Algorithm 1: Serial method for simulating an FT-ICR time-domain signal using the ‘sum of sines’ approach. The inputs are as follows: ${N}_{\mathrm{ions}}$ is the number of ions in the Penning trap; f and ${\sigma}_{f}$ define the ion eigenfrequency normal distribution; A and ${\sigma}_{A}$ define the single-ion amplitude normal distribution; ${\varphi}_{0,\mathrm{max}}$ is the maximum eigenmotion phase; ${A}_{\mathrm{noise}}$ is the noise amplitude; $\Delta t$, ${N}_{\mathrm{samples}}$ are the sample time and number of samples, respectively. The method UniformRandom (low, high, N = 1) samples N instances from a uniformly random distribution within the range defined from low (inclusive) to high (exclusive), and NormalRandom ($\mu ,\sigma $) randomly samples a Gaussian/normal distribution defined by a mean value $\mu $ and standard deviation $\sigma $. |

^{100}Sn

^{+}and

^{78}Ni

^{+}, the approximate single-ion signal amplitude is ${A}_{\mathrm{single}}$∼ 0.02 mV. This is used as a point of reference when determining a reasonable amplitude for

^{85}Rb, for which a large dataset of signals have been collected. Furthermore, combining these relationships provides a correlation between the relative ion radius in the Penning trap to the simulated amplitude for a single ion:

## 4. Identification of Ion Signals

#### 4.1. Dataset Construction for Signal Classifiers

#### 4.2. Optimal Classifier Hyperparameters

#### 4.3. Classification of Ion Signals

^{100}Sn

^{+}is expected to have a precision of about 60–70% for single ions.

^{100}Sn

^{+}was studied: simulations of 1000 background and single-ion signals each were classified, and the frequency extracted from the fitted Lorentzian distribution. The average recovered frequency agreed with the simulation input within the prediction uncertainty. The potential for adding additional systematic and statistical uncertainties remains a topic for further investigation when performing high-precision measurements.

## 5. Extracting Ion Characteristics with Neural Networks

#### 5.1. Construction of Datasets for Predicting Ion Characteristics

#### 5.2. Network Architecture and Training

#### 5.3. Model Uncertainty and Sensitivity Analysis

## 6. Applications to Experimental Data

^{85}Rb

^{+}ions produced by the LEBIT offline test ion sources. A brief description of the dataset is presented here; further details regarding the LEBIT experimental facility and methods for collecting these data can be found in [36]. Before injecting ions into the SIPT, a 0.25 mm collimator placed within the drift tube before the trap was used to minimize spread in the ions’ radii. The time-domain signal was collected at 24,000 points, each 5 μs apart, for a total of 120 ms. A total of 33,297 signals were collected while the rubidium source was active, and an additional 18,000 signals were collected while no ions were allowed into the trap. The ion source was operated at low rates such that the majority of ion signals were likely from single ions. Following the experiment, and based on the findings of Section 4.1, each time-domain signal was transformed to the frequency domain using an FFT; a zero-padding of twice the length of the time-domain set was appended to both sides before taking the FFT. The frequency data were then truncated to $\pm 25$ Hz of the expected eigenfrequency.

#### 6.1. Classification of Experimental Signals

#### 6.2. Predicted Ion Characteristics

^{85}Rb

^{+}signals produced by different numbers of ions in SIPT are calculated: $79\left(5\right)\%$ correspond to background, $18\left(4\right)\%$ are single-ion measurements, and $2.3\left(1.1\right)\%$ are measurements of at least two ions.

## 7. Conclusions

^{100}Sn

^{+}the classifiers were shown to have accuracies of ≈65%, which are sufficient to successfully perform a mass measurement.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Brown, B.A.; Richter, W.A. New “USD” Hamiltonians for the sd shell. Phys. Rev. C
**2006**, 74, 034315. [Google Scholar] [CrossRef] - Schatz, H.; Ong, W. Dependence of X-ray Burst Models on Nuclear Masses. Astrophys. J.
**2016**, 844, 139. [Google Scholar] [CrossRef] - Lunney, D.; Pearson, J.M.; Thibault, C. Recent trends in the determination of nuclear masses. Rev. Mod. Phys.
**2003**, 75, 1021–1082. [Google Scholar] [CrossRef] - Burbidge, E.M.; Burbidge, G.R.; Fowler, W.A.; Hoyle, F. Synthesis of the Elements in Stars. Rev. Mod. Phys.
**1957**, 29, 547–650. [Google Scholar] [CrossRef] - Mumpower, M.; Surman, R.; McLaughlin, G.; Aprahamian, A. The impact of individual nuclear properties on r-process nucleosynthesis. Prog. Part. Nucl. Phys.
**2016**, 86, 86–126. [Google Scholar] [CrossRef] - Baumann, T.; Hausmann, M.; Sherrill, B.; Tarasov, O. Opportunities for isotope discoveries at FRIB. Nucl. Instruments Methods Phys. Res. Sect. B Beam Interact. Mater. Atoms
**2016**, 376, 33–34. [Google Scholar] [CrossRef] - Blaum, K. High-accuracy mass spectrometry with stored ions. Phys. Rep.
**2006**, 425, 1–78. [Google Scholar] [CrossRef] - Brown, L.; Gabrielse, G. Geonium Theory: Physics of a single electron or ion in a penning trap. Rev. Mod. Phys.
**1986**, 58, 233. [Google Scholar] [CrossRef] - Bollen, G.; Moore, R.; Savard, G.; Stolenzberg, H. The accuracy of heavy-ion mass measurements using time of flight-ion cyclotron resonance in a Penning trap. J. Appl. Phys.
**1990**, 68, 4355. [Google Scholar] [CrossRef] - Eliseev, S.; Blaum, K.; Block, M.; Dörr, A.; Droese, C.; Eronen, T.; Goncharov, M.; Höcker, M.; Ketter, J.; Ramirez, E.M.; et al. A phase-imaging technique for cyclotron-frequency measurements. Appl. Phys. B
**2014**, 114, 107–128. [Google Scholar] [CrossRef] - Haxel, O.; Jensen, J.H.D.; Suess, H.E. On the “Magic Numbers” in Nuclear Structure. Phys. Rev.
**1949**, 75, 1766. [Google Scholar] [CrossRef] - Mayer, M.G. On Closed Shells in Nuclei. Phys. Rev.
**1948**, 74, 235–239. [Google Scholar] [CrossRef] - Marshall, A.; Hendrickson, C.; Jackson, G. Fourier transform ion cyclotron resonance mass spectrometry: A primer. Mass Spectrom. Rev.
**1998**, 17, 1–35. [Google Scholar] [CrossRef] - Kotsiantis, S.B.; Zaharakis, I.D.; Pintelas, P.E. Machine learning: A review of classification and combining techniques. Artif. Intell. Rev.
**2006**, 26, 159–190. [Google Scholar] [CrossRef] - Schmidhuber, J. Deep learning in neural networks: An overview. Neural Netw.
**2015**, 61, 85–117. [Google Scholar] [CrossRef] - Niu, Z.; Liang, H. Nuclear mass predictions based on Bayesian neural network approach with pairing and shell effects. Phys. Lett. B
**2018**, 778, 48–53. [Google Scholar] [CrossRef] - Lovell, A.E.; Mohan, A.T.; Sprouse, T.M.; Mumpower, M.R. Nuclear masses learned from a probabilistic neural network. Phys. Rev. C
**2022**, 106, 014305. [Google Scholar] [CrossRef] - Dong, X.X.; An, R.; Lu, J.X.; Geng, L.S. Novel Bayesian neural network based approach for nuclear charge radii. Phys. Rev. C
**2022**, 105, 014308. [Google Scholar] [CrossRef] - Jiang, W.G.; Hagen, G.; Papenbrock, T. Extrapolation of nuclear structure observables with artificial neural networks. Phys. Rev. C
**2019**, 100, 054326. [Google Scholar] [CrossRef] - Du, Y. Signal Enhancement and Data Mining for Chemical and Biological Samples Using Mass Spectrometry. Ph.D. Thesis, Purdue University, Ann Arbor, MI, USA, 2015. [Google Scholar]
- Nampei, M.; Horikawa, M.; Ishizu, K.; Yamazaki, F.; Yamada, H.; Kahyo, T.; Setou, M. Unsupervised machine learning using an imaging mass spectrometry dataset automatically reassembles grey and white matter. Sci. Rep.
**2019**, 9, 13213. [Google Scholar] [CrossRef] - Williams, D.K.; Kovach, A.L.; Muddiman, D.C.; Hanck, K.W. Utilizing Artificial Neural Networks in MATLAB to Achieve Parts-Per-Billion Mass Measurement Accuracy with a Fourier Transform Ion Cyclotron Resonance Mass Spectrometer. J. Am. Soc. Mass Spectrom.
**2009**, 20, 1303–1310. [Google Scholar] [CrossRef] [PubMed] - Williams, D.K., Jr. Exploring Fundamental Aspects of Proteomic Measurements: Increasing Mass Measurement Accuracy, Streamlining Absolute Quantification, and Increasing Electrospray Response. Ph.D. Thesis, North Carolina State University, Ann Arbor, MI, USA, 2009. [Google Scholar]
- Boiko, D.A.; Kozlov, K.S.; Burykina, J.V.; Ilyushenkova, V.V.; Ananikov, V.P. Fully Automated Unconstrained Analysis of High-Resolution Mass Spectrometry Data with Machine Learning. J. Am. Chem. Soc.
**2022**, 144, 14590–14606. [Google Scholar] [CrossRef] [PubMed] - Nesterenko, D.A.; Eronen, T.; Ge, Z.; Kankainen, A.; Vilen, M. Study of radial motion phase advance during motion excitations in a Penning trap and accuracy of JYFLTRAP mass spectrometer. Eur. Phys. J. A
**2021**, 57, 302. [Google Scholar] [CrossRef] - Jeffries, J.; Barlow, S.; Dunn, G. Theory of space-charge shift of ion cyclotron resonance frequencies. Int. J. Mass Spectrom. Ion Process.
**1983**, 54, 169–187. [Google Scholar] [CrossRef] - Duhamel, P.; Vetterli, M. Fast fourier transforms: A tutorial review and a state of the art. Signal Process.
**1990**, 19, 259–299. [Google Scholar] [CrossRef] - Payne, T.G.; Southam, A.D.; Arvanitis, T.N.; Viant, M.R. A signal filtering method for improved quantification and noise discrimination in fourier transform ion cyclotron resonance mass spectrometry-based metabolomics data. J. Am. Soc. Mass Spectrom.
**2009**, 20, 1087–1095. [Google Scholar] [CrossRef] - Chiron, L.; van Agthoven, M.A.; Kieffer, B.; Rolando, C.; Delsuc, M.A. Efficient denoising algorithms for large experimental datasets and their applications in Fourier transform ion cyclotron resonance mass spectrometry. Proc. Natl. Acad. Sci. USA
**2014**, 111, 1385–1390. [Google Scholar] [CrossRef] - Kanawati, B.; Bader, T.; Wanczek, K.P.; Li, Y.; Schmitt-Kopplin, P. FT-Artifacts and Power-function Resolution Filter in Fourier Transform Mass Spectrometry. Rapid Commun. Mass Spectrom.
**2017**, 31, 1607–1615. [Google Scholar] [CrossRef] - Mathur, R.; O’Connor, P.B. Artifacts in Fourier transform mass spectrometry. Rapid Commun. Mass Spectrom. RCM
**2009**, 23, 523–529. [Google Scholar] [CrossRef] - Comisarow, M.B.; Marshall, A.G. Frequency-sweep fourier transform ion cyclotron resonance spectroscopy. Chem. Phys. Lett.
**1974**, 26, 489–490. [Google Scholar] [CrossRef] - Kilgour, D.P.A.; Wills, R.; Qi, Y.; O’Connor, P.B. Autophaser: An Algorithm for Automated Generation of Absorption Mode Spectra for FT-ICR MS. Anal. Chem.
**2013**, 85, 3903–3911. [Google Scholar] [CrossRef] [PubMed] - Brustkern, A.M.; Rempel, D.L.; Gross, M.L. An electrically compensated trap designed to eighth order for FT-ICR mass spectrometry. J. Am. Soc. Mass Spectrom.
**2008**, 19, 1281–1285. [Google Scholar] [CrossRef] [PubMed] - Lincoln, D.; Baker, R.; Benjamin, A.; Bollen, G.; Redshaw, M.; Ringle, R.; Schwarz, S.; Sonea, A.; Valverde, A. Development of a high-precision Penning trap magnetometer for the LEBIT facility. Int. J. Mass Spectrom.
**2015**, 379, 1–8. [Google Scholar] [CrossRef] - Hamaker, A. Mass Measurement of the Lightweight Self-Conjugate Nucleus Zirconium-80 and the Development of the Single Ion Penning Trap. Ph.D. Thesis, Michigan State University, East Lansing, MI, USA, 2021. [Google Scholar]
- Johnson, J.B. Thermal Agitation of Electricity in Conductors. Phys. Rev.
**1928**, 32, 97–109. [Google Scholar] [CrossRef] - Barry, J.R.; Lee, E.A.; Messerschmitt, D.G. Digital Communication; Springer: New York, NY, USA, 2004. [Google Scholar]
- Marshall, A.G. Theoretical signal-to-noise ratio and mass resolution in Fourier transform ion cyclotron resonance mass spectrometry. Anal. Chem.
**1979**, 51, 1710–1714. [Google Scholar] [CrossRef] - Dahl, D.A. Simion for the personal computer in reflection. Int. J. Mass Spectrom.
**2000**, 200, 3–25. [Google Scholar] [CrossRef] - Hossin, M.; Sulaiman, M.N. A review on evaluation metrics for data classification evaluations. Int. J. Data Min. Knowl. Manag. Process.
**2015**, 5, 1. [Google Scholar] - Refaeilzadeh, P.; Tang, L.; Liu, H. Cross-Validation. In Encyclopedia of Database Systems; Liu, L., Özsu, M.T., Eds.; Springer: Boston, MA, USA, 2009; pp. 532–538. [Google Scholar] [CrossRef]
- Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; et al. Scikit-learn: Machine Learning in Python. J. Mach. Learn. Res.
**2011**, 12, 2825–2830. [Google Scholar] - Guyon, I.; Gunn, S.; Nikravesh, M.; Zadeh, L.A. Feature Extraction: Foundations and Applications; Springer: Berlin/Heidelberg, Germany, 2008; Volume 207. [Google Scholar]
- Pechenizkiy, M. The impact of feature extraction on the performance of a classifier: kNN, Naïve Bayes and C4.5. In Proceedings of the Conference of the Canadian Society for Computational Studies of Intelligence, Victoria, BC, Canada, 9–11 May 2005; Springer: Berlin/Heidelberg, Germany, 2005; pp. 268–279. [Google Scholar]
- Smith, J.O. Mathematics of the Discrete Fourier Transform (DFT): With Audio Applications; W3K Publishing, 2008. [Google Scholar]
- Diamantidis, N.; Karlis, D.; Giakoumakis, E. Unsupervised stratification of cross-validation for accuracy estimation. Artif. Intell.
**2000**, 116, 1–16. [Google Scholar] [CrossRef] - Snoek, J.; Larochelle, H.; Adams, R.P. Practical Bayesian Optimization of Machine Learning Algorithms. In Proceedings of the Advances in Neural Information Processing Systems, Tahoe, NV, USA, 3–6 December 2012; Pereira, F., Burges, C., Bottou, L., Weinberger, K., Eds.; Curran Associates, Inc.: Red Hook, NY, USA, 2012; Volume 25. [Google Scholar]
- Head, T.; MechCoder; Louppe, G.; Shcherbatyi, I.; Fcharras, Z.V.; cmmalone; Schröder, C.; nel215; Campos, N. scikit-optimize v0.5.2, 2018. [CrossRef]
- Svozil, D.; Kvasnicka, V.; Pospichal, J. Introduction to multi-layer feed-forward neural networks. Chemom. Intell. Lab. Syst.
**1997**, 39, 43–62. [Google Scholar] [CrossRef] - Chollet, F. Keras. 2015. Available online: https://keras.io (accessed on 18 May 2022).
- Agarap, A.F. Deep Learning using Rectified Linear Units (ReLU). arXiv
**2018**, arXiv:1803.08375. [Google Scholar] - Srivastava, N.; Hinton, G.; Krizhevsky, A.; Sutskever, I.; Salakhutdinov, R. Dropout: A simple way to prevent neural networks from overfitting. J. Mach. Learn. Res.
**2014**, 15, 1929–1958. [Google Scholar] - Kingma, D.; Ba, J. Adam: A Method for Stochastic Optimization. Int. Conf. Learn. Represent.
**2014**, arXiv:1412.6980. [Google Scholar] [CrossRef] - Gurney, K. An Introduction to Neural Networks; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Gal, Y.; Ghahramani, Z. Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning. In Proceedings of the 33rd International Conference on Machine Learning, New York, NY, USA, 20–22 June 2016; Balcan, M.F., Weinberger, K.Q., Eds.; PMLR: New York, NY, USA, 2016; Volume 48, pp. 1050–1059. [Google Scholar]
- Yeung, D.S.; Cloete, I.; Shi, D.; wY Ng, W. Sensitivity Analysis for Neural Networks; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]

**Figure 1.**Frequency spectra for Fourier transform ion cyclotron resonance signals from the single-ion Penning trap (SIPT) for various measured and simulated

^{85}Rb

^{+}ion signals. The black line indicates the experimental dataset, the dashed red line shows one possible Lorentzian distribution fit to the dataset.

**Top Left**: Example of a measured SIPT signal, likely from real ions, easily distinguished from noise as a strong peak is present near the expected frequency.

**Top Right**: A measured signal which is ambiguous as the identified peak frequency may also just be noise.

**Bottom Left**: Example of a simulated noise signal which exhibits a strong peak near the expected frequency, easily mistaken for a signal produced by an ion.

**Bottom Right**: A simulated signal created by two ions whose peak frequency is easily mistaken for noise due to destructive interference.

**Figure 2.**A cartoon illustration of the single-ion Penning trap ring electrode configuration and detection scheme. The shown configuration is optimal for dipole pickup of the ${\nu}_{+}$ eigenfrequency. The red electrodes labeled ‘P’ are those used for signal pickup, and those labeled ‘0’ are grounded or used for radio frequency excitations.

**Figure 3.**Power spectral density (PSD) for measured background data, compared to simulated white noise (of Equation (7)). The average of 1500 samples of each of the measured and simulated data are plotted above. The PSD is calculated over a 100 Hz bandwidth, centered about the reduced cyclotron frequency of

^{85}Rb

^{+}. The mean values of the averaged measured signal and white noise amplitude were optimized to be as close as possible.

**Figure 4.**A comparison of the signal-to-noise (SNR) ratio distributions for $N=1,2,3$ ions in the SIPT. Three different phase spread distributions are investigated, with each plot comparing the ‘sum of sines’ approach (

**top**) of Equation (11) to the rigorous method of [9] (

**middle**), and the residuals between the two methods (

**bottom**). For each ion number, ${10}^{4}$ simulated signals make up each distribution.

**Figure 5.**Impact of the included frequency range for the feature set on classification precision. Each feature set is a sampling from the discrete Fourier transform of the SIPT time-domain signals, which had a zero padding of twice the signal length on both the left and right sides. Note that typical SIPT signals are expected to have an FWHM of ≲10 Hz. Classifier precision was determined from a 10-fold cross-validation approach with a total of 10,000 samples. Four classifiers were considered: k-nearest neighbors (kNN), Gaussian naive Bayes (GNB), support vector machine (SVM), and logistic regression (LR).

**Figure 6.**Precision of classifiers for different simulated single-ion signal-to-noise (SNR) ratios. Four classifiers were considered: k-nearest neighbors (kNN), Gaussian naive Bayes (GNB), support vector machine (SVM), and logistic regression (LR). SNR values ranging from ∼$0.25$ to ∼$0.55$ over a 100 Hz spectral bandwidth are investigated, as these are expected values for single-ion signals. Precision was determined by training each classifier in a 5-fold cross-validation approach with a dataset of size 20,000.

**Figure 7.**Investigation of the neural network sensitivity for six unique simulated datasets. Sensitivity is determined by the residual between a given input ‘A’ and another input ‘B’ generated using the output parameters predicted by the initial input. Small residuals are desired. Note that the input to the neural network is a histogram of Lorentzian areas, defined by Equation (6), for a large set of SIPT FT-ICR frequency-domain signals. The reconstruction for input ‘B’ uses 80,000 ion signals to produce each histogram shown. A summary of the corresponding output parameters can be found in Table 2.

**Figure 8.**Comparison of a histogram of experimental

^{85}Rb

^{+}SIPT signal Lorentzian areas (defined by Equation (6)) to a histogram of 80,000 simulated

^{85}Rb

^{+}SIPT signals which was generated from the parameters predicted by the neural network; summarized in Table 3. Small residuals correspond to both reasonable predictions from the neural network as well as reasonable simulations of the SIPT FT-ICR signals.

**Table 1.**Summary of the optimal model hyperparameters found using a Bayesian optimization for four classifiers: k-nearest neighbors (kNN), Gaussian naive Bayes (GNB), support vector machine (SVM), and logistic regression (LR). Classifier precision was determined from a cross-validation approach with 10,000 samples. The signals with ions present were chosen to have $\mathrm{SNR}=0.23$ normally distributed with an uncertainty of ${\sigma}_{\mathrm{SNR}}=0.05$.

Classifier | Parameter | Value | Precision |
---|---|---|---|

kNN | Algorithm | Ball Tree | 0.711 |

Leaf Size | 46 | ||

k Neighbors | 35 | ||

GNB | Variable Smoothing | 0 | 0.704 |

SVM | C | 100 | 0.715 |

$\gamma $ | 1.895 | ||

Degree | 8 | ||

Kernel | rbf | ||

LR | C | 39 | 0.735 |

Solver | Saga | ||

Penalty | ℓ2 |

**Table 2.**A summary of the true and predicted parameters from the neural network for six unique simulated datasets. All predicted values are determined to 1 σ uncertainty, calculated using the MC Dropout method [56] for 100 iterations. Summaries for each output parameter can be found in Section 5.2.

Sim. Dataset 0 | Sim. Dataset 1 | Sim. Dataset 2 | ||||
---|---|---|---|---|---|---|

True | Prediction | True | Prediction | True | Prediction | |

μ_{p} | 0.54 | 0.54(8) | 0.37 | 0.43(8) | 0.14 | 0.18(6) |

A_{single} (mV) | 0.053 | 0.053(5) | 0.051 | 0.051(3) | 0.046 | 0.044(2) |

σ(A_{single}) (mV) | 0.000835 | 0.0017(9) | 0.00174 | 0.0023(8) | 0.000357 | 0.0024(7) |

V_{offset} (μV) | 8.89 | 8.1(1.1) | 9.79 | 8.4(1.0) | 1.85 | 0.9(8) |

${\varphi}_{0,\mathrm{max}}$ (rad) | 4.85 | 4(1) | 1.91 | 1.8(1.0) | 4.47 | 3.8(9) |

Sim. Dataset 3 | Sim. Dataset 4 | Sim. Dataset 5 | ||||

True | Prediction | True | Prediction | True | Prediction | |

μ_{p} | 0.36 | 0.37(7) | 0.12 | 0.18(7) | 0.29 | 0.34(8) |

A_{single} (mV) | 0.027 | 0.029(3) | 0.054 | 0.049(3) | 0.052 | 0.053(5) |

σ(A_{single}) (mV) | 0.000404 | 0.000264(12) | 0.000377 | 0.003(6) | 0.000078 | 0.00015(10) |

V_{offset} (μV) | 8.66 | 7.7(9) | 4.45 | 4.0(5) | 9.73 | 8.8(1.1) |

${\varphi}_{0,\mathrm{max}}$ (rad) | 3.8 | 4.1(7) | 3.44 | 3.2(9) | 5.53 | 4.2(1.1) |

**Table 3.**Neural network predictions on the experimental

^{85}Rb

^{+}dataset. Summaries for each output parameter can be found in Section 5.2.

Prediction | |
---|---|

${\mu}_{p}$ | 0.26(6) |

${A}_{\mathrm{single}}$ (mV) | 0.022(4) |

$\sigma \left({A}_{\mathrm{single}}\right)$ (mV) | 0.0024(5) |

V_{offset} (μV) | 2.0(7) |

${\varphi}_{0,\mathrm{max}}$ (rad) | 4.3(1.4) |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Campbell, S.E.; Bollen, G.; Hamaker, A.; Kretzer, W.; Ringle, R.; Schwarz, S.
Applications of Machine Learning and Neural Networks for FT-ICR Mass Measurements with SIPT. *Atoms* **2023**, *11*, 126.
https://doi.org/10.3390/atoms11100126

**AMA Style**

Campbell SE, Bollen G, Hamaker A, Kretzer W, Ringle R, Schwarz S.
Applications of Machine Learning and Neural Networks for FT-ICR Mass Measurements with SIPT. *Atoms*. 2023; 11(10):126.
https://doi.org/10.3390/atoms11100126

**Chicago/Turabian Style**

Campbell, Scott E., Georg Bollen, Alec Hamaker, Walter Kretzer, Ryan Ringle, and Stefan Schwarz.
2023. "Applications of Machine Learning and Neural Networks for FT-ICR Mass Measurements with SIPT" *Atoms* 11, no. 10: 126.
https://doi.org/10.3390/atoms11100126