#
Two-Scale Deep Learning Model for Polysilicon MEMS Sensors^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Model of the Polysilicon MEMS and Intrinsic Uncertainties

#### 2.1. Oscillation Amplitude of the Lorentz Force MEMS Magnetometer

^{3})/12. The beam is made of a polycrystalline silicon film with columnar structure, and the elastic properties governing its in-plane vibrations are assumed to be obtained through homogenization over a statistical volume element (SVE) of the polysilicon film.

_{1}and K

_{3}), amplitude of the oscillating external Lorentz force (${F}_{0}$), frequency of the forcing term (ω), and natural frequency of the beam (${\omega}_{1}$). The dynamics of this system is governed by weakly coupled thermo-electro-magneto-mechanical multi-physics, noticeable when writing explicit expressions for each of the previous terms. Nonetheless, the key aspect to highlight is that, since the solution for ${\mathsf{\nu}}_{max}$ depends on K

_{1}and K

_{3}which, in turn, depend on the flexural (EI) and axial (EA) rigidities of the beam, uncertainties in the values of the homogenized Young’s modulus, $\overline{E}$ and in-plane width, h (induced by defects such as over-etch, O), produce a scattering in the expected value of ${\mathsf{\nu}}_{max}$.

#### 2.2. Sources of Uncertainty in Polysilicon MEMS

## 3. Methodology

#### 3.1. Representation of the Resonant Structure

#### 3.2. The Neural Network-Based Model

_{1}branch, are fully connected to the eight-neuron input layer of the MLP

_{2}.

_{1}is composed of a sequence of fully connected layers. The specific sequence features an 8-node hidden layer followed by a 4-node and a 1-node output layer. As in the case of the CNN, only the output neuron was activated by a linear activation function while the rest of the units use ReLU activations. An identical configuration was chosen for MLP

_{2}.

^{−3}, the optimizer to Adam and the loss function to MSE. Furthermore, the implementation was completed, making use of Keras API. Regarding the hardware, a GeForce GTX 1050 Ti GPU was used.

## 4. Results

^{−8}$\mathsf{\mu}$m

^{2}was attained after 328 epochs. After this epoch, no considerable improvement was observed on the validation set over the next 100 epochs (set as the patience parameter), inducing the early stopping.

^{2}and the $5\times 5$ $\mathsf{\mu}$m

^{2}SVEs, respectively. In light and dark green we have included the corresponding identity mapping, which represents the ideal behavior we could expect from the model.

^{2}are reported in the plots: R

^{2}values are all close to 1, indicating a good performance. Moreover, as it could have been foreseen from the imbalance of the datasets, a slightly better result is observed for the smaller SVE samples.

^{2}values, the results are comparable to the performance obtained on the training set, which is a clear indication of the good generalization capability of the model.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Parity plots for the model at the device level: (

**a**) Training set; (

**b**) Validation set; (

**c**) Test set.

**Table 1.**Statistical indicators characterizing $\overline{E}$ for different SVE sizes, obtained with uniform strain boundary conditions.

SVE Size | μ [GPa] | σ [GPa] |
---|---|---|

2$\mathsf{\mu}$m × 2$\mathsf{\mu}$m | 150.1 | 5.5 |

5$\mathsf{\mu}$m × 5$\mathsf{\mu}$m | 149.3 | 2.4 |

Size | SVEs | $\mathbf{Mean}\text{}{\mathit{\nu}}_{\mathit{m}\mathit{a}\mathit{x}}$$\text{}\left[\mathsf{\mu}m\right]$ | $\mathbf{Standard}\text{}\mathbf{Deviation}\text{}{\mathit{\nu}}_{\mathit{m}\mathit{a}\mathit{x}}$$\left[\mathsf{\mu}m\right]$ | |
---|---|---|---|---|

Training | 2$\text{}\mathsf{\mu}$m × 2$\text{}\mathsf{\mu}$m | 1898 | 0.028 | 0.004 |

5$\text{}\mathsf{\mu}$m × 5$\text{}\mathsf{\mu}$m | 800 | 0.01 | 0.002 | |

Validation | 2$\text{}\mathsf{\mu}$m × 2$\text{}\mathsf{\mu}$m | 400 | 0.028 | 0.004 |

5$\text{}\mathsf{\mu}$m × 5$\text{}\mathsf{\mu}$m | 200 | 0.01 | 0.002 | |

Test | 2$\text{}\mathsf{\mu}$m × 2$\text{}\mathsf{\mu}$m | 198 | 0.028 | 0.004 |

5$\text{}\mathsf{\mu}$m × 5$\text{}\mathsf{\mu}$m | 99 | 0.01 | 0.002 |

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**MDPI and ACS Style**

Quesada-Molina, J.P.; Mariani, S.
Two-Scale Deep Learning Model for Polysilicon MEMS Sensors. *Comput. Sci. Math. Forum* **2022**, *2*, 12.
https://doi.org/10.3390/IOCA2021-10888

**AMA Style**

Quesada-Molina JP, Mariani S.
Two-Scale Deep Learning Model for Polysilicon MEMS Sensors. *Computer Sciences & Mathematics Forum*. 2022; 2(1):12.
https://doi.org/10.3390/IOCA2021-10888

**Chicago/Turabian Style**

Quesada-Molina, José Pablo, and Stefano Mariani.
2022. "Two-Scale Deep Learning Model for Polysilicon MEMS Sensors" *Computer Sciences & Mathematics Forum* 2, no. 1: 12.
https://doi.org/10.3390/IOCA2021-10888