# Deep Learning Based Multiresponse Optimization Methodology for Dual-Axis MEMS Accelerometer

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. MEMS Accelerometer Design

## 3. Basics of Deep Learning Model

_{1}, x

_{2},…, x

_{n}, where each x corresponds to an input variable, and each has a corresponding weight (w

_{1}, w

_{2},…, w

_{n}), as shown in Figure 2b. For a simpler notation, the perceptron and activation function can be presented as a combined unit, as illustrated in Figure 2c.

## 4. Proposed Deep-Neural-Network-Based Framework

#### 4.1. Design, Response, and Desirability Value Details

_{1}, x

_{2},…, x

_{8}) considered for the multiphysics design optimization of the MEMS accelerometer. These design parameters are the geometric parameters and the MEMS accelerometer operating conditions. The significance of the low and high levels for the design parameters has been discussed in [17]. The output responses considered for the optimization are natural frequency (y

_{1}), proof mass displacement (y

_{2}), pull-in voltage value (y

_{3}), capacitance change corresponding to the input acceleration (y

_{4}), and Brownian noise equivalent acceleration (BNEA) (y

_{5}).

#### 4.2. General Working of the Proposed Optimization Framework

_{1}, y

_{2},…, y

_{5}) using the input design parameters (x

_{1}, x

_{2},…, x

_{8}). The second model is implemented for the simultaneous optimization of the output characteristics of the MEMS accelerometer with respect to the input design parameters, and is referred to as the D model. While the Y model enables a simultaneous prediction of the five output characteristics of the MEMS accelerometer, it does not allow to simultaneously optimize these five output characteristics with respect to the design parameters. The simultaneous optimization of the output characteristics is achieved through the D model, which is based on maximizing the desirability function corresponding to the optimization objective function [25,30]. Based on the output of the D model, the values of the eight input design parameters are ranked and the combination which gives the maximum desirability values is presented as the optimized solution. Figure 4 provides a high-level pictorial overview of the working of the proposed framework.

#### 4.3. Output Response Prediction Model

_{1}, x

_{2},…, x

_{8}) corresponding to design variables and the output layer contains the corresponding output responses (y

_{1}, y

_{2},…, y

_{5}) that are to be predicted.

_{P}values for a set of X values, we used the data provided by [17]. The data has 80 rows of values, each row has a set of X values generated using Latin hypercube sampling and will be represented as X

_{S}; Ref. [17] obtained the Y values after performing simulations and these values will are represented as Y

_{TS}. For our work we have normalized the values between 0 and 1 to standardize the scale of each input and output value. A split of 80/20 was made for hyper-parameter tuning and training of the model. Here, the assumption is that the simulated data (as provided by [17]) used for training the Y model was generated taking into account the realistic design conditions of the MEMS accelerometer. Figure 6 shows the steps involved in the training process as well as the evaluation of the Y model.

#### 4.4. Effect of Design Parameters on the Output Responses

_{1}, x

_{2},…, x

_{8}) on the output responses (y

_{1}, y

_{2},…, y

_{5}) is also observed to obtain a deeper insight into their respective behaviors. In this regard, each input parameter is varied across its range while keeping all of the remaining input parameters fixed at the average of the low and high levels, as defined in Table 1. Since each output response has a different range and unit, they are normalized between 0 and 1 for comparisons.

_{1}) on y

_{1}, y

_{2},…, y

_{5}. The graphs show that there is a much stronger impact of the variation of x

_{1}on the pull-in voltage (y

_{3}) and BNEA (y

_{5}) than on the natural frequency (y

_{1}), proof mass displacement (y

_{2}), and capacitance change (y

_{3}). The results show that with an increase in the x

_{1}, the pull-in voltage value decreases and BNEA increases for the MEMS accelerometer.

_{2}) on y

_{1}, y

_{2},…, y

_{5}. The results show that with an increase in the x

_{2}value, the natural frequency and pull-in voltage value for the MEMS accelerometer decreases while the proof mass displacement and capacitance change for an input acceleration increases. Moreover, the effect of change in the x

_{2}on the MEMS accelerometer BNEA value is negligible.

_{3}) on y

_{1}, y

_{2},…, y

_{5}. The strongest effect of the variation of x

_{3}is clearly visible on the natural frequency (y

_{1}) that matches with the findings of [17]. Additionally, the graph also shows that x

_{3}also contributes at varying levels towards the proof mass displacement (y

_{2}), pull-in voltage (y

_{3}), and capacitance change (y

_{4}).

_{4}) on y

_{1}, y

_{2},…, y

_{5}. The graph shows that there is a significant impact of variation of x

_{4}on the natural frequency (y

_{1}), proof mass displacement (y

_{2}), pull-in voltage (y

_{3}), and capacitance change (y

_{4}).

_{5}) on y

_{1}, y

_{2},…, y

_{5}. It is evident from the graphs that x

_{5}strongly impacts the proof mass displacement (y

_{2}) and capacitance change (y

_{4}).

_{6}) on y

_{1}, y

_{2},…, y

_{5}. The graphs that only BNEA (y

_{5}) is impacted by the variation of x

_{6}, whereas the remaining output responses are not perturbed.

_{7}) on y

_{1}, y

_{2},…, y

_{5}. The behavior is similar to that observed for the case of x

_{6}, i.e., x

_{7}also impacts only the BNEA (y

_{5}), without really perturbing the remaining output responses.

_{8}) on y

_{1}, y

_{2},…, y

_{5}. There is not a strong impact of variation of x

_{8}on any output response that is in line with observations of [17]; though y

_{2}and y

_{4}appear slightly perturbed.

_{1}, y

_{2},…, y

_{5}) obtained using the proposed Y model with those obtained using the method in [17], based on MAE and RMSE scores (Table 2). Moreover, unlike the procedure adopted in [17] for generating the simulated data that was extremely time consuming (taking extended periods of time to complete), the proposed Y model (once trained) offers an alternative for generating more data (where needed) in the design space in an accurate and time-efficient manner without the need to perform simulations over longer periods of time. In fact, in the next section, the trained Y model is used to generate a larger dataset as required for training the D model.

## 5. Multiresponse Optimization Using D Prediction Model

#### 5.1. Training of the D Prediction Model

_{G}values (Figure 15). The X

_{G}values are passed to the Y model to obtain the corresponding Y

_{G}values. For each Y

_{G}value, the D

_{G}value was estimated using the same approach used in [17], and this obtained D value is represented as the D

_{TG}value, or true value of desirability for the generated set of X

_{G}values. A total of 3125 rows of values were obtained. All the values were normalized between 0 and 1 to standardize the scale of each input and output value. A split of 80/20 was made for hyperparameter tuning and training of the model. For training of the D model, the input layer therefore contains y

_{1}, y

_{2},…, y

_{5}and the output layer contains only a desirability value. The manual formula-based calculation of D (as in [17]) is thus replaced with a robust deep-learning-based D model.

#### 5.2. Multi-Response Optimization

_{R}. The X

_{R}is fed through the Y model to obtain the Y

_{R}. The obtained Y

_{R}values are fed to the D model to obtain the D

_{R}, which are the D values for the corresponding design parameters. Then, the index of the maximum D value is searched, and the corresponding Y and X to this maximum index are considered as the optimized x

_{1}, x

_{2},…, x

_{8}values. Table 3 presents the values obtained from the proposed method and the values reported in [17].

_{1}, y

_{2},…, y

_{5}) predicted using the optimized X values (x

_{1}, x

_{2},…, x

_{8}) based on the proposed method are listed in Table 3. For comparison, we computed the Y values (referred to as observed Y values) by performing FEM simulations in CoventorWare

^{®}software (Coventor, Raleigh, NC, USA) using the same X values as obtained based on the proposed method. The computed observed Y values are as follows: y

_{1}= 3096.43 Hz, y

_{2}= 0.676 μm, y

_{3}= 7.0303 V, y

_{4}= 521 fF, and y

_{5}= 0.7959 μg/$\sqrt{\mathrm{Hz}}$. Here, the t-test is performed to test the null hypothesis that the data in the two samples (predicted Y values and observed Y values) is derived from independent random samples having normal distributions of equal means and equal but unknown variances. The results of the t-test show that the null hypothesis is not rejected with p-value = 0.98, thus confirming that the data in the two samples is statistically highly similar.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The 2-DoF MEMS capacitive MEMS accelerometer design [17].

**Figure 2.**(

**a**) Perceptron with single value input, (

**b**) perceptron with multi-value input, and (

**c**) merging perceptron and activation function.

**Figure 4.**General working of the proposed parameter optimization methodology for the MEMS accelerometer. It is based on calculating the desirability (D) value for optimization using a cascade of two Deep Neural Networks (i.e., Y model and D model). X values correspond to (x

_{1}, x

_{2},…, x

_{8}); Y values correspond to (y

_{1}, y

_{2},…, y

_{5}); and D value refers to the desirability value.

**Figure 7.**Effect of variation of the overlap length of comb (x

_{1}) on the output responses y

_{1}, y

_{2},…, y

_{5}. Key. y

_{1}: natural frequency; y

_{2}: proof mass displacement; y

_{3}: pull-in voltage; y

_{4}: capacitance change; and y

_{5}: BNEA.

**Figure 8.**Effect of variation of the length of the suspension beam 1 (x

_{2}) on the output responses y

_{1}, y

_{2},…, y

_{5}. Key. y

_{1}: natural frequency; y

_{2}: proof mass displacement; y

_{3}: pull-in voltage; y

_{4}: capacitance change; and y

_{5}: BNEA.

**Figure 9.**Effect of variation of the length of suspension beam 2 (x

_{3}) on the output responses y

_{1}, y

_{2},…, y

_{5}. Key. y

_{1}: natural frequency; y

_{2}: proof mass displacement; y

_{3}: pull-in voltage; y

_{4}: capacitance change; and y

_{5}: BNEA.

**Figure 10.**Effect of variation of the width of the suspension beam (x

_{4}) on the output responses y

_{1}, y

_{2},…, y

_{5}. Key. y

_{1}: natural frequency; y

_{2}: proof mass displacement; y

_{3}: pull-in voltage; y

_{4}: capacitance change; and y

_{5}: BNEA.

**Figure 11.**Effect of variation of the input acceleration (x

_{5}) on the output responses y

_{1}, y

_{2},…, y

_{5}. Key. y

_{1}: natural frequency; y

_{2}: proof mass displacement; y

_{3}: pull-in voltage; y

_{4}: capacitance change; and y

_{5}: BNEA.

**Figure 12.**Effect of variation of the operating temperature (x

_{6}) on the output responses y

_{1}, y

_{2},…, y

_{5}. Key. y

_{1}: natural frequency; y

_{2}: proof mass displacement; y

_{3}: pull-in voltage; y

_{4}: capacitance change; and y

_{5}: BNEA.

**Figure 13.**Effect of variation of the operating pressure (x

_{7}) on the output responses y

_{1}, y

_{2},…, y

_{5}. Key. y

_{1}: natural frequency; y

_{2}: proof mass displacement; y

_{3}: pull-in voltage; y

_{4}: capacitance change; and y

_{5}: BNEA.

**Figure 14.**Effect of variation of the frequency ratio (x

_{8}) on the output responses y

_{1}, y

_{2},…, y

_{5}. Key. y

_{1}: natural frequency; y

_{2}: proof mass displacement; y

_{3}: pull-in voltage; y

_{4}: capacitance change; and y

_{5}: BNEA.

Notation | Design Parameters | Low Level | High Level |
---|---|---|---|

x_{1} | Overlap length of comb | 150 μm | 250 μm |

x_{2} | Length of suspension beam 1 | 400 μm | 500 μm |

x_{3} | Length of suspension beam 2 | 500 μm | 500 μm |

x_{4} | Width of suspension beam | 6 μm | 8 μm |

x_{5} | Input acceleration | 1 g | 25 g |

x_{6} | Operating temperature | 233.15 K | 373.15 K |

x_{7} | Operating pressure | 100 Torr | 760 Torr |

x_{8} | Frequency ratio | 0.1 | 0.5 |

**Table 2.**Comparison of the predicted output responses (y

_{1}, y

_{2},…, y

_{5}) obtained using the proposed Y model with those obtained using the method in [17], in terms of MAE and RMSE.

Output Response | MAE | RMSE | ||
---|---|---|---|---|

Proposed | [17] | Proposed | [17] | |

Natural frequency (y_{1}) | 12.67 Hz | 29.64 Hz | 15.41 Hz | 41.19 Hz |

Proof mass displacement (y_{2}) | 0.004 μm | 0.024 μm | 0.004 μm | 0.034 μm |

Pull-in voltage (y_{3}) | 0.065 V | 0.085 V | 0.072 V | 0.134 V |

Capacitance change (y_{4}) | 5.292 fF | 10.179 fF | 6.28 fF | 14.05 fF |

BNEA (y_{5}) | $0.004\text{}\mathsf{\mu}\mathrm{g}/\sqrt{\mathrm{Hz}}$ | $0.019\text{}\mathsf{\mu}\mathrm{g}/\sqrt{\mathrm{Hz}}$ | $0.005\text{}\mathsf{\mu}\mathrm{g}/\sqrt{\mathrm{Hz}}$ | $0.029\text{}\mathsf{\mu}\mathrm{g}/\sqrt{\mathrm{Hz}}$ |

**Table 3.**Comparison of the obtained optimized X values (x

_{1}, x

_{2},…, x

_{8}) using the proposed method with those reported in [17]. The corresponding Y values (y

_{1}, y

_{2},…, y

_{5}) are also listed.

Optimized Values Reported in [17] | ||||||||
---|---|---|---|---|---|---|---|---|

x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | x_{7} | x_{8} | |

Design Parameters (X values) | 153.6 μm | 403.6 μm | 500 μm | 6.26 μm | 25 g | 300 K | 760 Torr | 0.50 |

y_{1} | y_{2} | y_{3} | y_{4} | y_{5} | ||||

Output Responses (Y values) | 3036.4 Hz | 0.903 μm | 6.76 V | 676.2 fF | $0.81\text{}\mathsf{\mu}\mathrm{g}/\sqrt{\mathrm{Hz}}$ | |||

Optimized Values Obtained Using the Proposed Method. | ||||||||

x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | x_{7} | x_{8} | |

Design Parameters (X values) | 150.0 μm | 430.0 μm | 500 μm | 6.40 μm | 25 g | 300 K | 760 Torr | 0.45 |

y_{1} | y_{2} | y_{3} | y_{4} | y_{5} | ||||

Output Responses (Y values) | 3160.0 Hz | 0.723 μm | 7.22 V | 571.0 fF | $0.83\text{}\mathsf{\mu}\mathrm{g}/\sqrt{\mathrm{Hz}}$ |

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**MDPI and ACS Style**

Mattoo, F.A.; Nawaz, T.; Saleem, M.M.; Khan, U.S.; Hamza, A.
Deep Learning Based Multiresponse Optimization Methodology for Dual-Axis MEMS Accelerometer. *Micromachines* **2023**, *14*, 817.
https://doi.org/10.3390/mi14040817

**AMA Style**

Mattoo FA, Nawaz T, Saleem MM, Khan US, Hamza A.
Deep Learning Based Multiresponse Optimization Methodology for Dual-Axis MEMS Accelerometer. *Micromachines*. 2023; 14(4):817.
https://doi.org/10.3390/mi14040817

**Chicago/Turabian Style**

Mattoo, Fahad A., Tahir Nawaz, Muhammad Mubasher Saleem, Umar Shahbaz Khan, and Amir Hamza.
2023. "Deep Learning Based Multiresponse Optimization Methodology for Dual-Axis MEMS Accelerometer" *Micromachines* 14, no. 4: 817.
https://doi.org/10.3390/mi14040817