# Machine Learning Identification of Piezoelectric Properties

^{*}

## Abstract

**:**

_{11}, c

_{13}, c

_{33}, c

_{44}and e

_{33}were predicted using a neural network numerically trained by using finite element simulations. Close to one million simulations were performed by changing the value of the selected parameters by ±10% around the starting point. To train the network, the values of a PZT 27 piezoelectric ceramic with a diameter of 20 mm and thickness of 2 mm were used as the initial seed. The first results were very encouraging, and provided the original parameters with a difference of less than 0.6% in the worst case. The proposed approach is extremely fast after the training of the neural network. It is suitable for manufacturers or end users that work with the same material and a fixed number of geometries.

## 1. Introduction

## 2. Forward Optimization vs. Neural Network Approach

_{11}, c

_{12}, c

_{13}, c

_{33}, c

_{44}, e

_{31}, e

_{15}, e

_{33}, ε

_{11}, ε

_{33}) (see Equation (1)) is needed, and the mass density and geometric information must also be known. Using this information, the FEM simulation allows the electrical impedance response and the mechanical displacement in the nodal points to be obtained. Thus, this is called forward optimization, the computation is a simulation from the parameter space to the impedance response space. Figure 1 shows the flow chart for the forward optimization approach.

#### 2.1. Initial Seed

#### 2.2. FEM Simulation

#### 2.3. Optimization Algorithm

#### 2.4. Objective Function

#### 2.5. Exit Criteria

## 3. Materials and Methods

#### 3.1. Selected Sample

^{6}kg/m

^{3}. Table 1 shows the real constitutive parameters obtained from [10].

#### 3.2. Finite Elements

#### 3.3. Neural Network Implementation

#### 3.4. Database Creation

_{11}, c

_{13}, c

_{33}, c

_{44}, e

_{33}) and another 100,000 to evaluate the performance in the full parameters’ space.

## 4. Results

_{11}, c

_{13}, c

_{33}, c

_{44}, e

_{33}) while the others remained fixed $\left({c}_{12},{e}_{31},{e}_{15},{\epsilon}_{11},{\epsilon}_{33}\right)$. In the first case, the results were evaluated using samples of the same subspace.

#### 4.1. Results in the Restricted Subspace

_{11}, c

_{13}, c

_{33}, c

_{44}, e

_{33}). In this case, it was possible to follow each constitutive parameter over the training stage. Table 2 shows the mean values obtained for the validation set and the test set.

_{11}, c

_{13}, c

_{33}, c

_{44}, e

_{33}) and the results are in ppm. Table 3 shows the mean and the maximum $er{r}_{P}$ for each parameter.

#### 4.2. Results with the Full Set of Parameters

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Flow chart of the forward optimization approach using Finite Element Method (FEM). Blue indicates the input data, green the calculus performed at each step in the loop and red the results.

**Figure 2.**Neural network (NN) approach. Blue indicates the input data, green the calculus and red the results of each phase. Each phase has an input, computation, and a result.

**Figure 5.**Impedance results. The black curve is the impedance of the Test set, the red curve is the NN result.

**Figure 6.**Results for conductance and resistance. The black curve is the impedance of the Test set, the red curve is the NN result.

**Figure 7.**Results Impedance for the full set. The black curve is the impedance of the Test set, the red curve is the NN result.

**Figure 8.**Results of conductance and resistance for the full set. The black curve is the impedance of the Test set, the red curve is the NN result.

C_{11} | C_{12} | C_{13} | C_{33} | C_{44} | E_{31} | E_{15} | E_{33} | ε_{11}/ε_{0} | ε_{33}/ε_{0} |
---|---|---|---|---|---|---|---|---|---|

118.1 | 74.9 | 73.8 | 110.4 | 20.3 | –5.1 | 11.2 | 16.0 | 984 | 830 |

^{2}.

C_{11} | C_{13} | C_{33} | C_{44} | E_{33} | |
---|---|---|---|---|---|

MAPE_{val} | 0.48 | 0.43 | 0.31 | 0.57 | 0.59 |

MAPE_{test} | 0.48 | 0.43 | 0.32 | 0.57 | 0.59 |

C_{11} | C_{13} | C_{33} | C_{44} | E_{33} | |
---|---|---|---|---|---|

Err_{P} (Aver) | 3.5 | 2.9 | 3.3 | 4.5 | 6.6 |

Err_{P} (Max) | 15 | 5.8 | 5.1 | 11.7 | 21.4 |

C_{11} | C_{13} | C_{33} | C_{44} | E_{33} | |
---|---|---|---|---|---|

MAPE_{test} | 1.3 | 1.3 | 0.38 | 0.83 | 2.6 |

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**MDPI and ACS Style**

del Castillo, M.; Pérez, N.
Machine Learning Identification of Piezoelectric Properties. *Materials* **2021**, *14*, 2405.
https://doi.org/10.3390/ma14092405

**AMA Style**

del Castillo M, Pérez N.
Machine Learning Identification of Piezoelectric Properties. *Materials*. 2021; 14(9):2405.
https://doi.org/10.3390/ma14092405

**Chicago/Turabian Style**

del Castillo, Mariana, and Nicolás Pérez.
2021. "Machine Learning Identification of Piezoelectric Properties" *Materials* 14, no. 9: 2405.
https://doi.org/10.3390/ma14092405