# Characterizing Lossy Dielectric Materials in Shock Physics by Millimeter-Wave Interferometry Using One-Dimensional Convolutional Neural Networks and Nonlinear Optimization

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## Abstract

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## 1. Introduction

- A more robust 1D-CNN is proposed to address the inverse problem of lossless/lossy shocked wavefront (ILSW-Net), which embeds information about the complex refractive index of the material at rest to boost the prediction performance. The model’s performance is empirically validated on simulated validation data, demonstrating ability to significantly reduce the prediction error.
- A physics-informed loss between the input signal and the approximated signal computed through the estimation of the velocities and the complex refractive index is introduced. Moreover, this loss is minimized using a nonlinear optimization technique, namely the Nelder–Mead algorithm, enabling estimation even in the case of experimental data in which the measured complex refractive index is unavailable. The results show out that the obtained approximated signal fits better with the original one, and the estimation error is reduced.

## 2. Proposed Method to Derive the Refractive Index of Lossy Dielectric Materials Subjected to Shock

#### 2.1. Data Preprocessing

#### 2.1.1. Normalization of the Signal Delivered by the Millimeter-Wave Interferometer during a Shock Experiment

#### 2.1.2. Normalization of Output Variables

#### 2.2. Architecture of ILSW-Net

#### 2.3. Physics-Informed Loss and the Nelder–Mead Optimization Algorithm

## 3. Learning Procedure and Hyper-Parameter Setting

#### 3.1. Learning Procedure

#### 3.2. Hyper-Parameter Setting

## 4. Results and Discussions

#### 4.1. Lossless Material

#### 4.2. Lossy Material

#### 4.2.1. Validation on Simulated Data

- ILSW-Net with only the waveform as input (one-input ILSW-Net);
- One-input ILSW-Net with the data normalization process;
- ILSW-Net with two inputs, namely the waveform and the complex refractive index (${n}_{1}$) (two-input ILSW-Net);
- Two-input ILSW-Net with the data normalization process.

#### 4.2.2. Application to Experimental Data

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Architecture of ILSW-Net used to estimate the shock wavefront velocity (${V}_{1}$), particle velocity (${V}_{2}$), and the complex refractive index (${n}_{2}$) of the shocked dielectric material from simulated waveforms. Note that using the complex refractive index (${n}_{1}$) is a way to incorporate physical knowledge into the model, advantageously boosting its performance.

**Figure 2.**Performance comparison for different ILSW-Net configurations: (

**a**,

**b**) using a normalization (norm.) step; (

**c**) using real and imaginary parts of the refractive index (${n}_{1}$) as input; (

**d**) using a norm. step and real and imaginary parts of the refractive index (${n}_{1}$) as input.

**Figure 3.**Scatter plot of four values, namely ${V}_{1}$, ${V}_{2}$, and the real part (${r}_{2}$) and the imaginary part (${\kappa}_{2}$) of the refractive index (${n}_{2}$). The original values are depicted on x-axis, while predicted values are displayed on the y-axis.

**Figure 4.**Distribution of ground-truth values of ${\kappa}_{2}$ and the corresponding absolute percentage error (on the y-axis).

**Figure 5.**Comparison of theoretical waveforms (simulated using Equation (1), which accounts for the dielectric losses with the parameters given Table 4) with experimental waveforms delivered by a 94 GHz interferometer during impact experimentation. (

**a**,

**b**) High-density polyethylene (HDPE) material; (

**c**,

**d**) polymethyl methacrylate (PMMA) material.

**Table 1.**Dynamic range of ground-truth values used for the generation of training and validation datasets.

Ground Truth | Minimum Value | Maximum Value |
---|---|---|

$\mathrm{Real}\text{}\mathrm{part}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{refractive}\text{}\mathrm{index}\text{}({n}_{1}$) | 1 | 2 |

$\mathrm{Imaginary}\text{}\mathrm{part}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{refractive}\text{}\mathrm{index}\text{}({n}_{1}$) | 0.001 | 0.2 |

$\mathrm{Real}\text{}\mathrm{part}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{refractive}\text{}\mathrm{index}\text{}({n}_{2}$) | 1 | 3 |

$\mathrm{Imaginary}\text{}\mathrm{part}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{refractive}\text{}\mathrm{index}\text{}({n}_{2}$) | 0.001 | 0.2 |

$\mathrm{Shock}\text{}\mathrm{wavefront}\text{}\mathrm{velocity}\text{}({V}_{1}$) | 2000 m/s | 6000 m/s |

$\mathrm{Particle}\text{}\mathrm{velocity}\text{}({V}_{2}$) | 200 m/s | 600 m/s |

**Table 2.**The mean absolute percentage error (%) of ILSW-Net and the CNN model used in [9] for velocities ${V}_{1}$ and ${V}_{2}$ and the real part of ${n}_{2}$ on the validation set.

Model | ${\mathit{V}}_{1}$ | ${\mathit{V}}_{2}$ | ${\mathit{r}}_{2}$ |
---|---|---|---|

Model [9] | 2.46 | 3.59 | 0.75 |

ILSW-Net | 0.68 | 1.37 | 0.5 |

**Table 3.**The mean absolute percentage error (%) of ILSW-Net for velocities ${V}_{1}$ and ${V}_{2}$ and the real/imaginary parts of ${n}_{2}$ on the validation set.

Method | ${\mathit{V}}_{1}$ | ${\mathit{V}}_{2}$ | ${\mathit{r}}_{2}$ | ${\mathit{\kappa}}_{2}$ |
---|---|---|---|---|

One-input ILSW-Net (baseline) | 14.59 | 16.2 | 13.81 | 70.65 |

One-input ILSW-Net using norm. step | 9.49 | 9.82 | 9.22 | 136.62 |

Two-input ILSW-Net | 4.29 | 10.38 | 2.75 | 77.62 |

Two-input ILSW-Net using norm. step (final model) | 0.69 | 1.28 | 0.79 | 16.95 |

**Table 4.**Estimation of ${V}_{1}$, ${V}_{2}$, and ${n}_{2}$ by ILSW-Net for both HDPE and PMMA dielectric materials under mechanical impact with or without application of the optimization method. The accuracy is obtained for ${V}_{1}$ and ${V}_{2}$ from the experiments, as deduced from the experimental impact velocities.

HDPE | PMMA | |||
---|---|---|---|---|

Prediction | Accuracy (%) | Prediction | Accuracy (%) | |

${V}_{1}$ without refinement | 3616 m/s | 99.06 | 3397 m/s | 89.81 |

${V}_{1}$ with refinement | 3631 m/s | 99.47 | 3375 m/s | 90.40 |

${V}_{2}$ without refinement | 589 m/s | 85.57 | 251 m/s | 82.67 |

${V}_{2}$ with refinement | 565 m/s | 89.20 | 270 m/s | 88.93 |

$\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}\left[{n}_{2}\right]$ without refinement | 1.63 | 1.616 | ||

$\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}\left[{n}_{2}\right]$ with refinement | 1.60 | 1.614 | ||

$\mathrm{I}\mathrm{m}\left[{n}_{2}\right]$ without refinement | 0.036 | 0.0232 | ||

$\mathrm{I}\mathrm{m}\left[{n}_{2}\right]$ with refinement | 0.031 | 0.0224 |

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**MDPI and ACS Style**

Pham, N.T.; Lefrançois, A.; Aubert, H.
Characterizing Lossy Dielectric Materials in Shock Physics by Millimeter-Wave Interferometry Using One-Dimensional Convolutional Neural Networks and Nonlinear Optimization. *Electronics* **2024**, *13*, 1664.
https://doi.org/10.3390/electronics13091664

**AMA Style**

Pham NT, Lefrançois A, Aubert H.
Characterizing Lossy Dielectric Materials in Shock Physics by Millimeter-Wave Interferometry Using One-Dimensional Convolutional Neural Networks and Nonlinear Optimization. *Electronics*. 2024; 13(9):1664.
https://doi.org/10.3390/electronics13091664

**Chicago/Turabian Style**

Pham, Ngoc Tuan, Alexandre Lefrançois, and Hervé Aubert.
2024. "Characterizing Lossy Dielectric Materials in Shock Physics by Millimeter-Wave Interferometry Using One-Dimensional Convolutional Neural Networks and Nonlinear Optimization" *Electronics* 13, no. 9: 1664.
https://doi.org/10.3390/electronics13091664