# Lightweight Machine-Learning Model for Efficient Design of Graphene-Based Microwave Metasurfaces for Versatile Absorption Performance

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Graphene-Based Metasurface Absorber Model

#### 2.2. Machine-Learning Model

#### 2.3. Inverse Design System

## 3. Experiments and Results

#### 3.1. Dataset Collection

#### 3.2. Performance of Forward Prediction

#### 3.3. Results of Inverse Design System

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of forward prediction and inverse design of graphene-based microwave absorbers using machine-learning model.

**Figure 2.**Illustration of forward prediction machine-learning network. (

**a**) The five design parameters are normalized as a 5-dimension vector input for the machine-learning network. The output points correspond to 281 sample points of reflective spectrum in 6–20 GHz. The network consists of two fully connected layers and a transposed convolution block. (

**b**) The working mechanism of transposed convolution block considering batch size in training process with three transposed convolution layers. Here, ${n}_{b}$ is the batch size. The kernel size for each layer and feature-map evolution are demonstrated. This block eventually turns information from the second fully connected layer into 281 dimensions’ output which can reproduce the reflective spectrum.

**Figure 3.**Illustration of inverse design system optimization process. Before training begins, an initial seed ${\mathit{x}}_{0}$ is generated randomly. In iterative step k, loss L and the its gradients to $\mathit{x}$: ${g}_{k}={\nabla}_{\mathit{x}}L\left(\mathit{x}\right)$ are firstly computed through the pre-trained model. ${\widehat{g}}_{k}$ is derived by the adaptive subgradient method [52] based on ${g}_{k}$ in all iterations before k-th to determine the desecending direction ${d}^{k}$, which is the same dimension as $\mathit{x}$. $\lambda $ is the descending step of each iterative, which is a scalar in (0, 1). $\mathit{x}$ is updated with the iterative paradigm. After a fixed number of iterative steps, the optimized result can be then solved. Combinations of optimized parameters of any design requirements can be obtained within seconds.

**Figure 4.**Illustration of dataset generation. S1, S2, S3, S4 are randomly selected sample spectra from dataset.

**Figure 5.**(

**a**) Train and validation loss in the training process with and without normalization. (

**b**) Average percentage error of each 281 spectrum-point visualizations in the training process. (

**c**,

**d**) Prediction performance with and without transposed convolution layers in two extreme situations when the parameters are at the boundaries of sampling space. (

**e**,

**f**) Two examples of prediction performance of our FPN model.

**Figure 6.**Versatile absorber designs based on different requirements. (The reflectance are shown in logarithmic coordinate) Cases 1–4: inverse design in all five degree of freedom. Cases 5–8: inverse design with thickness of substrate h fixed at 3.5 mm. Cases 9–12: inverse design with sheet resistance of graphene ${R}_{g}$ fixed at 250 $\mathsf{\Omega}$. The colored area is the optimization area. FPN indicates the optimized reflection spectra given by inverse design system while CST represents the reflection spectra from CST simulations with the optimized parameter combinations.

Design Parameters | Start | End |
---|---|---|

${R}_{g}$ ($\mathsf{\Omega}$) | 132 | 300 |

d (mm) | 1 | 6 |

l (mm) | 5 | 11 |

p (mm) | 8 | 14 |

h (mm) | 2 | 4 |

Hyperparameters | Values |
---|---|

Learning rate | 3 × 10${}^{-4}$ |

Optimization method | Adam |

Learning-rate decay | 5 × 10${}^{-6}$ |

Loss function | MSELoss |

**Table 3.**Parameters combinations for cases in Figure 6.

Parameters | ${\mathit{R}}_{\mathit{g}}$ ($\mathsf{\Omega}$) | d (mm) | l (mm) | p (mm) | h (mm) |
---|---|---|---|---|---|

Case 1 | 136.42 | 1 | 6.36 | 11.63 | 2.6 |

Case 2 | 149.48 | 5.29 | 7.23 | 14 | 3.57 |

Case 3 | 153.8 | 3.43 | 6.49 | 12.26 | 3.14 |

Case 4 | 132 | 3.73 | 7 | 10.62 | 2.91 |

Case 5 | 171.48 | 4.42 | 7.58 | 13.17 | 3.5 |

Case 6 | 154.39 | 4.85 | 7.04 | 14 | 3.5 |

Case 7 | 179.47 | 2.38 | 7.35 | 13.18 | 3.5 |

Case 8 | 132 | 4.75 | 7.58 | 11.38 | 3.5 |

Case 9 | 250 | 1 | 6.28 | 10.43 | 3.16 |

Case 10 | 250 | 2.3 | 7.3 | 14 | 3.41 |

Case 11 | 250 | 1 | 6.86 | 12.5 | 3.16 |

Case 12 | 250 | 1.11 | 7.72 | 10.84 | 2.83 |

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**MDPI and ACS Style**

Chen, N.; He, C.; Zhu, W.
Lightweight Machine-Learning Model for Efficient Design of Graphene-Based Microwave Metasurfaces for Versatile Absorption Performance. *Nanomaterials* **2023**, *13*, 329.
https://doi.org/10.3390/nano13020329

**AMA Style**

Chen N, He C, Zhu W.
Lightweight Machine-Learning Model for Efficient Design of Graphene-Based Microwave Metasurfaces for Versatile Absorption Performance. *Nanomaterials*. 2023; 13(2):329.
https://doi.org/10.3390/nano13020329

**Chicago/Turabian Style**

Chen, Nengfu, Chong He, and Weiren Zhu.
2023. "Lightweight Machine-Learning Model for Efficient Design of Graphene-Based Microwave Metasurfaces for Versatile Absorption Performance" *Nanomaterials* 13, no. 2: 329.
https://doi.org/10.3390/nano13020329