# Scalable Graphene Defect Prediction Using Transferable Learning

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Molecular Dynamics Simulation of Defect-Containing Graphene

^{9}s

^{−1}. The small tensile pre-strain was applied to imitate the experimental setup of graphene vibration in [36]. Finally, the pre-strained graphene sheet was set to vibrate in the NVT ensemble at $T=300\mathrm{K}$ for 30 picoseconds, during which the out-of-plane displacement of all vibrating atoms were extracted for subsequent data processing. The sampling frequency of atom trajectories was 20 THz. Using the graphene sheet in Figure 1b as an example, the distributions of vibration amplitudes 15 ps and 30 ps after the initialization of the NVT ensemble are plotted in Figure 1e. The initial graphene configuration is also provided for comparison.

#### 2.2. Machine Learning

## 3. Results and Discussion

#### 3.1. One-Defect Scenarios

#### 3.2. Multiple-Defect Scenarios

#### 3.3. Weighted Cost Function

#### 3.4. Discussion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematics of defect-containing graphene sheets. Smaller graphene sheets with (

**a**) one defect and (

**b**) multiple defects for machine learning training. Larger graphene sheets with (

**c**) one defect and (

**d**) multiple defects for machine learning testing. “A” and “Z” denote the armchair and zigzag directions, respectively. (

**e**) An example of vibration amplitude distributions in a defect-containing graphene sheet. Distributions at three different time instances are plotted: before vibration, 15 ps, and 30 ps after the initialization of the NVT ensemble. $t$ denotes the time passed in the NVT ensemble where the vibrational responses are recorded.

**Figure 2.**Data preparation procedure for machine learning. Boxes in red are properties of each individual atoms. (

**a**) Illustration of query point and grid construction. (

**b**) Displacement time series $z\left(t\right)$ and (

**c**) the corresponding frequency response $z\left(f\right)$ of a vibrating atom in the graphene sheet.

**Figure 3.**Machine learning results of validation sets for one-defect scenarios. (

**a**) Schematic of discretized graphene domain for one-defect graphene sheets. The entire vibrating graphene domain is discretized by an $M$-by-$M$ mesh (in this case, $M=10$, $a=4.37\mathsf{\AA}$). Defect-containing grid is highlighted in light blue. (

**b**) Validation accuracies (TA, TPR, TNR) as a function of the grid size $a$ for 9-grid and 25-grid approaches.

**Figure 4.**Test accuracies for one-defect scenarios. (

**a**) TA, (

**b**) TPR, and (

**c**) TNR as a function of the grid size $a$ based on three test sets. (

**d**) Average accuracies over all three test sets as a function of $a$.

**Figure 5.**Test accuracies for multiple-defect scenarios. (

**a**) TA, (

**b**) TPR, and (

**c**) TNR as a function of the grid size $a$ based on three test sets. (

**d**) Average accuracies over all three test sets as a function of $a$.

**Figure 6.**Accuracies when a weighted cost function is used (weight ratios ${w}_{0}/{w}_{1}=0.5,1,2$). Average (

**a**) TA, (

**b**) TPR, and (

**c**) TNR of three test sets as a function of grid size $a$ for one-defect graphene sheets. Average (

**d**) TA, (

**e**) TPR, and (

**f**) TNR of three test sets as a function of grid size $a$ for multiple-defect graphene sheets.

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**MDPI and ACS Style**

Zheng, B.; Zheng, Z.; Gu, G.X.
Scalable Graphene Defect Prediction Using Transferable Learning. *Nanomaterials* **2021**, *11*, 2341.
https://doi.org/10.3390/nano11092341

**AMA Style**

Zheng B, Zheng Z, Gu GX.
Scalable Graphene Defect Prediction Using Transferable Learning. *Nanomaterials*. 2021; 11(9):2341.
https://doi.org/10.3390/nano11092341

**Chicago/Turabian Style**

Zheng, Bowen, Zeyu Zheng, and Grace X. Gu.
2021. "Scalable Graphene Defect Prediction Using Transferable Learning" *Nanomaterials* 11, no. 9: 2341.
https://doi.org/10.3390/nano11092341