Deep Learning on Atomistic Physical Fields of Graphene for Strain and Defect Engineering

Strain and defect engineering have profound applications in two‐dimensional materials, where it is important to determine the equilibrated atomistic structures with defect conditions under mechanical deformations for computational materials design. Nevertheless, how to efficiently predict relaxed atomistic structures and the associated physical fields on each atom or bond under evolving mechanical deformations remains as an essential challenge. To address this issue, a deep neural network architecture is designed to embed the state of applied strains into the defect‐engineered atomistic geometry, so that deformation‐coupled physical fields of interests on atoms or bonds can be predicted or derived over continuous state of deformations. For demonstration, the combination of applied tensile strains and shear strain on monolayer graphene with random distribution of Stone–Wales defects and vacancy defects is considered. The unique advantage of this framework is the development of strain‐embedding concept combined with generative adversarial network, which can be feasibly extended to other material and other conditions. The computational approach sheds light on boosting the efficiency of evaluating physical properties of 2D materials under complex strain and defect states.


Introduction
[3][4][5][6] Various electromechanical properties of 2D materials [7] depend on deformation states and defect configurations, offering rich opportunities for advanced functional design via strain and defect engineering, [8][9][10][11][12][13] where determining the relaxed atomistic configurations in the presence of mechanical deformations and atomistic defects is fundamental for the subsequent screening or design on structure-dependent physical-chemical properties.
Computational methods are usually used to achieve the above goal for accelerated materials design.However, traditional atomistic modeling methods such as density functional theory (DFT) and molecular dynamics may incur significant computational costs when dealing with numerous candidate atomistic models under different strain and defect conditions.The main reason is that the design space spanned by defect types, defect locations, defect numbers, and loading conditions is highly dimensional and highly complex.Particularly, the atomistic structure would change continuously as the applied stains evolve.[16][17][18][19][20][21][22][23] For example, convolutional neural networks (CNNs) are demonstrated useful for finding defect-engineered atomistic configurations that optimize mechanical properties such as stretchability, [20] thermoconductivity, [14] Young's modulus. [21]A generative adversarial network (GAN) can predict stress fields from geometry with high accuracy. [16,17,22,23]Recently, U-Net-based CNN can predict the von Mises stress fields of elastoviscoplastic grain microstructures. [15]Nevertheless, how to efficiently predict relaxed atomistic structures and the associated physical fields on each atom or bond under continuous mechanical deformations has remained unclear.
[29][30] We design a neural network architecture to embed the state of applied strains into the defect-engineered atomistic geometry. [30]Our goal is to predict the deformation-coupled physical fields of interests on each bond or on each atom over a continuous range of applied strains.For demonstration, we investigate displacement fields on each atom due to relaxation, strain fields, and hopping energy fields on each bond for electronic band structure calculations, as well as stress tensor fields on each atom.

Overview of Model Development and Workflow
The main goal of the current work is to develop a deep learning framework to produce deformation-coupled physical fields so that the predicted field values on each atom or bond can closely match those results from atomistic simulations (Figure 1).
Our deep learning framework is based on GAN that consists of two neural networks: a generator and a discriminator. [31]pecifically, we employ a conditional GAN that consists of two key components: the Generator U-Net [32] and the Discriminator PatchGAN. [33]Structural images and strain embeddings (SEs) (see Experimental Section) are fed to the generator to generate field images of interest.The discriminator evaluates these generated field images by comparing them to real field images obtained from atomistic simulations.With the proposed strainembedding method, we show that a well-trained generator can predict deformation-coupled atomistic fields of interests.
Four neural networks for displacement fields due to structural relaxation, bond strain field, hopping energy field, and atomic stress tensor field (σ xx , σ yy , and σ xy ) after structural relaxation are trained using the same neural network architecture.The input of the neural network is the state of externally applied strains (ε xx , ε yy , and ε xy ) (Figure 1a) which is processed in the block of SE, and the initial position field of atoms (Figure 1b) which is processed into structural images.Multiple blocks of SE are repeatedly fed into the neural network at deeper levels.This is the main feature of our neural network architecture for predicting deformation-coupled properties.
For demonstration, we study graphene with S-W defects and vacancies (Figure 1b).We use atomistic simulations to generate ground truth labels, which are performed in LAMMPS [34] with classic forcefield AIREBO potential. [35]Images of the initial atomistic structure, displacement field, bond strain field, hopping energy field, and stress tensor field are collected from simulations to create our dataset containing 10 000 data points, which are divided into 80% for training, 10% for testing, and 10% for validation.The zigzag (armchair) direction of graphene is labeled as the x(y)-direction in our notations.The defected graphene is stretched in both xand y-directions and shearing in the xy-direction (perpendicular to the y-direction and pointing toward the x-direction), as displayed in Figure 1a.
The out-of-plane displacement fields are not involved in our deep learning study, because in the context of in-plane strain engineering, the out-of-plane displacement is a higher-order small quantity to affect the bond lengths (therefore, the internal atomistic stress state) as the graphene morphology is gradually flattened under in-plane straining (see Supporting Information Figure S1).For example, if not considering the out-of-plane displacement, then at 2% uniaxial tensile strain, the maximum relative error (RE) on calculating bond length from our atomistic simulation is about 6.5%, which typically happens at defective sites, while the averaged RE on calculating bond length over the total number of bonds is 0.045%.Given that the deep learning method is essentially statistical learning carrying predictive errors, the consideration of out-of-plane displacement does not provide essentially new insights or compromise the validity of the main results of current work.
To accurately match the field values on each atom (bond) in the structural representation, we choose a small system to illustrate the process of pixelating our structure, as shown in Figure 1c-f.In the tensor representing the structure, an atom is represented by a square block of 3 Â 3 pixels that centers on the atom location.These on-atom pixel values are all equal to 1 (white).The non-atom pixel values are set to 0 (black).In the tensor representing training labels, nonatom pixel values remain zero, and on-atom pixel values are replaced with the same normalized field values (i.e., the normalized field values are the labels for training) on those atoms.In the examples shown in Figure 1c-f, all on-atom pixel values are 0.7, representing a training label value of 0.7 for that atom.For fields related to bonds (e.g., bond length, hopping energy), the centers of these 3 Â 3 pixels are then positioned at the centers of the bonds.To the best of our knowledge, our treatment is the first to emphasize on high-fidelity atom-wise or bond-wise comparison of field values based on GANs. [22]

Deformation-Coupled Displacement Field Due to Structural Relaxation
To capture the atomistic displacement fields (along x and y directions) due to structural relaxation, we prepare the labels of displacements by subtracting the background linear displacement field due to the deformation of the simulation box, as shown in Equation ( 1): where capitalized X and Y denote the initial coordinates, while lowercased x and y denote the deformed coordinates.u 0 x and u 0 y represent the processed displacement values in the xand y-directions, respectively, which are used for training.u s x refers to the linear interpolation displacement along x direction generated by the shear component ε xy (note that there is no such component along y direction due to our notation on deformation pattern as in Figure 1a).R x and R y represent the ratios of length change in the xand y-directions after deformation, respectively Our neural network's ability to predict structural relaxation displacement fields is significantly improved after using such labels of displacements.
Our neural network can accurately predict in-plane displacements.One illustration of a particular test example is provided in Figure 2a,b, while the statistics of comparison for all test models under various defect and strain conditions are shown in Figure 2c.When analyzing the results, the field images generated by the neural network are converted back into numerical values using the color mapping that was initially used to create images from atomistic simulation results, where each color is associated with a specific numerical value.Then, we extract the numerical values on the atoms for analysis.We use the numerical values on atoms to calculate R 2 score since we are interested in atom-wise validation.The meaning of the applied strains in Figure 2a is described in Figure 1a.For both displacement components, the neural network predictions closely align with the atomistic simulation results, effectively capturing the displacement patterns in the field maps (Figure 2a).The atom-wise comparison on randomly selected 100 atoms shows excellent agreement (Figure 2b).For most cases in the test set, the R 2 score when compared with the ground truth fields reaches over 0.96 (Figure 2c).

Deformation-Coupled Bond Strain Field and Hopping Energy Field
In this section, we deal with predicting bond properties (i.e., the bond strain field and hopping energy field) after structural relaxation using two approaches.Apparently, such fields can be either derived by the predicted displacement fields, or directly predicted from neural network.
The bond strain is defined as: where l represents the bond length after deformation, and L represents the bond length before deformation.In this work, L is equal to 1.42 Å. [36] Using predicted displacement field images, the derived bond strain field using Equation (2) achieves mean R 2 value of 0.971 in the test set.This is an example showing that once the deformation-dependent displacement fields due to structural relaxation are successfully predicted, many geometric properties can be straightforwardly derived with high accuracy.
To illustrate the capacity of our neural network architecture, we train another network with the same architecture but different parameters to directly predict the deformation-coupled bond strain field.Figure 3a-h showcases the effectiveness of our methods on the same structure and same loading conditions, as depicted in Figure 2a, but focused on predictions on bond properties.For our particular example (Figure 3a), both the predictions from neural network trained directly using the bond strain field (Figure 3b) and the derivation using predicted displacement field (Figure 3c) achieve extremely high accuracy.The errors on most atoms are very small, as exemplified by Figure 3g.For overall performance in test set, both methods predict bond strain with an average R 2 value greater than 0.97 (Figure 3i), indicating an excellent performance due to our method of SE.
Bond strain is related to the hopping energies which are used frequently in tight-binding calculation on electronic properties of graphene sheets [37] as well as determining the formation of pseudomagnetic fields. [38]Any changes in the equilibrium positions of atoms within a graphene sheet due to strain will be directly reflected in corresponding changes in the hopping energies as: where t 0 = À2.8 eV is the equilibrium hopping energy, a 0 = 0.142 nm is the length of the unstrained C-C bond, and r ij = jr i À r j j is the length of the strained bond between atom i and j.The decay factor β ¼ ∂ log t ∂ log a a¼a 0 ≈ 3.37 describes the change of the hopping energy with the modulation of the bond length. [39]opping energy field is a nonlinear function with respect to bond strain, as described in Equation ( 3), so that it can also be derived from the displacement field.For comparison, we also directly train a neural network to predict the deformationcoupled hopping energy field.Similar to the discussion of bond strain field, the results are displayed in Figure 3d-f,h,j.Again, our network shows excellent predictive power.Both the field derived from predicted displacement field and the field predicted directly from network match the ground truth with high accuracy.

Deformation-Coupled Stress Tensor Field
In this section, we discuss the application of our neural network to simultaneously predict three stress components.We also present the results from two approaches.For the derivation approach, we invoke the atomistic simulation to compute stress tensor using the predicted relaxed structure from displacement field.For the prediction approach, we directly use the three stress components obtained from atomistic simulations as the prediction targets.The neural network is trained to predict all three stress components simultaneously.
Figure 4a-l showcases the effectiveness of our methods on the same structure and same loading conditions, as depicted in Figure 2a, but focused on predictions on atomistic stress tensor properties.Figure 4a-i visually shows the degree of agreement labeled with absolute errors (AEs) for each stress component.Particularly, Figure 4j-l reveals the quantitative agreement with atomistic simulations for each stress component from 100 randomly selected atoms.For performance in test set, as shown in Figure 4m-o, the lowest R 2 error from the derivation approach is about 0.9, which is slightly lower than that of the prediction approach (0.97).The results show that our trained network has learned the mapping on deformation-coupled physical information, which provides a new approach for using neural networks as an alternative to atomistic simulations for stressstrain relationships.

Surrogate for Atomistic Simulations for Stress-Strain Relationships
Our neural network's ability to deliver high-fidelity field values on atoms or bonds makes it promising to serve as surrogate for atomistic simulations to obtain stress-strain relationships under complex loading combinations (Figure 5a).In this section, we conduct two experiments to showcase the effectiveness of our methods on the same structure, as depicted in Figure 2a, with the loading condition changes continuously.In the first experiment, all the externally applied strains (ε xx , ε yy , and ε xy ) linearly increase with the loading step simultaneously.The strain in the x-direction increases from 0 to 0.1 with a step size of 0.001.The strain in the y-direction also increases from 0 to 0.1 with a step size of 0.001.The shear strain Δ xy increases from 0 to 5 Å with a step size of 0.05 Å.All three directions are loaded simultaneously.In the second experiment, only the ε xx linearly increases with the load step so that the deformation protocol is uniaxial loading.Again, we use atomistic simulations to obtain the ground-truth values for performance evaluations, which include atomistic displacements due to relaxation, bond strain, and atomistic stresses after relaxation at all loading steps.
In terms of the acceleration of the computation speed, the predictions of pretrained neural network take about averagely 0.2 s on an Intel i7-12700 (2.10 GHz) CPU core for a data point, Derived values obtained from displacement and AE are in (c) and (f ).Comparisons of these values for randomly selected 100 bonds are in (g) and (h).Note that results in (a)-(h) are for the same defect and strain input as in Figure 2. The statistical analysis of the R 2 across the entire test set is shown in (i) and (j) for various defect and strain conditions.In the legend, the values in parentheses represent the corresponding mean and standard deviation of the R 2 score.
whereas the atomistic simulation using classic forcefield takes 6.5 s (Figure 5b).Note that this is only an illustration on the modeling system in current work.If the atomistic simulation is based on DFT calculations, then the acceleration effect would be even larger.The model size also matters.
In terms of accuracy, our approaches almost produce the precise results obtained from the atomistic simulations.Figure 5c shows that the R 2 values of various fields on atoms or bonds using direct prediction approach at every incremental loading step are all larger than 0.9, demonstrating high-accuracy match among various fields predicted from the neural network and computed from atomistic simulations.Note that the averaged R 2 value over all loading steps is greater than 0.97.For uniaxial loading along x direction, we report the uniaxial stress value as the average of σ xx of all atoms.5d shows the power of our neural network to produce the uniaxial stress-strain curve, which almost coincides with the ground truth (R 2 = 0.997 for prediction).These results demonstrate the potential of our neural network and SE method as an alternative approach for calculating deformation-coupled atomistic fields.

Remarks on Performance
Normally, converting a continuous field from images to specific value on atoms or bonds may be a challenging task.The subtlety is to collect a few suitable field values on pixels over a region covering the location of the atom or bond, out of which a single field value is then computed and assigned to an atom.Continuous mechanical loading poses more challenges as the location of atoms also evolves.Our work shows that it is possible to directly predict field values on each atom or bond purely based on geometry and loading conditions.
An important contribution of this work is the concept of SE (see Experimental Section) which allows the neural network to comprehend state of complex strains and predict its action on structure.The block of SE is repeatedly fed into the neural network at deeper levels, which is the main feature of our method for mapping deformation to properties.
Our SE methods have excellent scaling of data efficiency compared with the case of training individual neural network for every single strain condition.Admittedly, due to the inclusion of three strain components, each covering a continuous range, the space of strain combinations is vast.Previous literature studies focus fixed strain use about several thousand images. [22,23]ur neural network architecture only uses 8000 images in our training set, which have been proven to be highly effective to capture complex defects and strain states, and the computational cost is acceptable compared to training individual neural network for every single strain. [22,23]evertheless, our model is purely data-driven and relies on supervised learning.While our model achieves high accuracy in predicting many physical fields, it lacks certain generalization.In future research, introducing physical constraints to enhance the model's generalization capability and further improve its accuracy would be beneficial.
The focus of our work is about a unifying deep learning framework that is able to learn from complex strain states and defected configurations for the prediction of spatial field values on atoms or bonds.The studied field values are ultimately connected to the underlying atomistic configurations, emphasizing on establishing the structure-property relationship, which is central to discipline of materials science.It remains intriguing to investigate in the future if our deep learning framework is suitable for other local fields such as local density of states (for a particular energy level), and atomistic charges that can be generated using more expensive first-principles calculations, although the computational cost may be prohibitive considering that our atomistic models contain more than one thousand atoms.

Conclusion
A machine learning framework is developed to predict deformation-coupled fields on atoms or bonds in defectengineered graphene.Displacement fields due to relaxation, bond strain field, hopping energy field, and stress tensor fields after relaxation, for a wide range of complex loading conditions, are studied.The successful prediction on field values of atoms or bonds allows our neural network to work as a surrogate for atomistic simulations with extraordinary efficiencies.Our method is generally applicable to grasp the structure-deformation-field relationship for a wide range of materials.Particularly, the proposed SE framework, which plays the key role for capturing the deformation state, can be extended to embed a wide range of dynamically evolving factors, which can be jointly fed to the neural network for training.Although the current work is based on two-dimensional materials, it could potentially inspire future investigations on bulk materials.

Experimental Section
Graphical Representation of Defected Graphene and Image Processing: The graphene structures in this study contain two types of defects: typical S-W defects and vacancy defects.S-W defects have three possible orientations.We randomly introduce defects in the graphene.We randomly select an atom on the pristine graphene (with sizes of about 55 Å Â 55 Å) and rotate one of its three neighboring bonds by 90°to create an S-W defect.In our study, at most 15 S-W defects are introduced in a single structure.The total number of atoms remains the same as in perfect graphene after that.Next, without disturbing the S-W defects, we randomly select a carbon atom and remove n (where n is randomly chosen within the range of 1-4) atoms around it to create vacancy defects.The maximum number of vacancy defects is also set at 15.After the above operations, we obtain the initial configuration of the defected graphene.
For representing the structure and output fields in the neural network, we use gray images with a size of 256 Â 256 pixels.In the structure representation, the positions of atoms are depicted using solid white squares (pixel value 1), while empty positions are represented by black color (pixel value 0).Both the structure and displacement fields (similarly for other fields) are depicted on the undeformed configuration of the graphene, as paired images are suitable for training deep learning model.The images used for training are gray images, where the pixel values of the field images represent the magnitudes of field values.This allows us to efficiently utilize computational resources within the PyTorch neural network framework.The input structure images in PyTorch consisted of a single channel tensor of size (1, 256, 256).The training labels are tensors of size (2, 256, 256), with two channels representing the two displacement fields (similarly, one channel for the bond strain field and three for the stress tensor).
Atomistic Simulations for Ground Truth Dataset Generation: We consider graphene with periodic boundary conditions in both the x and y directions.To ensure that there are no interatomic forces in the z-direction, we use a larger height for the simulation box in the z-direction.The AIREBO [35] potential is used to calculate the interatomic forces and energies between atoms in the system, implemented in LAMMPS. [34]he system allows for atomic deformations in the x and y directions under tension and in the xy-direction under shear.We first follow the approach mentioned in Section 4.1 to create defected graphene.Then, within a specified strain range, we randomly select three strain combinations.The structure and strain combinations are the input information for atomistic simulations, as well as the input information for neural network.
The input and output on the field values constitute a data point for the training of neural network.The tensile strain ranges for ε xx and ε yy are set in the range of 0 to 10%, while ε xy is defined as a displacement Δ xy of the box in the xy direction (within the range of 0 to 5 angstroms).
All simulations are performed using LAMMPS and involve a series of energy minimizations using the conjugate gradient method.We utilize the LAMMPS command to directly modify the shape of the simulation box and perform energy minimization to obtain the output data file.Subsequently, Python code is used for postprocessing to prepare the data for training neural networks.
Data Composition: We use atomistic simulation to compute a total of 10 000 data points and perform postprocessing to obtain paired structure images (as inputs) and displacement fields images (as labels).Then, we divide the dataset into training, validation, and testing sets using an 8:1:1 ratio.The same division is applied to the bond strain fields and stress fields.Evaluation Metrics: To evaluate the accuracy of our network in predicting structural field properties, we make atom-wise or bond-wise comparisons, as enabled by our preprocessing treatment.We use the R 2 values to assess the agreement between the two sets of field values.The calculation for R 2 values is shown in Equation ( 4).Note that when calculating R 2 values, we exclude structures without defects since, after energy minimization, the field value on atom or bond in the pristine graphene is almost equal, resulting in zero variance and making it impossible to calculate R 2 .
We use AE (Equation ( 5)) to evaluate our model's performance.We do not use the RE (Equation ( 6)) because it becomes meaningless when the field value is zero or close to zero.
Atom-wise R 2 is defined as: where ûðx i Þ represents the predicted field value for the i-th point, uðx i Þ is the ground truth, and uðx i Þ is the average value of the uðx i Þ.
Atom-wise AE and RE are defined as: Deep Learning Approach: We use a general deep learning framework PyTorch to build our GAN model, which can transform structures and strains into displacement fields and other field spaces.The GAN consists of two main components: a generator and a discriminator.In our model, the generator is built using a neural network model called U-Net, originally designed for medical image segmentation. [32]It comprises an encoder and a decoder, with skip connections between them to facilitate information flow.We utilize the discriminator of PatchGAN, which evaluates the generated images and label images in small patches, ensuring that the generated images closely match the corresponding patches of the label images, rather than just the overall appearance. [33]ur models are trained on two NVIDIA RTX 3090 (24 GB) GPUs.After adjusting and optimizing the training process, we perform 300 epochs of training.We adopt a warm-up training strategy, [40] where the first 10 epochs are used for warm-up, with a batch size of 64.To prevent overfitting, we incorporate dropout with a probability of 0.2 in the neural network layers of both the generator and discriminator.Additionally, we employ a cosine learning rate schedule. [41]During training, we use the validation set to search for the best model performance without updating the model parameters.The parameter comparisons and evaluations presented in this article are conducted on the test set.
The architecture of the generator is illustrated in Figure 1.Both the encoder and decoder consist of eight layers.The loss function employed in our model is as follows: here ℒ gan is a sigmoid cross-entropy loss of the output image of discriminator and an array of ones, and ℒ image 1 is the mean AE between the generated image and the label image.λ 1 is proportion coefficient that is equal to 100.Note that in this work, although the analysis of the field results is based on images, we do not calculate the differences across the entire image.Due to the presence of many single white pixels in the images, we hypothesize that the neural network has a stronger ability to predict the white regions on the entire image, and this hypothesis is proved to be true in practice.Therefore, we specifically design a penalty term ℒ pixel 1 in the loss function of the generator to address this issue.In this penalty term, we select the pixels corresponding to the positions of the atoms in the image and calculate the L 1 loss, which is then multiplied by a coefficient λ 2 (=500) and added to the generator's loss function.This step improves the precision of our model's predictions at the atom level.
The Discriminator PatchGAN evaluates the generated field images by classifying individual patches (slice the image into 37 Â 37 patches in our case) in the image as "real" or "fake."The Discriminator loss function is defined as: here ℒ real is a sigmoid cross-entropy loss of real images and an array of ones, and ℒ fake is a sigmoid cross-entropy loss of the generated images and an array of zeros.Strain Embedding: Previous studies have based on a fixed strain value instead of considering evolving strains.The present study utilizes a SE method to adopt for deep learning model to investigate the consecutive strain changes.While there are various ways to encode strain, [42] we adapt and modify the sinusoidal encoding approach [43] to encode strain, based on which we propose a SE method.This embedding allows the neural network to predict the fields of the structure under continuous strains.The encoding formula used in this work is as follows: 1000ε dm (9)   where SE i represents the i-th element of the SE vector, which has a length of d.The ε corresponds to the strain value that needs to be encoded, and d m and d are hyperparameters.In this work, d m = 64.For the strains in the x and y directions, we used d = 101, while for the shear strain in the xy direction, we used d = 51.After encoding, the three vectors are concatenated to form a vector of length 253.This concatenated vector is then passed through two learnable linear layers and incorporated into the U-Net for training.Denote the batch size during training as B, then the strain vector's shape transforms from (B, 253) to (B, 512), and finally to (B, C), so that B batches of embedding vector of length C are obtained, where C matches the number of channels of the PyTorch tensor of hidden layer to be applied with the action of plus symbol, as shown in Figure 1.Further assume the shape of such tensor is (B, C, H, W), where B is batch size, C is channel size, H is the height, and W is width.Then the plus action works as follows.The B batches of embedding vector change dimension from (B, C) to (B, C, 1, 1), then element-wise additions are performed with the tensor of hidden layer with dimension (B, C, H, W) using broadcasting.

Figure 1 .
Figure 1.Overview of the computational framework in this work.a) The deformation state contains tensile strain (ε xx and ε yy ) and shear strain (ε xy , represented by Δ xy ).A carbon atom with initial coordinates (X, Y) transfers to coordinates (x, y) after deformation with a displacement vector μ.The bottom-left corners of the simulation box before and after deformation are fixed at coordinates (0, 0).b) Illustration on to-be-deformed graphene with Stone-Wales defects and vacancy defects that are studied in this work.c-f ) Brief illustrations on atom-wise pixilation.See main text for elaboration.The atomistic structure (c) is represented by a square matrix tensor (d).The normalized field value (illustrated by different colors) on each atom (e) ispresented by a square matrix tensor (f ).g) With initial structure and deformation state, real field images are generated using atomistic simulations, while predicted field images are produced from Generator.The goal of training is to make the predicted image as real as possible as judged by the Discriminator.The deformation state is repeatedly fed into the deep neural network through SE.Our work deals with four types of atomistic physical fields that can be directly predicted or derived from the Generator.

Figure 2 .
Figure 2. a) Comparison of predicted displacement components u and v with ground truth, for a particular defect and strain input, where the comparison of the ground truth with predictions for both displacement components for randomly selected 100 atoms are shown in (b).c) Histogram of atom-wise R 2 scores between the ground truth and the predictions in the test dataset with various defect and strain conditions.The average R 2 and standard deviation are indicated.

Figure 3 .
Figure 3. Performance for bond-wise bond strain and hopping energy.Ground truth values are in (a) and (d).Predicted values and AEs are in (b) and (e).Derived values obtained from displacement and AE are in (c) and (f ).Comparisons of these values for randomly selected 100 bonds are in (g) and (h).Note that results in (a)-(h) are for the same defect and strain input as in Figure2.The statistical analysis of the R 2 across the entire test set is shown in (i) and (j) for various defect and strain conditions.In the legend, the values in parentheses represent the corresponding mean and standard deviation of the R 2 score.

Figure 4 .
Figure 4. Performance for component-wise atom-wise values for all the three stress components.Ground truth values are in (a), (d), and (g).Predicted values and AEs are in (b), (e), and (h).Derived values obtained from displacement and AE are in (c), (f ), and (i).The comparisons of these values for randomly selected 100 atoms are in (j), (k), and (l).Note that results in (a)-(l) are for the same defect and strain input as in Figure 2. The statistical analysis of the R 2 score distribution across the entire test set is shown in (m), (n), and (o).In the legend, the values in parentheses represent the corresponding mean and standard deviation of the R 2 score.

Figure 5 .
Figure 5. a) Schematic showing the application of our neural network as surrogate of atomistic simulation under simultaneous loading in three directions.Three strains increase simultaneously with the loading step.b) High efficiency of neural network prediction on mechanical response of graphene, with a 33-fold acceleration from the calculation using atomistic simulations with classic forcefields.Note that this is only an illustration based on the modeling system in current work.c) Evolving atom-or bond-wise R 2 scores for atomistic physical fields of interests in our work as the load step increases.In the legend, the values in parentheses represent the corresponding mean and standard deviation of the R 2 score for all load steps.d) Strain-stress curve under uniaxial tension from atomistic simulation and neural network prediction.R 2 score for entire strain range is shown.
Our training processes are conducted on the training and validation sets.The training set is used to update the model parameters, while the validation set is used to evaluate the model's performance during training and find the best model for our task.The validation set do not update the model parameters.Finally, the test set is used to evaluate the performance of the trained model.