Transferring predictions of formation energy across lattices of increasing size

In this study, we show the transferability of graph convolutional neural network (GCNN) predictions of the formation energy of the nickel-platinum solid solution alloy across atomic structures of increasing sizes. The original dataset was generated with the large-scale atomic/molecular massively parallel simulator using the second nearest-neighbor modified embedded-atom method empirical interatomic potential. Geometry optimization was performed on the initially randomly generated face centered cubic crystal structures and the formation energy has been calculated at each step of the geometry optimization, with configurations spanning the whole compositional range. Using data from various steps of the geometry optimization, we first trained our open-source, scalable implementation of GCNN called HydraGNN on a lattice of 256 atoms, which accounts well for the short-range interactions. Using this data, we predicted the formation energy for lattices of 864 atoms and 2048 atoms, which resulted in lower-than-expected accuracy due to the long-range interactions present in these larger lattices. We accounted for the long-range interactions by including a small amount of training data representative for those two larger sizes, whereupon the predictions of HydraGNN scaled linearly with the size of the lattice. Therefore, our strategy ensured scalability while reducing significantly the computational cost of training on larger lattice sizes.


Introduction
The description of mixtures of multiple chemical elements forming a solid is of great importance for a wide area of condensed matter physics and material sciences.The interactions that lead to the formation of ordered compounds, that might foster different functional or mechanical behaviors or the disordered arrangement in alloys that are responsible for their technologically important properties can be described at many levels of theoretical treatment, ranging from highly accurate quantum chemical methods (e.g.first principles density functional calculations) that are intractable except for a small number of atoms, to effective atomistic models using classical mechanics and to continuum descriptions.The step from quantum mechanical to classical models is particularly challenging and time-consuming.Here we are exploring a machine learning (ML) approach to this step in the context of solid solution alloys as example systems.
Due to their enhanced functional properties, such as resistance to high temperatures and external stresses, solid solution alloys are an important class of alloys used in a wide range of applications, from aerospace and automotive industries to medical devices and electronics.At the atomic scale, solid solution alloys are characterized by a disordered arrangement of atoms of different pure elements within the crystal structure.The disordered atomic configuration results in improved macroscopic behavior of the materials compared to intermetallic (ordered) compounds.
The experimental study of solid solution alloys employs various techniques to investigate the atomic structure of the material, including x-ray diffraction and transmission electron microscopy.However, these techniques are limited in their ability to provide a complete picture of the atomic arrangement of the material.In fact, the behavior of the electrons and atoms in solid solution alloys is difficult to understand and predict using experimental techniques alone.
Density functional theory (DFT) [1,2] is an important computational method for studying solid solution alloys [3][4][5] that compensates for the limitations and impracticality of experimental approaches, as it allows researchers to study the properties of solid solution alloys at the electronic level, including how the composition affects the properties of the resulting alloy [6,7].Studying solid solution alloys at the atomic and electronic levels requires considering crystal structures sufficiently large in size (i.e. containing thousands of atoms) to accurately capture the effect of the configurational entropy on the material properties.The influence of atomic disorder on thermodynamic and mechanical properties has already been explored computationally [6][7][8][9] for various families of crystalline materials including solid solution alloys.Unfortunately, DFT calculations can fast become computationally expensive or infeasible, especially for systems with thousands of atoms.Within the context of material design where several chemical compositions and atomic configurations need to be explored in the search of materials with desired functional properties, this computational challenge increases further due to the combinatoric complexity of the chemical space.A thorough and time-efficient exploration of such a high-dimensional space requires fast (but still accurate) evaluations of the target property for every possible atomic arrangement.Due to the dimensionality of the chemical space that scales exponentially with the size of the crystal structure, using DFT calculations to thoroughly explore such chemical space in the context of material design becomes prohibitive.
ML models have shown the potential to greatly accelerate DFT calculations while maintaining a high degree of accuracy [10][11][12].In particular, deep learning (DL) models have shown the ability to effectively capture relevant non-linearities caused by disordered atomic configurations of an atomic system [13][14][15][16][17][18][19][20][21][22].In fact, DL models can be trained on a subset of the data generated from DFT calculations and then used to predict the properties of interest for new inputs.The training and use of the DL models for inference can be performed in a fraction of the time it would take to run a full DFT calculation, while still producing quite accurate results.This drastic reduction of time to predict material properties of solid solution alloys using atomic information results in a promising path towards an effective acceleration of material discovery and design [23][24][25][26][27][28].Among the several material properties of solid solution alloys that can be computed using the atomic structure, the formation energy plays an important role as its minimization allows to categorize stable and metastable atomic configurations at a given composition and temperature via classical Monte Carlo calculations.
Graph convolutional neural networks (GCNNs) are effective DL surrogate models for material science [29][30][31][32][33][34][35][36][37].In particular, GCNNs have been shown to accurately predict the formation energy of disordered atomic configurations for solid solution alloys [38][39][40][41] because they take advantage of the topological information of the data samples by representing the atomic configurations of each crystal structure as graphs.In this representation, each atom is represented as a node in the graph, and the edges between nodes represent the bonding interactions between the atoms.This graph representation can capture the local environment and coordination of the atoms, which are important factors that contribute to the formation energy of the material.The GCNN is then trained on a dataset of known formation energies of disordered atomic configurations for solid solution alloys.During training, the GCNN learns to map the graph representations of the atomic configurations to their corresponding formation energies.Once trained, the GCNN can be used to predict the formation energy of new, previously unseen disordered atomic configurations.
Recent results showed that GCNN models can accurately predict the formation energy of solid solution alloys using training data that thoroughly samples different crystal phases, chemical compositions, atomic disorder, and atom displacements from the ideal crystal structure due to geometry optimization [41].However, the existing studies for solid solutions are limited to small atomic systems, thereby neglecting the effect of long-range interactions on the formation energy.Long-range interactions refer to the interactions between atoms that are separated by more than one or two atomic distances and they may play a non-negligible role in determining the properties of solid solution alloys.However, incorporating long-range interactions in GCNN models is challenging due to the computational cost to run DFT calculations on crystal structures large enough to accurately capture them.Recent work described in [42] has developed GCNN models for transferable potential energy surface prediction of alumina polymorphs across different lattice sizes and crystal structures and by fixing the chemical composition of the material.This work sheds light on promising transferability property of GCNN models, and encourages further research to assess the transferability of GCNN predictions for other classes of materials, such as solid solution alloys, and to extend the transferability study by spanning several chemical compositions of the material.
In order to limit the computational cost for collecting DFT training data for solid solution alloys that captures both short-range and long-range interactions, we propose a strategy that collects more data of small crystal sizes and a lower amount of data of crystals of larger size.The choice of this sampling strategy is motivated by the following reasons: • short-range interactions are the dominating factor in the formation energy of solid solution alloys and can be learned by GCNN models from DFT data calculated for small crystal structures; • GCNN models can learn local inter-atomic interactions from small crystal structures which then can be used for predictions on crystal structures of larger size; and • the DFT data collected on larger crystals is needed only to correct the GCNN predictions by including longrange interactions.
As an illustration, we train the GCNN model on an open-source dataset for the nickel-platinum (NiPt) solid solution alloy [43] that provides the formation energy calculated with large-scale atomic/molecular massively parallel simulator (LAMMPS) using the second nearest-neighbor modified embedded-atom method (2NN MEAM) empirical potential (used in this work for illustration purposes) for a number of different atomic configurations.Note that while our ultimate goal is to provide a model for first principles DFT data, in the present work we choose to substitute these results with those from classical force fields, to reduce the computational cost of data collection, while retaining qualitative aspects of the atomic interactions.The dataset comprises data for three different sizes: 256 atoms, 864 atoms, and 2048 atoms, respectively.The first set of numerical results shows the performance with respect to the accuracy and training time of the GCNN model when it is trained and tested on crystal structures of the same size.These results show the algorithmic scalability of GCNN models, meant as the property of the model to have a training cost that scales linearly with the size of crystal structures while maintaining the same validation performance.The second set of numerical results shows the efficacy of our proposed sampling strategy.To this aim, we compare the accuracy of a GCNN model trained only on crystal structures of 256 atoms with another GCNN model where the training data is augmented with a small percentage (gradually incremented up to 20%) of crystal structures containing 864 atoms and 2048 atoms, respectively.The numerical results show that adding a small percentage of data with crystal structures containing 864 and 2048 atoms helps the GCNN model correct the predictions of the formation energy by including the effect of long-range interactions, thereby ensuring the transferability of the GCNN model to crystal structures of larger sizes.The rest of this work is organized as follows.Section 2 reviews briefly relevant related work.Section 3 describes the methodology adopted to collect the atomic modeling data and the construction of the GCNN architecture.Section 4 provides the details of the numerical results we obtained in terms of (i) algorithmic scalability of GCNN in handling disordered atomic configurations of varying sizes, and (ii) effectiveness of our proposed sampling strategy to ensure the transferability of GCNN predictions across lattices of increasing sizes at a reduced computational cost for data collection.Section 5 elaborates on existing ML and DL studies to develop surrogate models for predictions of material properties of solid solution alloys, and the significance of our contribution in this field.Section 6 summarizes our work and the future directions in which the existing effort will be further developed.

Related work
State-of-the-art computational approaches to simulate and study the properties of solid solution alloys include methods such as special quasi-random structure (SQS) [44][45][46] as well as various types of ML models.SQS was developed to capture the local atomic arrangements and short-range order in disordered systems, such as alloys, in a more efficient manner compared to fully random structures.Since describing the atomic arrangement in a disordered alloy can be challenging due to the lack of long-range order and the large number of possible atomic configurations, the SQS method addresses this challenge by approximating the disordered alloy structure using a set of SQSs.These structures are derived from a periodic crystal lattice, typically a supercell, in which the positions of atoms are arranged based on certain rules and constraints.The key idea behind the SQS model is to ensure that the short-range order in the alloy, such as pair correlations between neighboring atoms, matches statistically that of a fully random structure.For small-sized to medium-sized systems, typically containing hundreds of atoms, the SQS model is generally considered computationally efficient.However, as the size of the system increases to thousands of atoms or more, the computational cost of using the SQS model becomes prohibitive even for existing US-DOE supercomputing facilities because the number of needed special quasi-random structures increases exponentially with the system size [47].
Previous works on ML for solid solutions have addressed the challenge of capturing the effects of disordered atomic configurations using cluster expansion (CE) [10,12,48,49] models.Albeit these models have the advantage of generating interpretable features, the inclusion of higher-order interactions to model the disordered mixing of multiple atom species makes them prone to overfitting [50].In particular, CE models struggle when they predict the formation energy of solid solution ferromagnetic alloys across a vast compositional range [40].Generic DL models, being specifically designed to accurately capture non-linear trends, provide fast and accurate estimates of material properties from atomic information [24,50,51] for materials with disordered atomic configurations.However, generic DL models often improve accuracy with respect to other data-driven models by sacrificing interpretability.Moreover, the computational training of DL models generally does not scale with the size of the crystal structures.GCNN models have shown significant advantages over other types of DL architectures both in terms of interpretability (the topological structure of a crystal lattice is naturally embedded into a graph) as well as in terms of scalable training.We discuss the scalability of GCNN training across lattices if increasing size in section 5.

Dataset description 3.1.1. NiPt
This dataset describes ground state properties of the NiPt solid solution binary alloy, which is interesting due to its magnetic and charge transfer properties [52].The two constituent elements, nickel (Ni) and platinum (Pt), are randomly placed on the face-centered cubic crystal structure (figure 1 shows an example of disordered atomic arrangement).The lattice constant for this structure is 3.840 angstroms.The dataset encompasses three distinct sizes of the crystal structure: 256 atoms, 864 atoms, and 2048 atoms, each of which comprises 1900 configurations.For each lattice size, the data spans concentrations from 0% of Pt to 100% of Pt in the NiPt binary system, with increasing the concentration of Pt in the system every 5%.A total of 100 random configurations were generated for each chemical composition, each with a different random seed.The output files for each configuration provide details such as mass, type, atomic coordinates, energy per atom, and forces in the x, y, and z directions.For each atomic configuration, the output was collected every 150 steps during the minimization stage and every 1000 steps during the replica exchange stage.LAMMPS [53], which is a molecular dynamics code, was used to generate data for NiPt alloy.The simulation used the interatomic potential for NiPt binary system 'MEAM_LAMMPS_KimSeolJi_2017_PtNi__MO_ 02 084 0179 467_001' [3] from the OpenKIM library (Open Knowledgebase of Interatomic Models) [54][55][56][57].This potential was developed based on the 2NN MEAM.The simulation process starts generating the random NiPt structure and continues with the short minimization and a simulation employing replica exchange.During the optimization step, atomic coordinates are adjusted and energy minimization takes place, typically resulting in a local minimum of potential energy.The conjugate gradient algorithm was employed for this minimization process.The subsequent replica exchange simulation, known as parallel tempering, involves four separate replicas or ensembles of the system and occurs after the optimization stage.Multiple snapshots of the configuration were gathered throughout both the optimization and replica exchange phases.In figure 2 we show the distribution of values of the formation energy in histograms and we also plot the formation energy as a function of the chemical concentration of Ni atoms for lattice sizes of 256 atoms, 864 atoms, and 2048 atoms.The total number of data samples for the atomic system with 256 atoms is 65 046.The total number of data samples for the atomic system with 864 atoms is 63 936.The total number of data samples for the atomic system with 2048 atoms is 61 997.
For each atomic configuration, we provide the formation energy at the ground state (0 Kelvin).In the ground state, the total energy E of an alloy can be written as where N elem is the total number of distinct elements in the system, c i is the molar fraction of each element i, E i is the molar energy of each element i, and ∆E form is the formation energy.We predict the formation energy for each sample by subtracting the internal energy from the EAM-computed total energy.The formation energy is more directly related to the atomic configuration, and emphasizes the importance of an effective GCNN model for accurate predictions.

GCNN
A graph is usually represented by a set of nodes and a set of edges between these nodes that describe the connectivity of the graph [58].GCNNs [59] are DL models based on a message-passing framework to exchange information of a node in the graph with all the others within a prescribed neighborhood.The typical GCNN architecture is characterized by three different types of hidden layers: graph convolutional layers, graph pooling layers, and fully connected (FC) layers.

Graph convolutional layers
The convolutional layers are the backbone of the GCNN architecture and are used to iteratively propagate information between adjacent nodes (in this case atoms).Through consecutive steps of message passing, the graph nodes gather information from nodes with increasing mutual distance.The message passing framework enables the GCNN model to simultaneously handle graphs of different size by adjusting the size of the message vectors and the number of message passing iterations based on the size of the input graph.A variety of graph convolutional layers have been developed and they all differ for the mathematical operations used to perform the message passing.In this work, we will use the principal neighborhood aggregation (PNA) [60] as it has been shown to significantly outperform other graph convolutional layers for predictions of the formation energy of solid solution alloys [41].

Global pooling layers and FC layers
Global pooling layers aim at collapsing the node feature associated with each atom across a graph into a single feature.This is achieved by collecting the local interactions of each atom with its neighbors and use these values to estimate global properties (e.g.summation, maximum, minimum, standard deviation).FC layers are at the end of the architecture to take the results of the global pooling layer and generate the predictions as output.

HydraGNN: distributed PyTorch implementation of GCNNs
The GCNN model we use for this study is the open-source, scalable architecture HydraGNN [61] that uses Pytorch [62,63] as both a robust DL library, as well as a performance portability layer for running on multiple hardware architectures.This enables HydraGNN to run on CPUs and GPUs, from laptops to supercomputers.The Pytorch Geometric [64,65] library built on Pytorch is particularly important for our work and enables many graph convolutional layers to be used interchangeably.HydraGNN is available on GitHub (https://github.com/ORNL/HydraGNN).

Sampling across lattices of increasing size to capture long-range effects
Our proposed sampling strategy aims at minimizing the computational cost for data collection while still ensuring that HydraGNN captures both the short-range and the long-range interactions needed for accurate predictions of the formation energy in disordered atomic configurations of solid solution alloys.The sampling strategy takes advantage of the capability of HydraGNN to transfer learned short-range interactions across samples of variable size to concentrate most of the data in regions characterized by small crystals that are less expensive to collect.A small portion of more expensive data samples for larger crystals is included in the training data to primarily correct the predictions of the surrogate model by including long-range interactions that are not included in data samples at a small scale.This sampling strategy is a computationally effective alternative to the SQS method to thoroughly span the configurational entropy space of disordered atomic configurations of alloys with thousands of atoms.The efficacy of our sampling strategy relies on the ability of the model to transfer features that describe short-range and intermediate-range interactions across lattices of different size, thereby alleviating the amount of data required to accurately capture long-range interactions on crystals of large size (i.e.thousands of atoms or more).This results into an effective reduction of the computational cost for the collection of DFT data with respect to SQS models.

Results
In this section, we present the numerical results that illustrate the efficacy of the sampling strategy proposed in this work.Section 4.1 describes the hyperparameter configuration selected for training.Section 4.2 provides benchmark results based on traditional training approaches to illustrate the model's performance both in terms of accuracy and computational time when the model is trained and tested on crystal structures of the same size.These results are used to test the algorithmic scalability of the training, i.e. the property of HydraGNN to have a training cost that scales linearly with the size of crystal structures used for training while maintaining the same validation performance across crystals of the same size.Section 4.2.1 illustrates the model's performance both in terms of accuracy and computational time when the baseline training dataset with a crystal structure of 256 atoms is augmented with a small amount of data that describe larger crystal structures containing 864 atoms and 2048 atoms, respectively.

Training setup
The HydraGNN architecture used in this study comprises six PNA convolutional layers, each containing 50 neurons.Initially, we chose a cutoff radius of 5 Å to construct the local neighborhoods used by the PNA mask.Two FC layers, each consisting of 50 neurons, are appended to the stack of PNA layers.The models are trained using the Adam method [66] with a learning rate of 0.001, batch sizes of 32, and a maximum of 50 epochs.To prevent overfitting, early stopping is employed, wherein training is halted if the validation loss function fails to decrease for consecutive epochs, indicating that further epochs are unlikely to improve the loss function value.The training set accounts for 80% of the total dataset, while the remaining data is evenly divided for validation and testing.We adopted a compositional stratified splitting approach, similar to our previous work [22,38], to ensure equal representation of all compositions in the training, validation, and testing subsets of the dataset.The training of each GCNN model was executed on an NVIDIA A100 GPU.

Algorithmic scalability
We first train and test HydraGNN on crystal structures of the same size.For each size of the crystal structures considered (i.e.256 atoms, 864 atoms, and 2048 atoms), we assess the predictive performance of the model both in terms of accuracy and training time for increasing percentages of the amount of data from the training set, namely 10%, 20%, and 100%.The training results on different percentages of the training set for crystal structures with 256 atoms, 864 atoms, and 2048 atoms are shown in figure 3. Across all three sizes of the crystal structures, the MAE on the testing portion of the dataset diminishes by a 3x factor between using only 10% of the training set and 100%.In figure 4 we show the scatterplots of the GCNN models trained on 80% of the data and tested on the remaining 20% for crystal structures with 256 atoms, 864 atoms, and 2048.In figure 5 we show the training time as a function of the size of the crystal structure.To further qualify the scaling performance of the training, we included one additional set of crystal structures with 108 atoms [67] for the training, where the crystal structures have been obtained with the same methodology as already described in section 3.1.1.The scaling results show that the training time of the model scales linearly with the size of the crystal.Moreover, the model attains a final MAE on the testing set comparable in magnitude across all lattice sizes.For physical systems like solid solution alloys where the graph representation of the crystal is characterized by a high degree of connectivity to model the delocalization of the valence electron in metallic bonds, this result clearly shows the computational advantage of using HydraGNN over other types of DL architectures (e.g.multi-layer perceptrons or transformers) for which the training time on highly connected graphs would scale polynomially.

Sampling strategy for transferability of GCNN predictions across crystal structures of increasing size
We show the performance of HydraGNN both in terms of accuracy and training time when the baseline training set with crystal structures of 256 atoms is augmented with a small percentage (10% and 20%) of data from the training sets with crystal structures of 864 atoms and/or 2048 atoms.The accuracy values are reported in figure 6.We notice that the predictions on crystals with 2048 atoms benefit from the inclusion of additional training data both from 864 atoms and 2048 atoms.However, the predictions on crystal structures with 864 atoms benefit only from the inclusion of crystal structures with 864 atoms and not from the inclusion of crystal structures with 2048.We attribute this phenomenon to the fact that crystal structures with 864 atoms contain short-range and intermediate-range interactions that can be transferred to boost the  learning on crystals of larger size.In contrast, crystal structures with 2048 atoms contain long-range interactions that, albeit learned by the model, are not relevant and therefore not transferable to smaller size crystals.Finally, we notice that using both 20% of crystal structures with 864 atoms and 2048 atoms helps the HydraGNN model reach a comparable accuracy to the one reached when the model is trained on the entire training set of crystals containing 2048 atoms.In figure 7 we compare the training time between baseline training approaches where the entire training set with crystals of a fixed size is used versus the training that follows the guidelines of our sampling strategy.We notice that our approach leads to a 2x speed-up of the training by using only 20% of training data from larger crystals while still maintaining the final desired  accuracy of the trained model.In figure 8 we show the scatterplots for the GCNN predictions when the model is trained with data that follows our sampling strategy.
This result highlights a two-fold computational benefit from our sampling strategy: a significant reduction of the computational cost to collect the training data needed at intermediate and large crystal sizes to reach a prescribed accuracy, and an effective reduction of the GCNN training time.

Discussion
Accurately predicting the formation energy of solid solution alloys as a function of the atomic disorder is extremely important to study how the atomic structure affects the thermodynamic behavior of a solid solution alloy.In order to capture all the relevant contributions from an atomic scale that determine the formation energy of the atomic system, it is important to consider crystal structures with thousands of atoms to ensure that the effect of long-range interactions is considered.However, the expensive computational cost of state-of-the-art high-fidelity DFT approaches precludes the use of these techniques for a thorough analysis of atomic structure with such a large size.
To circumvent the challenges of the DFT approaches, a lot of work has been done for this type of application using various types of DL models to derive the properties of the material from its structural information.Despite the ability of DL models to provide fairly high accuracy in capturing non-linear structure-dependent properties, the models have been shown to carry two main advantages over other DL architectures [31,34,38].One advantage is physical interpretability by naturally embedding the topological structure of a crystal into a graph, where atoms are treated as nodes and interatomic bonds as edges.The message-passing operations performed by HydraGNN maintain a high degree of physical interpretability because the inter-node exchange of information models the interatomic interactions within an atomic system.Another advantage is the scalability of the training, evidenced by the fact that the computational time to train HydraGNN increases linearly with the number of atoms in the lattice.So far, existing studies that use GCNN models for solid solution alloys have been limited to crystal structures of small/moderate scale [31,34,38,41].To the best of our knowledge, a study that illustrates the predictive performance of GCNN models on solid solution alloys with crystal structures of thousands of atoms across a vast compositional range still needs to be conducted.
Our study extends previous work on GCNNs for solid solution alloys [41] by showing that GCNNs can transfer predictions across lattices of larger size up to thousands of atoms.This effective transferability allows GCNNs to model long-range interactions, thereby enabling more accurate estimates of material properties and boosting an effective AI-enabled material design.
Although GCNN models typically outperform other ML model with simpler architectures in terms of accuracy and ability to capture complex features, it is important to acknowledge that GCNNs have also noteworthy limitations and their use to more realistic scenarios beyond ideal optimized crystal structures is needed.Firstly, GCNNs often require a larger volume of training data.This factor may restrict the practicality of employing existing GCNNs in domains where data availability is limited.Secondly, GCNNs primarily focus on local neighborhood information and may struggle to effectively incorporate global information from the entire graph.This limitation can restrict their ability to capture complex dependencies and patterns that span the entire graph structure [41,68].Lastly, the predictive performance of GCNN models in more realistic situations including defects (e.g.atom vacancies) [69] and lattice site impurities [70] still needs to be studied.

Conclusion and future work
We proposed a sampling strategy to collect disordered atomic configurations for solid solution alloys that aim at minimizing the computational cost for data collection while still ensuring that the model captures both short-range and long-range interactions for accurate predictions of the formation energy.Our sampling strategy takes advantage of the capability of the model to transfer learned short-range interactions across samples of variable sizes.This allows one to concentrate on collecting most of the data for small crystals, which is less expensive to collect.A small portion of more expensive data samples for larger crystals is included in the training data to correct the predictions of the surrogate model by including long-range interactions that are not included in data samples at a small scale.
Our results set a promising path towards effectively sampling combinatorically complex chemical spaces by significantly reducing the computational cost for training without sacrificing the accuracy of the trained surrogate model.Although our results focused on the training performance of GCNN models as surrogates for atomic modeling simulations, our goal goes beyond using GCNNs as a replacement for the atomic modeling solvers.More generally, we aim to use GCNN models to construct well-educated initial guesses that could be used as a starting point for physics-based simulations, enabling the numerical study of large-scale atomic structures that otherwise would not be computationally affordable.
Future work will be dedicated to training the GCNN model described in this work, namely HydraGNN, on higher fidelity DFT data that describes the formation energy of high entropy alloys as a function of the configurational entropy, and integrate the trained GCNN surrogate within a Classical Monte Carlo framework to accelerate the acceptance-rejection steps of the Markov chains and speed-up the calculations of the thermodynamic properties such as specific heat and the completion of the temperature vs. chemical composition phase diagram.We will also extend the existing work to assess the capability of GCNN models to include the effect of defects (e.g, atom vacancies) and lattice site impurities.

Figure 1 .
Figure 1.NiPt solid solution binary alloy with 50% chemical concentration of Ni and 50% chemical concentration of Pt.

Figure 2 .
Figure 2. Plots of the formation energy as a function of the chemical composition (left) and histogram of the formation energy (right) for crystal structures with 256 atoms (top), 864 atoms (center), and 2048 atoms (bottom).

Figure 3 .
Figure 3. GCNN accuracy as a function of the percentage of training data.

Figure 4 .
Figure 4. Scatterplot of GCNN predictions.HydraGNN is trained and tested on crystal lattices of the same size: 256 atoms (top-left), 864 atoms (top-right), and 2048 atoms (bottom-center).The colormap describes the density of points.

5 .
GCNN training time as a function of the crystal structure size.

Figure 6 .
Figure 6.GCNN accuracy for predicting the formation energy on crystal structures of 864 atoms (top) and 2048 atoms (bottom).The GCNN model is trained on crystal structures of 256 atoms with an additional small percentage (10% and 20%) of data that describes crystal structures with 864 atoms and 2048 atoms.

Figure 7 .
Figure 7.Comparison of GCNN training time between the training set with crystals of fixed size and data augmentation according to the proposed sampling strategy.

Figure 8 .
Figure 8. Scatterplot of HydraGNN predictions when the training on crystal structures with 256 atoms is augmented with a small percentage of crystal structures with 864 atoms and/or 2048 atoms.The colormap describes the density of points.