Performance of two complementary machine-learned potentials in modelling chemically complex systems

Multicomponent alloys, particularly high-entropy alloys (HEAs), exhibit exceptional properties due to their complex compositional space. However, this complexity presents challenges in developing accurate interatomic potentials. Traditional potentials like EAM and MEAM lack the flexibility to model such systems effectively. Machine learning (ML)-based potentials offer a more systematic approach, encoding atomic environments through local representations that capture both vibrational and configurational degrees of freedom. This paper focuses on two ML potentials, MTP and GM-NN, to investigate their ability to simultaneously model vibrational and configurational entropy in the Ta-V-Cr-W alloy family across a wide temperature range. The study aims to evaluate the accuracy and efficiency of these potentials against density functional theory (DFT) calculations, considering both in-distribution (used for training) and out-of-distribution (not used for training) subsystems of the alloy family. This includes studying 0K energies and forces in binaries, ternaries, and quaternaries, as well as near-melting temperature energies and forces in disordered alloys. This comprehensive approach is crucial for accurately analyzing the thermodynamics of the alloy across the entire temperature range, from the high-temperature disordered phase to the low-temperature ordered phases.

Previous studies have explored the application of machine learning potentials to model HEAs, with MTPs being used for thermodynamic property prediction up to the melting point of a single disordered phase of TaVCrW and partial compositional spaces of TiZrHfTay. However, the application of GM-NNs to metallic HEA systems remains largely unexplored. While DFT calculations provide high accuracy, their computational cost restricts their application to larger system sizes needed to study short-range ordering, highlighting the need for accurate and efficient interatomic potentials. This paper addresses this gap by applying and comparing MTPs and GM-NNs for the complex Ta-V-Cr-W system.

The study employs two machine-learning potentials, the moment tensor potential (MTP) and the Gaussian moment neural network (GM-NN). Both potentials assume the locality of interatomic interactions, meaning the total energy of a system is the sum of individual atom energies, each calculated considering interactions within a cutoff radius (5.0 Å in this study). The atomic environment is encoded using symmetry-preserving representations: invariant polynomials for MTP and Cartesian Gaussian-type orbitals for GM-NN. These descriptors are invariant to translations and permutations but equivariant to rotations. The descriptors serve as input to a linear regression (MTP) or neural network (GM-NN) to predict the atom's energy. Both models are trained on DFT data (energies, forces, and stresses) using a combined loss function that balances the contribution of these properties. For active learning, uncertainty measures based on gradient feature maps are employed. For MTP, the maximum volume (MaxVol) principle is used; for GM-NN, the largest cluster maximum distance (LCMD) method and last-layer gradient features (FEAT(LL)) are employed. The training data encompasses various configurations: 0 K configurations in small, medium, and large supercells representing different ordering states (binaries, ternaries, quaternaries, strained structures); and high-temperature (2500 K) disordered structures obtained from molecular dynamics simulations. The models' performance is evaluated on in-distribution (training data) and out-of-distribution (non-equiatomic ternaries not included in the training data) subsystems. The accuracy is assessed using RMS errors in energies and forces. Thermodynamic properties such as isobaric heat capacity are also calculated using thermodynamic integration. The training and inference times for both models are compared, considering the effect of model parameters (number of descriptors for MTP, number of neurons and input features for GM-NN) and hardware (CPU vs. GPU).

Both MTP and GM-NN potentials demonstrate remarkable accuracy in predicting DFT values for in-distribution subsystems (0 K binaries, ternaries, quaternaries, and 2500 K disordered quaternary), with RMSEs in energies of less than a few meV/atom and RMSEs in forces of less than 0.15 eV/Å. The models are equally accurate, with MTP slightly better for high-temperature predictions and GM-NN showing marginal advantage for 0K predictions. The prediction of the heat capacity up to 2200 K also showed good agreement with DFT values. Even for out-of-distribution subsystems (non-equiatomic ternaries), the models showed good generalization ability. While absolute energies exhibited larger errors (up to 20 meV/atom), the relative energies (relevant for thermodynamic properties) remained accurate (below 5.5 meV/atom). In comparison, classical EAM/MEAM potentials showed significantly higher errors (one to two orders of magnitude). MTP models exhibited faster convergence with training set size compared to GM-NN models due to fewer trainable parameters. Active learning improved the convergence of forces for MTP but had minimal impact on energy errors. On CPUs, MTPs were faster in terms of inference times, while GM-NNs demonstrated faster performance on GPUs. The MTP demonstrated better performance on the 2500 K structure.

The results demonstrate the high accuracy and transferability of both MTP and GM-NN potentials in modeling complex multicomponent alloys. The ability to accurately predict both vibrational and configurational degrees of freedom enables modeling of the alloy's behavior across a wide temperature range, including low-temperature ordering and phase separation, and high-temperature disordered phases. The comparable performance of MTP and GM-NN, despite their different underlying formalisms, suggests a maturity in the field of machine-learning potentials. The better performance of MTP for high temperature is attributed to the fact that active learning in the training of the MTP preferentially selected high temperature configurations. The superior accuracy and efficiency compared to classical potentials makes these ML models suitable for simulating large-scale systems, enabling studies of phenomena such as short-range order and order-disorder transitions, which are crucial for understanding the mechanical properties of HEAs. The limitations of active learning in certain cases, particularly for energy, highlights the importance of careful data selection during training. The interplay between accuracy and computational cost emphasizes the need for model selection based on specific application requirements.

This study demonstrates the effectiveness of MTP and GM-NN potentials in modeling the complex Ta-V-Cr-W alloy system, accurately capturing both configurational and vibrational degrees of freedom. Both models provide comparable accuracy, with MTP exhibiting faster training convergence and GM-NN offering faster execution on GPUs. Active learning is beneficial for force prediction, particularly in MTP, but less so for energy. Future work could focus on including electronic and magnetic degrees of freedom, using advanced active learning techniques, and exploring the application of these potentials to predict other properties and processes in HEAs.

The study's accuracy relies on the DFT data used for training and validation. The DFT calculations themselves have inherent limitations, and potential errors in the DFT calculations could affect the accuracy of the trained potentials. Further limitations include the focus on the Ta-V-Cr-W system; the generalizability of the findings to other alloy systems needs further investigation. Additionally, the active learning strategies used might not be optimal and further improvements in active learning methodologies for this class of problem could enhance the performance of machine learned potentials.