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

High-entropy and compositionally complex alloys, such as Ta–V–Cr–W, exhibit remarkable mechanical properties but pose challenges for accurate atomistic modeling due to the need to simultaneously capture configurational (ordering, segregation) and vibrational (finite-temperature) effects across a vast compositional space. Ab initio simulations (DFT/AIMD) are accurate but computationally prohibitive for the large cells required to capture short-range order and phase separation. Conventional empirical potentials (EAM/MEAM) lack the flexibility to represent complex interactions. This work investigates whether two state-of-the-art machine-learned interatomic potentials—the moment tensor potential (MTP) and the Gaussian moment neural network (GM-NN)—can accurately and robustly model both configurational and vibrational degrees of freedom for the Ta–V–Cr–W alloy family across 0 K ordered phases and near-melting disordered phases, including assessment on out-of-distribution compositions and the role of active learning.

Conventional interatomic potentials (EAM/MEAM) rely on restrictive functional forms and struggle with complex many-body interactions typical of multicomponent alloys. Recent ML potentials employ symmetry-preserving local descriptors and flexible mappings to energies/forces, enabling near-DFT accuracy. MTPs have previously modeled thermodynamics up to melting for TaVCrW and parts of TiZrHfTa_y spaces. GM-NNs, based on Gaussian moment descriptors and neural networks, have shown success for molecular and surface systems (e.g., N and H2 on ices, magnetic anisotropy tensors in molecular crystals) but had not been tested for metallic HEAs. Active learning frameworks leveraging uncertainty estimates have been developed to efficiently sample informative configurations. This study positions MTP and GM-NN within this context, comparing their capabilities on a refractory HEA system and contrasting to EAM/MEAM baselines.

Models and representations: Both MTP and GM-NN assume locality with a cutoff radius R_cut = 5.0 Å, decomposing total energy into atomic site contributions. They use rotationally equivariant intermediate tensor representations followed by rotationally invariant contractions as features. - MTP: Site energy is a linear expansion in basis functions constructed from moment tensor descriptors with radial Chebyshev expansions and tensor contractions. Model complexity is controlled by a level parameter (lev_max); this work uses lev_max = 24. Parameters are fit using BFGS to minimize a combined loss on energies, forces, and stresses with weights C_E = 1/N, C_F = 0.001 Å^2, C_σ = 0.0001/N. - GM-NN: Uses Gaussian moment (GM) features with trainable chemical parameters and radial basis (8 Gaussians; N_basis = 6), contracted up to four-body terms, mapped by a fully connected NN (default 593-512-512-1, Swish activations, neural tangent parameterization). Trained with Adam (β1=0.9, β2=0.999, ε=1e-7), minibatch 32, 1000 epochs, layer-wise learning rates (0.0075 FC layers, 0.005 representation, 0.0125/0.00025 for shift/scale). Early stopping prevents overfitting. Active learning and uncertainty: - MTP: MaxVol-based selection using determinant of submatrix of basis-function derivatives. - GM-NN: LCMD batch selection using last-layer gradient features (FEAT(LL)) to estimate similarity/uncertainty. Data generation (DFT): VASP with PAW, GGA (PBE), spin-unpolarized; Methfessel–Paxton (order 1, α=0.1), plane-wave cutoff 350 eV, automatic k-mesh with KSPACING=0.12. Valence electrons: Ta/V=11, Cr/W=12. Dataset composition (total 6711 configurations): 1) 0 K small supercells (2–8 atoms): 4491 binaries across TaV, TaCr, TaW, VCr, VW, CrW (enumlib), plus 248 equiatomic quaternaries (from substitutions). Includes 595 ternaries and 594 quaternaries overall at 0 K. 2) 0 K medium/large supercells (16, 128, 432, 8192 atoms): Binary B2/B32, random solid solutions, and unary-unary interfaces; quaternary phase-separation variants (B2/B2, B2/B32, B32/B32, binary/random combinations). 3) High-T (2500 K) disordered structures (128, 432 atoms): NPT MD using prior MTP; 983 (128-atom) and 48 (432-atom) snapshots selected via active learning; DFT single-point energies/forces/stresses. 4) Deformed binaries (432 atoms): Strain along [100] to mimic phase-separated interfaces: TaV +3.7%, CrW −1.28%, TaCr +0.27%, VW −0.14%, TaW +4.06%, VCr −4.71%. Evaluation strategy: Assess RMS errors in energies (meV/atom) and forces (eV/Å) on in-distribution subsystems (binaries, ternaries, quaternary at 0 K; disordered TaVCrW at 2500 K; strained binaries) and on out-of-distribution non-equiatomic ternaries (0 K and 2000 K). Thermodynamic property tested: isobaric heat capacity Cp(T) via thermodynamic integration up to 2500 K, compared to DFT literature (electronic contribution removed). Baselines: EAM (and MEAM in SI). Training/inference times compared across MTP levels (16–28) and GM-NN architectures, CPU vs GPU.

- In-distribution accuracy: For 0 K subsystems, RMSEs are 1–4 meV/atom (energies) and 0.02–0.06 eV/Å (forces). For disordered TaVCrW at 2500 K, energy RMSE ≈ 2.4–2.7 meV/atom; force RMSE ≈ 0.156–0.179 eV/Å. Strained binaries’ energy errors are ≤ 4.5 meV/atom. MTP is marginally better on high-T forces; GM-NN slightly better on 0 K energies/forces. - Force correlation: Both ML models show excellent correlation with DFT for 0 K and 2500 K TaVCrW; EAM shows poor correlation and much larger scatter. - Heat capacity: Both ML models reproduce Cp(T) close to DFT (without electronic contribution) up to ~2200 K; beyond, ML curves diverge modestly. EAM fitted only to 2500 K structures deviates already after ~750 K. - Out-of-distribution (non-equiatomic ternaries): Absolute energy errors can be up to ~20 meV/atom, but relative energies (to relaxed reference) have RMSE < 5.5 meV/atom. Force RMSEs at 2000 K are ≤ ~0.16 eV/Å; 0 K force errors are similarly low except for one high-force configuration (Cr17W17V66). EAM errors are much larger (absolute energies off by 120–200 meV/atom; relative energies 60–80 meV/atom; forces 0.5–0.7 eV/Å). - Active learning: MTP converges to low energy RMSE with relatively small training sizes (~128 structures; ~11k atoms total) and benefits from active learning for force RMSE (due to higher fraction of 2500 K configurations selected). GM-NN shows nearly linear improvement with training size; active learning provides little additional benefit for energies/forces under the chosen strategy. - Performance trade-offs: On CPU, MTPs are faster than GM-NNs; on GPU, GM-NN is faster (≈2.5× faster than lev24 MTP on RTX 3090 Ti). Lev24 MTP attains near-saturated accuracy; increasing level raises cost with modest accuracy gains. EAM/MEAM are 1–2 orders faster but at least an order of magnitude less accurate. - Overall, both MTP and GM-NN achieve near-DFT accuracy across configurational and vibrational spaces for Ta–V–Cr–W, generalizing well to out-of-distribution compositions for relative energetics and finite-T forces.

The study demonstrates that two complementary ML interatomic potentials (MTP and GM-NN) can accurately capture both configurational (0 K ordered, strained, phase-separated structures) and vibrational (high-T disordered) degrees of freedom in a complex quaternary alloy. This directly addresses the central challenge of modeling HEAs across temperature regimes with near-DFT fidelity while enabling large-scale simulations. Despite differences in architecture and parameter counts, both models deliver similar accuracy on in-distribution data and robust generalization to out-of-distribution ternaries for relative energetics and high-T forces—key for reliable thermodynamics and MD sampling. The models reproduce Cp(T) up to ~2200 K, indicating accurate free energies to within a few meV/atom. Active learning offers limited benefits for energies but accelerates force convergence for MTP by prioritizing high-temperature configurations; human-guided dataset construction remains important. Practical considerations emerge: MTP converges quickly with smaller datasets and is CPU-efficient; GM-NN scales better on GPUs and may surpass MTP with larger datasets, though representation choice impacts accuracy more than network size reductions. Compared to EAM/MEAM, ML models are substantially more accurate and sufficiently fast for extensive simulations, making them preferable for HEA studies where classical potentials prove unreliable. Incorporating electronic and magnetic contributions via upsampling or magnetic extensions would further close the gap to full DFT thermodynamics.

This work develops and benchmarks two advanced ML interatomic potentials—MTP and GM-NN—for the Ta–V–Cr–W alloy family, achieving near-DFT accuracy for energies and forces over both 0 K ordered and high-T disordered phases. The models generalize well to out-of-distribution ternaries for relative energies and finite-temperature forces and enable accurate predictions of thermodynamic properties such as heat capacity up to ~2200 K. MTP converges rapidly with fewer data and is CPU-efficient, while GM-NN offers competitive accuracy with superior GPU performance; both far outperform classical EAM/MEAM in accuracy. Future directions include: expanding training data to better cover compositional/temperature spaces; integrating electronic free-energy contributions via DFT upsampling; incorporating magnetic degrees of freedom (e.g., magnetic MTP); and coupling ML-MD with Monte Carlo to enhance sampling of order-disorder transitions and phase separation in HEAs.

- Electronic and magnetic degrees of freedom are not included in the training data; thus, results represent vibrational and configurational contributions only. Electronic free energy and magnetism would need to be added (e.g., DFT upsampling, magnetic MTP) for full accuracy. - Out-of-distribution absolute energies show larger errors (up to ~20 meV/atom), reflecting difficulty in predicting absolute chemical potentials across compositions; single-point DFT corrections may be needed when absolute energies matter. - Potential dataset bias: high-T MD snapshots were sampled using an MTP-based active learning pipeline, which could favor MTP performance; a broader, model-agnostic sampling might mitigate this. - Active learning improvements were limited for energy accuracy in this complex system; careful batch selection and human-guided diversity remain important. - Classical baselines (EAM/MEAM) proved inaccurate for this system, limiting broader comparisons but underscoring the necessity of ML potentials.