Machine-learned interatomic potentials for alloys and alloy phase diagrams

The study addresses the need for accurate, transferable, and computationally efficient interatomic potentials (IPs) to enable large-scale, long-timescale simulations beyond the reach of density functional theory (DFT). Conventional IPs are fast but lack quantitative accuracy for many properties, whereas machine-learned IPs are more accurate but slower and may suffer from limited transferability. The paper investigates whether two machine-learned IP frameworks—the Gaussian approximation potential with SOAP descriptors (SOAP-GAP) and the moment tensor potential (MTP)—can achieve DFT-level accuracy across structures and compositions for a binary metallic alloy (Ag–Pd), including off-lattice behavior, and whether they can capture dynamic quantities such as phonon dispersions and reproduce temperature–composition phase behavior. The Ag–Pd system is a stringent test due to the chemical and size similarity of Ag and Pd which favors cluster expansion (CE) in on-lattice modeling; continuous-basis ML potentials must handle both structural and compositional degrees of freedom. The work aims to demonstrate accuracy, transferability, and practical applicability (e.g., phase diagrams) of ML IPs compared to CE.

GAP was introduced in 2010, leveraging kernel regression with invariant many-body representations (e.g., SOAP), and has been successfully applied to molecules, solids, defects, dislocations, and grain boundaries. SOAP-GAP can be systematically improved and has become a benchmark for ab initio-accurate surrogate models. Other ML potentials (e.g., neural networks, ACE, linear-invariant representations) have shown strong performance, though applications to alloys are still developing and comparative studies have not fully addressed dynamical quantities like phonons or temperature–composition phase diagrams. MTP provides an efficient polynomial-like basis over distances and angles, shown to reach accuracy comparable to GAP in various systems and offering significant speed advantages. CE remains a standard for alloy energetics on lattices due to speed and compositional basis suitability but cannot treat off-lattice dynamics or structural perturbations. Prior alloy studies show factors (e.g., lattice mismatch) governing CE reliability. Existing works on Ag–Pd include thermodynamic assessments and first-principles predictions of ordering, but experimental phase diagrams typically show only liquidus–solidus lines with limited evidence of ordered solid phases due to kinetic inhibition.

Datasets and DFT setup: - Active learning database construction used MTP tools. Seed catalog: small fcc- and bcc-based derivative superstructures (initial 58 structures up to 4 atoms). DFT energies, forces, and virials computed and used to fit an initial MTP, then structures relaxed; configurations triggering extrapolation (per MTP extrapolation grade) were recomputed with DFT and added. Iterated while expanding to larger unit cells until all enumerated structures up to cell size 12 (10,850 structures) could be reliably relaxed. Final active-learning dataset: 774 configurations. - DFT: VASP with PBE; Monkhorst–Pack or Wisesa–McGill–Mueller (WMM) k-point schemes. PREC=Accurate; SCF EDIFF=1e-4 during active learning. For final fits and phonon convergence: tighter settings; linear k-point density ≈0.015 k-points/Å⁻1. Final recomputation: MINDISTANCE=65 (Mueller scheme), EDIFF=1e-8. Liquid dataset for validation: - Ab initio MD at high T for compositions 25, 50, 75 at.% Ag in 32-atom cells. Target temperatures near linear interpolation of elemental melting points: 2766, 3063, 3360 K; thermostat SMASS=3; POTIM=1.0 fs; run 100,000 fs; snapshots every 50 fs; NELMIN=4; Γ-point during MD. Post-MD re-evaluation with 4×4×4 MP k-point grid. ~6000 snapshots subsampled for validation. Potential fitting: - GAP: Implemented in QUIP as a sum of two-body Gaussian kernel term and many-body SOAP kernel term. Regularization α (target errors): energy 1e-3 eV/atom, forces 10 eV/Å, virials 0.02 eV/atom. Two-body parameters include σ=1.0, r_cut=6.0 Å, transition width 1.0 Å, δ=2.0 eV, r_sparse=25. SOAP parameters include r_cut=4.5 Å, transition width 0.5 Å, δ=0.2 eV, n_sparse=500, Π_max=8, l_max=8, ζ=2, a_atom=0.5 Å. - MTP: Polynomial degree up to 16, 188 fitting parameters. Training weights: energy 10 eV, forces 1.4×10^2 eV/Å, stress 0.04 eV (correspond to different trade-offs vs GAP). Representative hyperparameters: cutoff 5.0 Å; radial functions 4; radial basis size 6; stress and force weights per Table 3; BFGS iterations 500. Dataset augmentation for nested sampling (NS): - To avoid unphysical dimer formation in high-T/gas regimes encountered by NS, augmented training data: 67 near-hull structures (within 5 meV/atom of convex hull) expanded to 32–64 atom supercells and used as starting configurations. Phonon calculations: - DFT workflow: relax each configuration (IBRION=2, ISIF=3). Generate supercells via phonopy choosing HNF matrices to maximize periodic-image separation (>32 atoms) with preference for larger point group if equivalent. Displacements computed with EDIFF=1e-8, ADDGRID=TRUE, ENCUT=400 eV, and MINDISTANCE=55 (Mueller k-grid). ML phonons: evaluate force constants with GAP and MTP, compare dispersions along special paths; integrated RMSE across Brillouin zone sampled on 13×13×13 grid. Transition pathway test: - Construct smooth high-energy path between two Ag–Pd configurations connected by atom swaps and slight cell distortion. Energy evaluated along 11 images. Compare GAP vs MTP behavior under extrapolation; monitor GAP predictive variance and MTP extrapolation grade (threshold ~10 indicates need to add to training set). Nested sampling for phase diagram: - Constant-pressure NS used to estimate isobaric–isothermal partition function; specific heat peaks identify transitions. Extended to semi-grand-canonical (sGC) NS: total atoms fixed, species counts fluctuate with chemical potential difference Δμ=μ_Ag−μ_Pd. NS applied to free energy F = E + Σ N_i μ_i. Monte Carlo steps include species swaps (pair exchanges and single-atom composition changes for sGC). Composition obtained as ensemble average vs temperature. Finite-size broadening expected; also performed constant-composition NS and a large-scale coexistence simulation for validation. Representative NS parameters (per ref. 41 notation): examples include N=96 (fixed composition) or 64 (sGC), K≈1080–1152 configurations, evaluations per walk L≈640–1166, position steps, cell steps (volume:shear:stretch), swap and composition steps as listed in Table 4. One configuration removed per NS iteration (K_s=1). Coexistence simulation: equimolar Ag–Pd, 16,384 atoms (8×8×64 fcc cells), combined MD+MC to sample disorder; melting point compared with NS.

- Accuracy on liquid validation set (severe extrapolation): • Energy RMSE (meV/atom): GAP 15.4; MTP 10.9. • Force RMSE (meV/Å): GAP 224; MTP 241. • Virial RMSE (meV/Å^3): GAP 8.3; MTP 12.7. These indicate strong transferability despite no liquid data in training. - Phonons across compositions and structures (65 fcc-derivative structures, 2–6 atoms/cell): • Integrated RMSE over 13×13×13 k-grid: GAP training 0.13±0.05 THz; prediction 0.13±0.04 THz. MTP training 0.12±0.03 THz; prediction 0.11±0.01 THz. • Both models reproduce dynamically unstable modes where present. - Transition pathway behavior: • Along a high-energy transformation path (~25 eV total change), GAP with strong regularization yields qualitatively correct barrier without spurious minima, albeit underestimating barrier height due to lack of training data. MTP, relying on global polynomial basis and active learning, exhibits a false local minimum under extrapolation; extrapolation grade flags unreliability (>~10). - Phase diagram via NS (with MTP): • Semi-grand-canonical NS traces composition vs temperature for different Δμ, revealing liquidus–solidus (L–S) gaps as horizontal composition jumps (finite-size broadened). • Melting behavior qualitatively matches experiment but is shifted by ~200 K (consistent with known PBE biases). Liquidus–solidus gaps agree reasonably after applying the uniform 200 K shift. • Coexistence simulation at 50% Ag predicts T_m ≈ 1315 K vs NS ≈ 1380 K; discrepancy consistent with NS finite-size effects (e.g., ~200 K for 64 particles). • Solid–solid order–disorder transition at 50% Ag observed at ~125 K; manifests as planar ordering along stacking planes, confirmed by simulated XRD. - Comparison to cluster expansion (CE) on a CE reference dataset: • Training error (meV/atom): CE 1.6; MTP 1.9; GAP 2.7. • Validation error (meV/atom): CE 2.1; MTP 4.2; GAP 5.7. • Basis sizes: CE 41; MTP 569; GAP 1075. CE excels on-lattice with compact basis, while ML IPs approach CE accuracy and additionally model off-lattice forces, virials, and phonons.

The results show that both SOAP-GAP and MTP can learn the Ag–Pd potential energy surface with near-DFT fidelity across a broad composition and structure space, including off-lattice configurations. Close agreement in phonon training vs prediction errors indicates appropriate bias–variance balance and limited overfitting. While CE remains superior for on-lattice energetics with minimal basis and speed, ML IPs provide access to derivatives (forces, virials, Hessians) and dynamic properties like phonon dispersions and temperature–composition phase behavior, which CE cannot capture. The MTP’s computational efficiency enables nested sampling of phase diagrams; the observed ~200 K uniform temperature shift relative to experiment is consistent with PBE tendencies, suggesting that trends and relative features are reliable even if absolute transition temperatures are offset. The transition pathway analysis highlights methodological differences: GAP’s localized, regularized kernel approach is more robust under extrapolation, avoiding spurious minima, whereas MTP requires active learning safeguards (extrapolation grade) to remain in the interpolation regime. Together, the methods illustrate a practical pathway: use active learning to curate robust training sets, employ MTP for large-scale thermodynamic exploration, and leverage GAP for challenging transformations demanding superior extrapolative behavior.

This work demonstrates that two machine-learned interatomic potentials—SOAP-GAP and MTP—can achieve DFT-level accuracy for energies, forces, virials, and phonon spectra across the Ag–Pd alloy’s configuration–composition space, and can be used to compute phase behavior via nested sampling. GAP exhibits superior extrapolation behavior due to strong regularization and local descriptors, while MTP’s efficiency enables practical phase diagram calculations. Compared with CE, these off-lattice models approach on-lattice energy accuracy while providing access to dynamical and structural properties, opening avenues for computational alloy design that require both accuracy and flexibility. Future directions include: enriching training data (e.g., along relevant transformation paths) via active learning to enhance transferability; employing higher-level electronic structure methods or corrected DFT functionals to reduce absolute temperature offsets; extending sGC nested sampling and IP training to multi-component and complex alloy systems; and improving implementations to further reduce computational cost, especially for GAP.

- Transferability limits: ML IPs are reliable primarily within the training data manifold; extrapolation can yield large errors or spurious features (e.g., MTP false minimum along a high-energy path). Active learning and uncertainty/extrapolation metrics (GAP variance, MTP extrapolation grade) are necessary safeguards. - Functional bias: Phase transition temperatures are uniformly shifted by ~200 K relative to experiment, consistent with PBE; absolute temperatures require correction or higher-level methods. - Finite-size effects: NS with small systems (e.g., 64–96 atoms) broadens transitions and shifts melting points; discrepancies vs large coexistence simulations are expected. - Dataset scope: Training omitted explicit liquid configurations (used only for validation) and required augmentation to avoid dimers in gas-like regimes encountered by NS. - Computational cost: GAP evaluations (SOAP overlaps) are more expensive than MTP, limiting feasibility for very large sampling tasks in current implementations. - CE comparison caveat: CE training dataset differs (on-lattice only), and k-point schemes may reduce DFT numerical error differently, affecting reported error comparisons.