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Machine-learned Interatomic Potentials for Alloys and Alloy Phase Diagrams

Conrad W. Rosenbrock, Konstantin Gubaev, Alexander V. Shapeev, Livia B. Pártay, Noam Bernstein, Gábor Csányi, Gus L. W. Hart

arXiv:1906.07816v2cond-mat.mtrl-sciphysics.comp-ph

TL;DR

First-principles methods are computationally demanding, while machine-learned potentials can have limited transferability, making alloy modeling across compositions challenging. This paper develops GAP and MTP models for Ag-Pd and evaluates their accuracy and use in phase-diagram calculations. The models match cluster expansion for on-lattice energies while also supporting off-lattice properties and structural changes.

  • Problem

    First-principles calculations are resource intensive, and machine-learned interatomic potentials may be limited in transferability, complicating their reliable use across alloy configurations and compositions.

    Source: §I · pp. 1–2

  • Method

    The paper fits GAP and MTP potentials for Ag-Pd and uses semi-grand-canonical nested sampling to calculate phase diagrams while allowing composition to vary.

    Source: §I · p. 2, B. Methodology · p. 3

  • Results

    The potentials are as good as cluster expansion for on-lattice energies, while also computing forces, virials, and Hessians across compositions and approximating phonon dispersion and temperature-composition phase diagrams.

    Source: §VI · p. 9

  • Takeaways & Limitations

    These machine-learned potentials offer a viable alternative to cluster expansion for alloy modeling, with the additional ability to represent structural perturbations and derivatives of the potential energy surface.

    Source: §VI · p. 9

    Cluster expansion cannot address dynamic processes involving structural perturbations, limiting its usefulness for those problems.

    Source: §VI · p. 9

Abstract

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We introduce machine-learned potentials for Ag-Pd to describe the energy of alloy configurations over a wide range of compositions. We compare two different approaches. Moment tensor potentials (MTP) are polynomial-like functions of interatomic distances and angles. The Gaussian Approximation Potential (GAP) framework uses kernel regression, and we use the Smooth Overlap of Atomic Positions (SOAP) representation of atomic neighbourhoods that consists of a complete set of rotational and permutational invariants provided by the power spectrum of the spherical Fourier transform of the neighbour density. Both types of potentials give excellent accuracy for a wide range of compositions and rival the accuracy of cluster expansion, a benchmark for this system. While both models are able to describe small deformations away from the lattice positions, SOAP-GAP excels at transferability as shown by sensible transformation paths between configurations, and MTP allows, due to its lower computational cost, the calculation of compositional phase diagrams. Given the fact that both methods perform as well as cluster expansion would but yield off-lattice models, we expect them to open new avenues in computational materials modeling for alloys.

I. INTRODUCTION

This work addresses the limited accuracy and transferability of interatomic potentials by testing machine-learned models for Ag-Pd, a demanding cluster-expansion benchmark. It evaluates their accuracy across configurations and compositions and their use for phonons, phase diagrams, and transitions between structures.

I. INTRODUCTION

  • First-principles simulations are resource intensive, while machine-learned potentials can be less transferable and are typically slower than conventional potentials.The introduction contrasts DFT’s sub-thousand-atom, picosecond reach with materials problems requiring millions of atoms over micro- to millisecond timescales.Source: §I · p. 1
  • Both GAP and MTP fit Ag-Pd energies, forces, and stresses with near-DFT accuracy across solid and liquid configurations and can model phonons and compositional phase diagrams.MTP is faster than GAP; GAP uses kernel regression with SOAP, while MTP uses a polynomial basis of interatomic distances and angles.Source: §I · p. 2
  • Ag-Pd is a stringent test because silver and palladium are chemically similar and have similar atomic sizes, making the system well suited to cluster expansion.The paper tests whether machine-learned potentials can compete with cluster expansion despite its simpler lattice-gas formalism.Source: §I · p. 2
  • The study compares phonon predictions on untrained alloy configurations, uses MTP for phase-diagram calculations, and tests GAP and MTP along a low-energy transition pathway.The pathway comparison examines the roles of regularization and active learning.Source: §I · p. 2

II. DATASETS AND FITTING

The potentials share an active-learned dataset built by iteratively fitting and relaxing enumerated structures. The final training data retain the discovered configurations but use more tightly converged DFT calculations.

A. Datasets

  • Both potentials used the active-learned dataset, while a liquid dataset was reserved for validation; some phonon prediction structures overlapped with seed configurations.The final training set retained the 774 discovered configurations but used MINDISTANCE=65 and EDIFF=1e-8; higher k-point density and tighter convergence were needed for good phonon-dispersion convergence.Source: A. Datasets · p. 2, §1 · p. 2

1. Active-Learned Dataset

  • Iterative active learning expanded enumerated fcc- and bcc-derived structures as needed until MTP could reliably relax structures up to 12 atoms-per-cell units, yielding 774 configurations.Structures with excessive estimated extrapolation error during relaxation were recomputed with DFT and added to the training set.Source: §1 · p. 2

2. Liquid Dataset

A separately generated liquid dataset provides validation data, while GAP and MTP are fitted to the active-learned dataset using distinct model forms and fitting weights.

2. Liquid Dataset

  • Liquid-like validation configurations came from high-temperature DFT molecular dynamics at 25, 50, and 75 at-% Ag in 32-atom cells.Snapshots were taken every 50 fs during 100,000 fs runs, then reevaluated with a 4 × 4 × 4 Monkhorst-Pack grid.Source: §2 · p. 3

B. Potential Fitting

  • GAP combines a two-body distance-kernel term with a many-body SOAP-kernel term, with σ values setting regularization and the accuracy tradeoff among energies, forces, and virials.The target σ values were 10−3 eV per atom for energy, 10−3 eV/Å for forces, and 0.02 eV per atom for virial stresses.Source: B. Potential Fitting · p. 3
  • MTP uses a polynomial basis up to degree 16 with 188 fitting parameters and was trained on the same dataset as GAP.Its fitting weights were 10−3 eV for energy, 1.4 × 10−2 eV/Å for force, and 0.04 eV for stress.Source: B. Potential Fitting · p. 3

A. Dataset Augmentation

The MTP training set was augmented for high-temperature nested sampling, and the phase-diagram method extends constant-pressure nested sampling to allow composition to vary at fixed particle number.

A. Dataset Augmentation

  • Because nested sampling reaches temperatures well outside the original training data, MTP was augmented to avoid dimers forming in the gaseous phase.The augmentation began from 67 structures within 5 meV/atom of the stable-structure convex hull and used on-the-fly training during molecular dynamics from near-zero to melting temperatures.Source: A. Dataset Augmentation · p. 3

B. Methodology

  • Semi-grand-canonical nested sampling samples a free energy that includes species counts and chemical potentials, while keeping total particle number constant and allowing species identities to change.Monte Carlo steps propose changing a randomly selected atom’s species; with two species, the simulation is characterized by the chemical-potential difference ∆µ.Source: B. Methodology · p. 3
  • Runs at a range of ∆µ values yield temperature-composition phase diagrams; the study also compares them with constant-composition runs and extends one 50% composition run to low temperature.Specific-heat maxima provide estimates of transition temperatures; constant-composition runs identify a transition temperature from the Cp(T) peak.Source: B. Methodology · pp. 3–4

IV. RESULTS AND DISCUSSION

GAP and MTP predict energies, forces, and virials for liquid configurations absent from their training data, providing evidence of transferability. Their error patterns differ across properties, while both also agree well with DFT phonon predictions.

A. Energy, Force and Virial Predictions

B. Phonon Predictions

Phonon band structures connect vibrational behavior with phase stability and probe how accurately a potential captures small deformations. The paper assesses this behavior across alloy configurations because prior potentials had reproduced spectra for selected cases, but not generally across an alloy system.

B. Phonon Predictions

  • Phonon eigenvalues describe the energy required to excite vibrational modes in a crystal and directly describe phase stability at moderate temperatures via the quasi-harmonic approximation.Accurate reproduction requires the potential to capture curvature near the relaxed structure.Source: B. Phonon Predictions · p. 5
  • Phonon validation probes second derivatives of the energy surface, complementing energy and force checks of specific points and slopes.The study compares DFT and potential predictions for 65 fcc-type derivative superstructures with cell sizes from 2 through 6.Source: B. Phonon Predictions · p. 5
  • Earlier potentials had reproduced phonon spectra for selected compositions and configurations, but not generally across an entire alloy system.The phonon comparisons are intended to assess accuracy across the Ag-Pd alloy system.Source: B. Phonon Predictions · p. 5

1. DFT Phonon Dispersion Curves

GAP and MTP reproduce phonon behavior across Ag-Pd structures, including a DFT-predicted dynamic instability. Their validation spans the composition range, with phonon errors close to 0.1 THz across the full validation set.

1. DFT Phonon Dispersion Curves

  • DFT phonon calculations used twice-relaxed structures with cell shape and volume allowed to change, then phonopy displacements and symmetry-selected supercells of at most 32 atoms.The supercells maximized the distance between periodic images, with larger point-group symmetry breaking ties.Source: §1 · p. 5

2. Machine-learn Phonon Dispersion Curves

  • For an untrained 4-atom structure at 50 at-% Ag, both potentials approximate the phonon eigenvalues well along the special path, with differing errors across k-points.The integrated RMS error samples all eigenvalues on a 13 × 13 × 13 grid in the Brillouin zone.Source: Figure 2 · p. 6
  • A single GAP or MTP potential approximates phonons with good accuracy across the full compositional space for many configurations.Additional phonon plots covering a broad structure-composition range are reported in the Supplementary Information.Source: §2 · p. 6
  • ≈0.1 THz integrated phonon error for both GAP and MTP across the 65-structure validation set, with good agreement with DFT across the full spectrum.Similar training and prediction errors indicate a good bias-variance trade-off.Source: §2 · p. 6
  • For a DFT-predicted dynamically unstable structure at 16 at-% Ag, both potentials recover the instability, with GAP more accurate.The seed configuration was included in the active-learned fitting dataset.Source: Figure 3 · p. 6

C. Transition Pathway

Along a transition path with untrained configurations, GAP gives qualitatively reasonable behavior while MTP produces a false local minimum. MTP’s lower computational cost also makes nested-sampling exploration of the phase space practical.

C. Transition Pathway

  • At the path’s highest point, the total energy is about 9.5 eV versus approximately −16 eV at the start, and GAP’s error is several eV.The qualitative behavior of GAP’s prediction remains correct despite the barrier error.Source: C. Transition Pathway · p. 7
  • MTP detects extrapolation but behaves more poorly than GAP; its extrapolation grade should be checked against representative expected configurations before large-scale molecular dynamics.The authors attribute the difference to GAP’s local basis and built-in regularization versus MTP’s global polynomial basis.Source: C. Transition Pathway · p. 7

V. PHASE DIAGRAM RESULTS

  • MTP’s lower cost makes nested-sampling exploration practical, while a single fixed-composition, fixed-pressure temperature slice requires more than 2 billion potential evaluations.That cost is described as presently prohibitive for GAP but reasonable with MTP.Source: §V · p. 7

C. Transition Pathway

  • GAP avoids a false minimum along the untrained transition path, whereas MTP’s extrapolative prediction produces a deep false minimum.GAP nevertheless significantly underestimates the barrier because training data are lacking along the path.Source: Figure 4 · p. 7

A. Liquid-Solid Transition

MTP nested sampling qualitatively reproduces the experimental melting behavior and liquidus-solidus gap after a roughly 200 K temperature shift. It also finds an order-disorder transition at 125 K for the 1:1 composition.

A. Liquid-Solid Transition

  • MTP’s melting behavior qualitatively matches the experimental line but is shifted by approximately 200 K.The liquidus-solidus gaps are also in reasonable agreement with experiment when the same shift is included.Source: A. Liquid-Solid Transition · p. 8, Figure 5 · p. 8
  • The liquidus-solidus gap is reasonably consistent with experimental data when the simulation results are shifted by 200 K.The figure caption indicates that the experimental liquidus and solidus are adapted from Refs. [103, 113, 126].Source: A. Liquid-Solid Transition · p. 8, Figure 5 · p. 8

B. Solid-Solid Transition

  • Reported experimental phase diagrams typically show solidus and liquidus lines and a solid solution below the solidus, while reliable evidence for first-order solid-state transformations is lacking.One proposed diagram suggests two solid-solid transitions, but the authors note the lack of reliable evidence for such transformations.Source: B. Solid-Solid Transition · p. 8
  • The 1:1 composition has an order-disorder transition at 125 K, with similar transition temperatures reported by other computational works.Sampling continued below the melting temperature because the melting transition was underestimated.Source: B. Solid-Solid Transition · p. 8

C. Comparison to Cluster Expansion

Against a state-of-the-art Ag-Pd cluster expansion, GAP and MTP perform well in the region where that model is valid, though the cluster expansion is slightly better. The low-temperature 50% Ag structure shows ordering confirmed by simulated XRD.

C. Comparison to Cluster Expansion

  • The cluster expansion is slightly better, but GAP and MTP both perform well in the configuration-space region where it is valid.The comparison uses the Ag-Pd cluster expansion described in.Source: Table IV · p. 8, C. Comparison to Cluster Expansion · p. 8
  • The lower-temperature structure shows clear ordering as stacking planes, confirmed by simulated XRD.Ag atoms were substituted for H atoms to artificially boost the simulated structure-factor difference.Source: C. Comparison to Cluster Expansion · p. 9

VI. CONCLUSION

Machine-learned interatomic potentials match cluster expansion for on-lattice alloy energies while also supporting derivatives and structural dynamics that cluster expansion cannot address.

VI. CONCLUSION

  • Cluster expansion supports ground-state searches across compositions and temperature-dependent phase mapping in certain systems, but cannot address dynamics involving structural perturbations.Source: §VI · p. 9
  • Machine-learned interatomic potentials match cluster expansion for on-lattice energy calculations, while also computing forces, virials, and Hessians across compositions.These derivatives are sufficiently accurate to approximate dynamic properties such as phonon dispersion curves.Source: §VI · p. 9
  • The authors describe machine-learned potentials as a viable alternative to cluster expansion, noting that software for dataset creation and fitting is readily available and easy to use.Source: §VI · p. 9
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