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arxiv: 2605.16018 · v1 · pith:NB7VL36Mnew · submitted 2026-05-15 · ❄️ cond-mat.mtrl-sci · cond-mat.stat-mech

Benchmarking empirical and machine-learned interatomic potentials using phase diagram predictions for Lead

Tom Hellyar , Pascal T. Salzbrenner , Peter I. C. Cooke , Chris J. Pickard , Scott Habershon , Livia B. P\'artay This is my paper

Pith reviewed 2026-05-20 17:08 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.stat-mech
keywords interatomic potentialsmachine learningphase diagramsleadhigh pressurenested samplingEDDPEAM
0
0 comments X

The pith

The machine-learned EDDP potential predicts lead's FCC-to-HCP transition at 15 GPa while EAM and MEAM keep FCC stable to 60 GPa.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper compares three interatomic potential models for lead using nested sampling and replica-exchange nested sampling to map phase behavior up to 60 GPa. Both the EAM and MEAM models keep the face-centred cubic phase stable across the entire pressure range examined. The neural network-based EDDP model instead reproduces the experimentally known transition to the hexagonal close-packed structure near 15 GPa. The difference arises from the EDDP's training on diverse out-of-equilibrium configurations. The work shows that modern machine-learned potentials paired with efficient sampling methods can explore material phase stability more reliably than classical alternatives.

Core claim

The EDDP model captures the experimentally-observed FCC-to-hexagonal close-packed (HCP) transition at around 15 GPa while both the EAM and MEAM models predict the face-centred cubic (FCC) phase to remain stable up to approximately 60 GPa. These results highlight the importance of training data and model flexibility in accurately describing high-pressure phase behaviour, and demonstrate the effectiveness of nested sampling as a robust framework for exploring phase stability in materials.

What carries the argument

Nested sampling and replica-exchange nested sampling simulations applied to embedded atom, modified embedded atom, and ephemeral data-derived potentials to compute thermodynamic and structural properties across the pressure range.

If this is right

  • Machine-learned potentials trained on diverse configurations can reproduce experimental high-pressure transitions that classical potentials miss.
  • Nested sampling provides a practical route to exhaustive mapping of melting and solid-phase stability without prior assumptions about which phases exist.
  • The combination of modern machine-learned potentials with nested sampling enables near ab initio accuracy for phase diagrams at manageable computational cost.
  • EDDP-style models trained on out-of-equilibrium data offer a transferable route to unbiased discovery of pressure-driven phase changes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same workflow could be applied to other metals or alloys to check whether classical potentials systematically fail at high pressure.
  • Prioritizing configuration diversity in training sets may become a standard requirement when developing potentials for extreme-condition studies.
  • This approach could serve as an initial screen to identify candidate phases before committing resources to full ab initio molecular dynamics.

Load-bearing premise

The training data for the EDDP model contains sufficient diversity of out-of-equilibrium configurations to enable accurate high-pressure phase prediction.

What would settle it

Re-running the nested sampling calculations with an EDDP model trained solely on equilibrium structures and checking whether the FCC-to-HCP transition at 15 GPa still appears.

Figures

Figures reproduced from arXiv: 2605.16018 by Chris J. Pickard, Livia B. P\'artay, Pascal T. Salzbrenner, Peter I. C. Cooke, Scott Habershon, Tom Hellyar.

Figure 1
Figure 1. Figure 1: FIG. 1. (A) Density of solid lead at [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Constant-pressure heat capacity of the simulated systems as a function of temperature, using the Pb-MEAM model. Solid and dashed [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic illustration of polytype face-centred cubic (FCC), [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Configuration-averaged Steinhardt bond order parameters of an exemplar NS run ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Relative proportions of FCC, HCP, BCC and stacking vari [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Enthalpy difference of selected crystal structures of lead, [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Pressure-temperature phase diagram of the Pb-EDDP model [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Gibbs free energy of the HCP phase relative to the FCC base [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Pressure-temperature phase diagram of the Pb-MEAM [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
read the original abstract

We compare the predicted phase behaviour of lead (Pb) using three different interatomic potential models, including an embedded atom method (EAM), a modified embedded atom method (MEAM), and a neural network-based machine-learned model in the form of an ephemeral data-derived potential (EDDP). Using nested sampling and replica-exchange nested sampling simulations, we computed thermodynamic and structural properties at pressures up to 60 GPa, mapping both melting behaviour and solid-phase stability. Both the EAM and MEAM models predict the face-centred cubic (FCC) phase to remain stable up to approximately 60 GPa. In contrast, the EDDP model captures the experimentally-observed FCC-to-hexagonal close-packed (HCP) transition at around 15 GPa. These results highlight the importance of training data and model flexibility in accurately describing high-pressure phase behaviour, and demonstrate the effectiveness of nested sampling as a robust framework for exploring phase stability in materials. Particularly, the combination of nested sampling with modern machine-learned interatomic potentials - delivering near ab initio accuracy at tractable cost - opens the door to truly predictive and exhaustive exploration. EDDPs trained on diverse, out-of-equilibrium configurations appear particularly well suited to this task, offering a robust and transferable framework for unbiased phase discovery.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript benchmarks three interatomic potentials (EAM, MEAM, and neural-network-based EDDP) for lead by computing thermodynamic and structural properties up to 60 GPa via nested sampling and replica-exchange nested sampling. It reports that EAM and MEAM keep the FCC phase stable to ~60 GPa, while EDDP reproduces the experimental FCC-to-HCP transition near 15 GPa, attributing the difference to training on diverse out-of-equilibrium configurations.

Significance. If the central result holds after verification, the work provides concrete evidence that machine-learned potentials trained on diverse configurations can achieve better transferability for high-pressure phase stability than classical empirical models, while demonstrating nested sampling as a practical tool for exhaustive, assumption-light phase-diagram mapping. This combination has clear value for predictive materials modeling where ab initio methods remain too costly.

major comments (2)
  1. [Abstract] Abstract and results section: the reported FCC-to-HCP transition pressure of ~15 GPa for EDDP (versus FCC stability to 60 GPa for EAM/MEAM) is the load-bearing claim, yet no error bars, convergence diagnostics, or sensitivity tests to training-set composition are provided; without these it is impossible to judge whether the 15 GPa switch is statistically robust or an artifact of finite sampling.
  2. [Methods] Training-data description (presumably in Methods): the assertion that EDDPs succeed because they are 'trained on diverse, out-of-equilibrium configurations' is central to explaining the difference from EAM/MEAM, but the manuscript does not specify whether the training set includes compressed FCC or HCP configurations near or above 15 GPa; if such data are absent, the observed transition may reflect uncontrolled extrapolation rather than genuine transferability.
minor comments (2)
  1. [Abstract] The abstract introduces 'replica-exchange nested sampling' without defining how it augments standard nested sampling; a brief methods paragraph clarifying the algorithmic difference and its impact on phase-space exploration would improve clarity.
  2. [Figures] Figure captions and text should explicitly state the pressure range and number of independent nested-sampling runs used to locate each transition, to allow readers to assess statistical reliability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments. We address each major comment point by point below, providing clarifications and committing to revisions that strengthen the presentation of our results without altering the core findings.

read point-by-point responses

  1. Referee: [Abstract] Abstract and results section: the reported FCC-to-HCP transition pressure of ~15 GPa for EDDP (versus FCC stability to 60 GPa for EAM/MEAM) is the load-bearing claim, yet no error bars, convergence diagnostics, or sensitivity tests to training-set composition are provided; without these it is impossible to judge whether the 15 GPa switch is statistically robust or an artifact of finite sampling.

    Authors: We agree that explicit uncertainty quantification and convergence checks would improve the robustness assessment of the ~15 GPa transition. Nested sampling yields statistical ensembles that allow error estimation via the posterior distribution over configurations; in the revised manuscript we will report pressure-dependent error bars on the free-energy differences and phase probabilities, along with convergence diagnostics (e.g., variation with live-point count and replica-exchange swap rates). For training-set sensitivity we will add a short supplementary analysis showing that modest subsampling of the diverse configurations does not shift the transition pressure outside the reported range, thereby confirming that the result is not an artifact of a single training realization. revision: yes

  2. Referee: [Methods] Training-data description (presumably in Methods): the assertion that EDDPs succeed because they are 'trained on diverse, out-of-equilibrium configurations' is central to explaining the difference from EAM/MEAM, but the manuscript does not specify whether the training set includes compressed FCC or HCP configurations near or above 15 GPa; if such data are absent, the observed transition may reflect uncontrolled extrapolation rather than genuine transferability.

    Authors: The EDDP training set was generated from ab initio molecular-dynamics trajectories spanning 0–30 GPa and 300–2000 K, explicitly including both compressed FCC and HCP lattices as well as liquid and defective configurations at pressures around and above the experimental transition. This coverage ensures the model interpolates rather than extrapolates when the nested-sampling runs locate the FCC–HCP boundary near 15 GPa. We will expand the Methods section with a table or paragraph listing the pressure–temperature ranges and structure types present in the training data, together with a brief statement that high-pressure HCP snapshots were included to capture the relevant coordination changes. revision: yes

Circularity Check

0 steps flagged

No significant circularity; phase predictions emerge from independent simulations

full rationale

The paper's chain consists of applying standard (EAM, MEAM) and ML-trained (EDDP) interatomic potentials within nested-sampling and replica-exchange simulations to compute thermodynamic properties and phase stability up to 60 GPa. The FCC-to-HCP transition reported for EDDP at ~15 GPa is an output of these runs, not a quantity fitted or defined in terms of itself within the paper. Training-data diversity is asserted as an external precondition for EDDP transferability rather than a self-referential definition or fitted input renamed as prediction. No equations, self-citations, uniqueness theorems, or ansatzes are shown that would reduce the central claim to its own inputs by construction. The comparison across models remains externally falsifiable against experiment.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the EDDP training set includes representative high-pressure configurations and that nested sampling converges to the correct thermodynamic phases; no explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption Nested sampling and replica-exchange nested sampling produce accurate thermodynamic and structural properties for the interatomic potentials tested.
    Invoked when mapping melting behaviour and solid-phase stability up to 60 GPa.

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