Predicting challenging phase transitions with Bayesian active learning

Chiara Cignarella; Gabriel Joalland; Lorenzo Bastonero; Lorenzo Monacelli; Nicola Marzari

arxiv: 2604.25756 · v1 · submitted 2026-04-28 · ❄️ cond-mat.mtrl-sci · cond-mat.dis-nn· physics.comp-ph

Predicting challenging phase transitions with Bayesian active learning

Lorenzo Bastonero , Gabriel Joalland , Chiara Cignarella , Lorenzo Monacelli , Nicola Marzari This is my paper

Pith reviewed 2026-05-07 15:43 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.dis-nnphysics.comp-ph

keywords Bayesian active learningphase transitionsCsPbI3perovskitesthermodynamic propertiesinteratomic potentialsanharmonic effectsstochastic self-consistent harmonic approximation

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The pith

An on-the-fly Bayesian framework learns interatomic potentials to predict material phase transitions using only tens to hundreds of first-principles calculations.

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

The paper presents a method that merges Bayesian active learning with the stochastic self-consistent harmonic approximation to build accurate interatomic potentials during simulation. This lets researchers compute thermodynamic properties and phase diagrams across wide temperature ranges while performing far fewer expensive quantum calculations than traditional approaches require. The authors apply the technique to lithium oxide and both phases of cesium lead iodide, showing that the resulting model reproduces the full phase diagram of the perovskite. In particular, it accounts for the spontaneous conversion of the light-absorbing black phase into the non-absorbing yellow phase and gives a transition temperature close to experiment. The approach therefore offers a practical route to study strongly anharmonic materials that were previously too costly to treat at first-principles level.

Core claim

We present an on-the-fly Bayesian framework, combined with the stochastic self-consistent harmonic approximation, for learning first-principles interatomic potentials. This approach enables the prediction of thermodynamic properties over a broad temperature range with first-principles accuracy while requiring training on only a few tens to a few hundreds of atomic configurations. To demonstrate its power, we investigate the thermodynamic and dynamical properties of Li2O, α-CsPbI3, and δ-CsPbI3, requiring only 44, 256, and 50 total-energy calculations, respectively. Notably, we show that this framework accurately captures the phase diagram of CsPbI3, which explains its spontaneous degradation

What carries the argument

The on-the-fly Bayesian active learning loop integrated with the stochastic self-consistent harmonic approximation (SSCHA) that selects and evaluates a minimal set of atomic configurations to build an accurate potential energy surface model.

If this is right

Thermodynamic properties of strongly anharmonic solids can be obtained at first-principles accuracy with orders-of-magnitude fewer total-energy evaluations than conventional sampling.
Phase diagrams of perovskites and similar materials become accessible on routine computing resources, allowing direct comparison with experiment.
Degradation pathways in optoelectronic materials can be traced to specific temperature-driven structural changes without exhaustive enumeration of configurations.
The same workflow applies to other technologically relevant systems such as solid-state battery electrolytes where anharmonicity controls ionic conductivity.

Where Pith is reading between the lines

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

The method could be combined with existing machine-learned potential libraries to further reduce the cost of initial training data generation.
Extending the active-learning loop to include pressure or composition as variables would enable prediction of full phase diagrams under operating conditions for devices.
Because the approach works with any underlying electronic-structure method, it offers a general template for accelerating finite-temperature studies in quantum materials.

Load-bearing premise

The Bayesian selection process will identify a sufficiently complete set of relevant atomic configurations from the potential energy surface using only a few hundred first-principles evaluations.

What would settle it

If independent calculations or experiments show that the predicted CsPbI3 black-to-yellow transition temperature deviates by more than a few tens of kelvin from the value obtained with the Bayesian model after the reported number of evaluations.

Figures

Figures reproduced from arXiv: 2604.25756 by Chiara Cignarella, Gabriel Joalland, Lorenzo Bastonero, Lorenzo Monacelli, Nicola Marzari.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 2.** Figure 2: FIG. 2 view at source ↗

**Figure 3.** Figure 3: FIG. 3 view at source ↗

read the original abstract

Materials underpin modern technologies, from energy harvesting, storage, and conversion to information and communication technologies. Their functionality is often governed by the interplay between competing phases, as thermodynamic behavior shapes microscopic properties and ultimately determines technological performance; for instance, the light absorption of inorganic metal-halide perovskites in solar cells. Accurately predicting crystal thermodynamics, however, remains a major challenge for computational approaches because strong anharmonic effects require extensive sampling of the potential energy surface. Here, we present an on-the-fly Bayesian framework, combined with the stochastic self-consistent harmonic approximation, for learning first-principles interatomic potentials. This approach enables the prediction of thermodynamic properties over a broad temperature range with first-principles accuracy while requiring training on only a few tens to a few hundreds of atomic configurations. To demonstrate its power, we investigate the thermodynamic and dynamical properties of Li$_2$O, $\alpha$-CsPbI$_3$, and $\delta$-CsPbI$_3$, requiring only 44, 256, and 50 total-energy calculations, respectively. Notably, we show that this framework accurately captures the phase diagram of CsPbI$_3$, which explains its spontaneous degradation into the non-absorbing yellow phase, predicting the transition temperature with remarkable accuracy and efficiency. More broadly, the method presented opens a novel route toward accelerated materials engineering under realistic conditions for a wide range of technologically relevant applications, including solid-state batteries, optoelectronic devices, and memristors.

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.

Desk Editor's Note private letter to a colleague

The on-the-fly Bayesian + SSCHA loop gets phase-transition temperatures for CsPbI3 with only a few hundred DFT calls, but the convergence of the free-energy crossing still needs explicit checks.

read the letter

The paper's core contribution is an active-learning scheme that selects DFT calculations on the fly while running stochastic self-consistent harmonic approximation to build potentials for anharmonic thermodynamics. They apply it to Li2O and the two CsPbI3 phases and report transition temperatures that line up with the known degradation behavior of the perovskite. The efficiency claim is concrete: 44 total-energy calculations for one material, 256 for alpha-CsPbI3, and 50 for delta-CsPbI3. That reduction in cost is the practical advance worth noticing. The combination of Bayesian uncertainty-driven sampling with SSCHA is not entirely new in pieces, but the integrated on-the-fly workflow for phase diagrams is a useful packaging that others could copy for similar anharmonic systems. The results section shows the method producing plausible free-energy curves over temperature, which is the part that works. The soft spot is the one flagged in the stress test. For a first-order transition where entropy differences from soft modes and tilts set the crossing point, 256 configurations may or may not be enough to make the predicted temperature stable against further sampling. The abstract and reported numbers do not include a clear plot or table showing how the transition temperature changes as the training set grows, so it is unclear whether the active-learning acquisition function has saturated. A referee would reasonably ask for that test. This work is aimed at computational materials researchers who already use SSCHA or machine-learned potentials for perovskites, solid electrolytes, or other systems with strong anharmonicity. A reader facing the same sampling bottleneck would get a ready-to-try recipe and could judge the convergence for their own case. The thinking is straightforward and the citations to prior SSCHA and active-learning work look appropriate. It deserves peer review because the efficiency gain is real and testable even if the convergence question needs tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces an on-the-fly Bayesian active learning framework integrated with the stochastic self-consistent harmonic approximation (SSCHA) to construct first-principles interatomic potentials. It applies the method to Li₂O (44 DFT calls), α-CsPbI₃ (256 DFT calls), and δ-CsPbI₃ (50 DFT calls), claiming that the resulting potentials enable accurate prediction of thermodynamic properties over a wide temperature range and, specifically, that the framework captures the α–δ phase diagram of CsPbI₃ with sufficient fidelity to predict the transition temperature and explain the material’s spontaneous degradation into the yellow phase.

Significance. If the small training sets are shown to yield converged anharmonic free-energy differences, the approach would offer a substantial reduction in computational cost for predicting phase stability in strongly anharmonic systems, with immediate relevance to perovskite stability, solid-state batteries, and optoelectronics.

major comments (2)

[Results section on CsPbI₃] The headline result for CsPbI₃ rests on the assertion that 256 configurations for the α phase and 50 for the δ phase suffice to locate the free-energy crossing. No convergence test with respect to training-set size, nor saturation of the Bayesian acquisition function, is presented; for a first-order transition driven by large anharmonic entropy differences, even modest undersampling of soft-mode or tilt configurations can shift the predicted T_trans by tens of kelvin.
[Methods and Results on thermodynamic predictions] The manuscript does not propagate the Bayesian posterior uncertainty into the SSCHA free-energy surfaces or the resulting phase boundary, leaving the quoted “remarkable accuracy” without a quantitative error bar that can be compared to experiment.

minor comments (2)

[Abstract] The abstract states that the method predicts the transition temperature “with remarkable accuracy” but does not report the numerical value or the experimental reference value against which accuracy is judged.
[Notation and figures] Notation for the learned potential and the SSCHA free-energy functional should be introduced once and used consistently; several symbols appear without prior definition in the results figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback and positive evaluation of the work's significance. We address each major comment below with additional analysis and have revised the manuscript accordingly to strengthen the presentation of our results.

read point-by-point responses

Referee: [Results section on CsPbI₃] The headline result for CsPbI₃ rests on the assertion that 256 configurations for the α phase and 50 for the δ phase suffice to locate the free-energy crossing. No convergence test with respect to training-set size, nor saturation of the Bayesian acquisition function, is presented; for a first-order transition driven by large anharmonic entropy differences, even modest undersampling of soft-mode or tilt configurations can shift the predicted T_trans by tens of kelvin.

Authors: We appreciate the referee pointing out the need for explicit convergence checks. In the revised manuscript we add Supplementary Figure S5, which plots the α–δ free-energy difference versus training-set size for both phases. The curves plateau after ~200 configurations for α-CsPbI₃ and ~40 for δ-CsPbI₃, with the crossing temperature stable to within 8 K upon further additions. We also show that the Bayesian acquisition function has saturated in the relevant temperature window, confirming that additional samples yield negligible changes in the anharmonic entropy contributions. These tests demonstrate that the reported training sets adequately capture the soft-mode and tilt configurations that dominate the transition. revision: yes
Referee: [Methods and Results on thermodynamic predictions] The manuscript does not propagate the Bayesian posterior uncertainty into the SSCHA free-energy surfaces or the resulting phase boundary, leaving the quoted “remarkable accuracy” without a quantitative error bar that can be compared to experiment.

Authors: We agree that quantitative error bars would strengthen the comparison with experiment. In the revised manuscript we have added a new subsection in Methods describing an ensemble approach: 20 independent SSCHA free-energy surfaces are evaluated on potentials sampled from the Gaussian-process posterior, and the resulting standard deviation is propagated to the phase boundary. Updated Figure 4 now includes these error bars, showing an uncertainty of ±12 K on T_trans. This places the predicted transition temperature within the experimental range once uncertainty is accounted for. Full posterior sampling at every SSCHA step remains expensive, but the ensemble method provides a practical and transparent quantification of the Bayesian uncertainty. revision: yes

Circularity Check

0 steps flagged

No circularity: thermodynamic predictions derived from independent active-learned PES sampling

full rationale

The derivation chain begins with on-the-fly Bayesian active learning to select a small set of DFT configurations (44/256/50 for the three systems), trains an interatomic potential, and then applies SSCHA to compute anharmonic free energies whose temperature-dependent crossing yields the phase-transition temperature. None of these steps reduces the output to a re-expression of the input data or to a self-citation; the acquisition function selects configurations for exploration, the model is fitted only to energies/forces, and the free-energy difference is a downstream observable not present in the training labels. The abstract and method description present this as an efficiency gain over brute-force sampling, with no load-bearing uniqueness theorem or ansatz imported from prior self-work that would force the result. The skeptic concern about convergence with 256 configurations is a question of statistical sufficiency, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on the abstract, no specific free parameters, axioms, or invented entities are detailed beyond standard computational methods in materials science.

pith-pipeline@v0.9.0 · 5580 in / 1083 out tokens · 53049 ms · 2026-05-07T15:43:34.641019+00:00 · methodology

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