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Accelerating Materials Discovery with Bayesian Optimization and Graph Deep Learning

Yunxing Zuo, Mingde Qin, Chi Chen, Weike Ye, Xiangguo Li, Jian Luo, Shyue Ping Ong

arXiv:2104.10242v1cond-mat.mtrl-sci

TL;DR

Structure-based ML needs equilibrium crystal structures, but obtaining them through DFT relaxation is costly. The paper introduces BOWSR, which uses symmetry-constrained Bayesian optimization and a graph energy model to approximate equilibrium structures, improve property predictions, and screen about 400,000 candidates.

  • Problem

    Structure-based ML property prediction requires equilibrium crystal structures as inputs, but obtaining them through relaxation typically relies on costly DFT calculations.

    Source: §1 · p. 2, §3 · p. 13

  • Method

    BOWSR uses Bayesian optimization over symmetry-constrained parameters with a MEGNet formation-energy model to approximate equilibrium crystal structures without DFT.

    Source: §1 · pp. 2–3, §3 · p. 13

  • Results

    BOWSR reasonably approximates equilibrium structures, substantially improves ML property predictions, and supports rapid screening of about 400,000 candidate crystals.

    Source: §3 · p. 13

  • Takeaways & Limitations

    The approach was used to screen candidate crystals for stability and exceptional mechanical properties.

    Source: §3 · p. 13

    BOWSR's effectiveness is limited by the accuracy of its energy evaluator.

    Source: §3 · p. 13

Abstract

from arXiv · show

Machine learning (ML) models utilizing structure-based features provide an efficient means for accurate property predictions across diverse chemical spaces. However, obtaining equilibrium crystal structures typically requires expensive density functional theory (DFT) calculations, which limits ML-based exploration to either known crystals or a small number of hypothetical crystals. Here, we demonstrate that the application of Bayesian optimization with symmetry constraints using a graph deep learning energy model can be used to perform "DFT-free" relaxations of crystal structures. Using this approach to significantly improve the accuracy of ML-predicted formation energies and elastic moduli of hypothetical crystals, two novel ultra-incompressible hard materials MoWC2 (P63/mmc) and ReWB (Pca21) were identified and successfully synthesized via in-situ reactive spark plasma sintering from a screening of 399,960 transition metal borides and carbides. This work addresses a critical bottleneck to accurate property predictions for hypothetical materials, paving the way to ML-accelerated discovery of new materials with exceptional properties.

1. Introduction

Structure-based ML can predict materials properties across diverse crystals, but obtaining the equilibrium structures it needs typically requires expensive DFT relaxation. The paper introduces BOWSR to relax crystals without DFT and demonstrates its use in screening and synthesizing hard materials.

1. Introduction

  • Using both structural and chemical features lets ML models distinguish polymorphs and apply across diverse crystal structures while predicting properties such as formation energies and elastic moduli.Graph neural networks represent crystal atoms and bonds as nodes and edges, and have shown state-of-the-art accuracy across energetic, electronic, and mechanical properties.Source: §1 · p. 2
  • Structure-based ML needs equilibrium crystal structures as inputs, but obtaining them requires relaxing initial structures along potential energy surfaces typically computed with expensive DFT.Earlier efforts to derive graph-based interatomic forces were limited to a few molecular systems or constrained chemical spaces.Source: §1 · p. 2
  • BOWSR uses Bayesian optimization with symmetry constraints and a MEGNet formation-energy model to obtain equilibrium crystal structures without DFT.The resulting relaxed structures serve as inputs to ML property models.Source: §1 · pp. 2–3
  • Screening ~400,000 transition-metal borides and carbides identified two new materials with relatively low predicted energy above hull, which were synthesized and showed exceptional mechanical properties.The materials are MoWC2 and ReWB.Source: §1 · p. 3

2. Results

The results describe BOWSR’s use of symmetry-constrained Bayesian optimization with a graph energy model to relax crystals and screen candidate materials. They also present the workflow for generating and DFT-verifying candidates, including the synthesis of MoWC2 and ReWB, whose measured moduli agreed with predictions and which exhibited ultra-incompressibility.

2.1. Bayesian Optimization With Symmetry Relaxation Algorithm

  • BOWSR parameterizes each crystal based on independent lattice parameters and uses Bayesian optimization to iteratively propose lower-energy geometries with an energy model.The MEGNet formation-energy model was trained on DFT-computed formation energies of 133,420 Materials Project crystals and had an MAE of 26 meV atom−1.Source: §2.1 · pp. 4–5

2.2. Properties Prediction

  • Using BOWSR-relaxed structures as inputs reduces energy MAE by a factor of four, from 363 meV atom−1 to 88 meV atom−1, and raises R2 from 0.67 to 0.96.Source: §2.2 · p. 7
  • The DFT-relaxed-input MAEs are a best-case baseline, not a measure of MEGNet performance, because the models were trained on data including DFT-relaxed structures.The reported MAEs are therefore much smaller than the models' previously reported cross-validation MAEs.Source: §2.2 · p. 6
  • BOWSR-relaxed structures showed consistently low prediction errors regardless of initial structure, while relaxed-structure errors correlated linearly with surrogate-model errors across RMSE values of 27 to 1000 meV atom−1.Linear correlations also held for elastic-modulus errors.Source: §2.2 · p. 7

2.3. Discovery of Ultra-incompressible Hard Materials

  • Candidates are generated by elemental substitution into structure prototypes, relaxed with BOWSR, screened for predicted thermodynamic stability and mechanical properties, then verified using DFT.Figure 3 presents this discovery workflow.Source: Figure 3 · p. 8, §2.3 · p. 8
  • BOWSR with MEGNet screened 399,960 generated candidates, and 1,603 candidates passing all three ML screening criteria received DFT relaxations.Candidates came from elemental substitutions to 5,555 ternary structure prototypes; screening evaluated stability and mechanical properties.Source: §2.3 · p. 9
  • MoWC2 (P63/mmc) and ReWB (Pca21) were synthesized as single phases and confirmed by XRD, while six other attempted compositions yielded multiple phases.Attempts covered ten candidates with eight unique compositions.Source: §2.3 · p. 9, Figure 4 · p. 11
  • Both materials had bulk and shear moduli in excellent agreement with MEGNet and DFT predictions, with bulk moduli close to or above 300 GPa.The moduli were measured using the pulse-echo method.Source: Figure 5 · p. 12, §2.3 · p. 12
  • MoWC2 had Hv 36.6 at 0.49 N and 20.9 GPa at 9.8 N; ReWB had 29.5 at 0.49 N and 17.6 GPa at 9.8 N.The 0.49 N hardness values were within 20–25% of estimates derived from MEGNet- and DFT-predicted shear moduli using Hv = 0.151G.Source: Figure 5 · p. 12, §2.3 · pp. 12–13

3. Discussion

The discussion presents symmetry-constrained Bayesian optimization with a MEGNet energy model as a way to approximate equilibrium crystal structures without ab initio relaxation. These structures improve ML property predictions and support large-scale screening, while the approach remains dependent on energy-model accuracy and reliable target-property models.

3. Discussion

  • The approach addresses a costly bottleneck because obtaining equilibrium structures for accurate ML property models requires expensive ab initio computations.Source: §3 · p. 13
  • Coupling a MEGNet energy model with Bayesian optimization over symmetry-constrained parameters approximates equilibrium structures.Source: §3 · p. 13
  • The resulting structures reasonably approximate equilibrium and substantially improve ML property predictions.The approach enabled rapid screening of approximately 400,000 candidate crystals for stability and exceptional mechanical properties.Source: §3 · p. 13
  • BOWSR effectiveness is limited by the energy evaluator's accuracy, and more accurate ML energy models may improve relaxed structures and subsequent property predictions.Source: §3 · p. 13
  • The method was demonstrated for ultra-incompressible materials because acquiring elastic moduli with standard DFT is costly, but it can extend to properties with reliable ML models.Source: §3 · p. 13
  • Expensive-property datasets tend to be smaller, making reliable ML models harder to build; higher errors should be factored into screening thresholds.Source: §3 · pp. 13–14

5. Material and Methods

The methods combine symmetry-constrained Bayesian optimization and MEGNet models with DFT calculations, spark plasma sintering, and experimental characterization of candidate materials, including repeated indentation measurements and ultrasonic velocity assessments.

5.1. Bayesian Optimization With Symmetry Relaxation Algorithm

  • Symmetry was held fixed to reduce independent variables, while Bayesian optimization searched lattice parameters and atomic coordinates for minimum energy using variable changes as inputs.The search used the energy model as a surrogate.Source: §5.1 · p. 14

5.2. Materials Graph Network Models

  • MEGNet models trained on Materials Project data achieved test MAEs of 26 meV atom−1 for Ef, 0.07 for log10(KVRH), and 0.12 for log10(GVRH).The training data comprised 133,420 formation-energy pairs and 12,179 bulk/shear-modulus pairs.Source: §5.2 · p. 16

5.3. DFT calculations

  • The DFT relaxations and energy calculations used VASP with PBE-GGA, a 520 eV cutoff, and energy and force convergence thresholds of 10−5 eV and 0.01 eV/Å, respectively.Elastic-tensor calculations used a tighter energy convergence criterion of 10−7 eV.Source: §5.3 · p. 16

5.4. Synthesis

  • Eight candidate compositions—ReOsB, ReOsB2, MoWC2, ReWB, Re13WB9, OsWB, Re6W7B8, and ReW2B2—were synthesized at composition-specific target temperatures of 1500–1800 °C and held isothermally for 10 min.During final densification, pressure was increased to 50 MPa at a ramp rate of 5 MPa/min.Source: §5.4 · p. 17

5.5. Experimental Characterization

  • Sintered specimens were ground to remove the carbon-contaminated surface layer from graphite tooling and polished; Rigaku Miniflex XRD used Cu Kα radiation at 30 kV and 15 mA, while LECO Vickers tests applied 0.49–9.8 N.Source: §5.5 · p. 18
  • Over 30 indentation measurements per specimen were conducted at 9.8 N across different locations to support statistical validity and reduce microstructural and grain-boundary effects.Source: §5.5 · p. 18
  • Young’s and shear moduli were calculated from ultrasonic velocities measured with a Tektronix TDS 420A digital oscilloscope, following ASTM standard A494-15.Source: §5.5 · p. 18

6. Data Availability

The raw and processed data needed to reproduce the findings cannot currently be shared for legal or ethical reasons.

6. Data Availability

  • The raw and processed data required to reproduce the findings cannot be shared at this time for legal or ethical reasons.Source: §6 · p. 18

Supplementary Material Accelerating Materials Discovery with Bayesian

Across the supplementary analyses, BOWSR relaxation lowers and stabilizes MEGNet formation-energy errors relative to unrelaxed substitution structures, while prediction errors correlate with energy-model error.

measured lattice parameters for the discovered ultra-incompressible crystals,

The passage names ReWB and MoWC2 and states that their lattice angles are fixed by their crystal systems.

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