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Low-Scaling Algorithm for Nudged Elastic Band Calculations Using a Surrogate Machine Learning Model

José A. Garrido Torres, Paul C. Jennings, Martin H. Hansen, Jacob R. Boes, Thomas Bligaard

arXiv:1811.08022v1physics.comp-ph

TL;DR

Classical NEB can require hundreds of force calls because it iteratively moves and evaluates every image, and first-principles force evaluations are costly. The paper uses a GPR surrogate with uncertainty-guided acquisition and an image-count-independent convergence criterion; it reports the same 1.060 eV barrier in 11 ML-NEB calls versus 243 classical force calls. The authors also report solver-specific convergence issues in some tested settings.

  • Problem

    Classical NEB moves and evaluates every image iteratively, requiring hundreds of force calls, while first-principles force evaluations can be computationally expensive.

    Source: p. 2

  • Method

    The method uses a GPR surrogate’s uncertainty estimates in acquisition decisions and defines convergence independently of the number of NEB images.

    Source: p. 2

  • Results

    11 ML-NEB function calls versus 243 classical force calls yielded the same 1.060 eV energy barrier.

    Source: §1 · pp. 4–5

  • Takeaways & Limitations

    ML-NEB evaluation cost is independent of moving-image count, allowing the number of images to be chosen without added evaluation cost.

    Source: §1 · pp. 7–8

    LBFGS struggled to find an optimal minimum for one tested transition except with 11 images, and MDMin showed convergence issues in DFT calculations.

    Source: §1 · p. 8

Abstract

from arXiv · show

We present the incorporation of a surrogate Gaussian Process Regression (GPR) atomistic model to greatly accelerate the rate of convergence of classical Nudged Elastic Band (NEB) calculations. In our surrogate model approach, the cost of converging the elastic band no longer scales with the number of moving images on the path. This provides a far more efficient and robust transition state search. In contrast to a conventional NEB calculation, the algorithm presented here eliminates any need for manipulating the number of images to obtain a converged result. This is achieved by inventing a new convergence criteria that exploits the probabilistic nature of the GPR to use uncertainty estimates of all images in combination with the force of the transition state in the analytic potential. Our method is an order of magnitude faster in terms of function evaluations than the conventional NEB method with no accuracy loss for the converged energy barrier values.

1. In this example, we used 9 moving im-

Across the tested transitions, ML-NEB produces paths and barrier values comparable to classical NEB while requiring fewer function evaluations. Its evaluation cost is independent of the number of moving images.

1. In this example, we used 9 moving im-

  • 11 ML-NEB function calls versus 243 classical force calls yielded the same 1.060 eV energy barrier.Source: §1 · pp. 4–5
  • ML-NEB paths were virtually identical to those from FIRE, LBFGS, and MDMin, and Acq. 2 consistently used fewer evaluations than classical algorithms.Source: §1 · pp. 6–7
  • As image count increased, classical CI-NEB evaluations grew exponentially, whereas ML-NEB evaluation count remained independent of the number of moving images.Source: §1 · p. 7
  • The maximum GPR energy-prediction error stayed below the 0.05 eV uncertainty threshold, while Acq. 2 and 3 outperformed Acq. 1 in evaluation count and path accuracy.Source: §1 · p. 8
  • LBFGS struggled to find an optimal minimum for one transition except with 11 images, and MDMin encountered convergence issues in DFT calculations.Source: §1 · p. 8
  • Acquisition-function performance depends on balancing exploration of path uncertainty with exploitation toward saddle-point convergence.Source: §1 · pp. 8–9

Supplemental Material for Low-Scaling Algorithm for Nudged Elastic Band

The supplemental material identifies the paper, its authors and affiliations, and its date.

I. BENCHMARK USING DENSITY FUNCTIONAL THEORY

The DFT benchmark compares ML-NEB with conventional optimizers on three reaction systems, using 11 images per path. ML-NEB (Acq. 2) required the fewest function evaluations in the extended examples.

I. BENCHMARK USING DENSITY FUNCTIONAL THEORY

  • The DFT benchmark tested keto-enol tautomerization of formamide and water dissociation on Pt3 and Pt(111), using 11 images per reaction path.Source: §I · p. 11, Figure S.1 · p. 11
  • Table I reports the number of function evaluations needed by FIRE, BFGS, and ML-NEB, including its three acquisition functions, across the three reactions.Source: Table I · p. 11
  • 10–46 times more function evaluations were required by FIRE and BFGS than by ML-NEB in the extended examples, where Acq. 2 required the fewest.The MDMin algorithm was excluded from this benchmark because of convergence problems.Source: §I · p. 11

II. COMPUTATIONAL DETAILS

The computational details specify the periodic DFT settings and describe the Pt surface model and constraints used in the calculations.

II. COMPUTATIONAL DETAILS

  • Periodic DFT used PAW, plane waves with a 400 eV cutoff, Gaussian smearing of 0.1 eV, and PBE with Grimme’s D3(BJ) dispersion correction.Source: §II · p. 11
  • The electronic self-consistent cycle was converged to 10^-5 eV, with force thresholds of 0.02 eV/Å for relaxations and 0.05 eV/Å for NEB calculations.Source: §II · p. 11
  • For the Pt(111) model, a three-layer (3×3) slab had its top layer relaxed, with about 10 Å of vacuum normal to the surface.Source: §II · p. 12
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