Abstract
We propose an algorithm for optimizations in which the gradients contain stochastic noise. This arises, for example, in structural optimizations when computations of forces and stresses rely on methods involving Monte Carlo sampling, such as quantum Monte Carlo or neural network states, or are performed on quantum devices that have intrinsic noise. Our proposed algorithm is based on the combination of two ingredients: an update rule derived from the steepest-descent method, and a staged scheduling of the targeted statistical error and step size, with position averaging. We compare it with commonly applied algorithms, including some of the latest machine learning optimization methods, and show that the algorithm consistently performs efficiently and robustly under realistic conditions. Applying this algorithm, we achieve full-degree optimizations in solids using ab initio many-body computations, by auxiliary-field quantum Monte Carlo with plane waves and pseudopotentials. A potential metastable structure in Si is discovered using density-functional calculations with synthetic noisy forces.
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Data availability
All data in this Article were generated with Quantum Espresso 5.0.1 and/or plane-wave AFQMC v26-2020.04.02 developed in our group (unpublished). Source data for Figs. 1–4 are available at https://github.com/schen24wm/geoopt-srcdata and Zenodo48 and also available with this Article. Detailed parameters of the Cmca structure (Fig. 5) we found are available in Supplementary Section 5.
Code availability
Code of FSSD × SET is available at https://github.com/schen24wm/fssd-set and Zenodo49, and available as Supplementary Software 1 with this Article.
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Acknowledgements
We thank B. Busemeyer, M. Lindsey, F. Ma, M. A. Morales, M. Motta, A. Sengupta, S. Sorella and Y. Yang for discussions. S.C. thanks the Center for Computational Quantum Physics, Flatiron Institute, for support and hospitality. We acknowledge financial support from the following grant: US Department of Energy (DOE) grant DE-SC0001303 (S.C.). The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript. We thank William & Mary Research Computing and Flatiron Institute Scientific Computing Center for computational resources and technical support. The Flatiron Institute is a division of the Simons Foundation.
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S.C. and S.Z. conceived the project. S.C. executed the research with support and supervision from S.Z. S.C. and S.Z. wrote the manuscript. S.Z. obtained funding.
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Supplementary Figs. 1–9 and Discussion.
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Scripts and an example for the FSSD-SET algorithm.
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Chen, S., Zhang, S. A structural optimization algorithm with stochastic forces and stresses. Nat Comput Sci 2, 736–744 (2022). https://doi.org/10.1038/s43588-022-00350-w
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DOI: https://doi.org/10.1038/s43588-022-00350-w