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Efficient learning of accurate surrogates for simulations of complex systems

Nature Machine Intelligence volume 6, pages 568–577 (2024)Cite this article

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A preprint version of the article is available at arXiv.

Abstract

Machine learning methods are increasingly deployed to construct surrogate models for complex physical systems at a reduced computational cost. However, the predictive capability of these surrogates degrades in the presence of noisy, sparse or dynamic data. We introduce an online learning method empowered by optimizer-driven sampling that has two advantages over current approaches: it ensures that all local extrema (including endpoints) of the model response surface are included in the training data, and it employs a continuous validation and update process in which surrogates undergo retraining when their performance falls below a validity threshold. We find, using benchmark functions, that optimizer-directed sampling generally outperforms traditional sampling methods in terms of accuracy around local extrema even when the scoring metric is biased towards assessing overall accuracy. Finally, the application to dense nuclear matter demonstrates that highly accurate surrogates for a nuclear equation-of-state model can be reliably autogenerated from expensive calculations using few model evaluations.

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Fig. 1: Schematic for automated generation of inexpensive surrogates for complex systems.
Fig. 2: Candidate surrogates for the 2D Rastrigin function, learned with a thin-plate RBF estimator using ‘sparsity’ sampling, a ‘strict’ tolerance and a test metric for validity based on the average graphical distance between the learned surrogate and sampled data.
Fig. 3: Convergence of test score versus model evaluations for different benchmark functions learned with a thin-plate RBF estimator using ‘sparsity’ sampling and a test metric for validity based on the average graphical distance between the learned surrogate and sampled data.
Fig. 4: Candidate surrogates for the 2D Rosenbrock function, learned with a thin-plate RBF estimator using ‘sparsity’ sampling, a ‘loose’ tolerance and a test metric for validity based on the average graphical distance between the learned surrogate and sampled data.
Fig. 5: Candidate surrogates for the 8D Rosenbrock function, learned with a thin-plate RBF estimator using ‘sparsity’ sampling, a ‘loose’ tolerance and a test metric for validity based on the average graphical distance between the learned surrogate and sampled data.
Fig. 6: Candidate surrogates for quark matter equation of state (EOS) demonstrate fidelity of thin-plate RBF with Nelder–Mead solver to capture the complex phase transition (PT).

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Data availability

The dataset to create the figure plots in the main text and Supplementary Information are available via Zenodo at https://doi.org/10.5281/zenodo.10908462 (ref. 50).

Code availability

The code, as well as the sampled data and learned surrogates, is available via Code Ocean at https://doi.org/10.24433/CO.1152070.v1 (ref. 51).

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Acknowledgements

Research supported by Los Alamos National Laboratory under the Laboratory Directed Research and Development programme (project nos. 20190005DR, 20200410DI and 20210116DR), by the Department of Energy Advanced Simulation and Computing under the Beyond Moore’s Law Program and by the Uncertainty Quantification Foundation under the Statistical Learning programme. Triad National Security, LLC operates the Los Alamos National Laboratory for the National Nuclear Security Administration of the US Department of Energy (contract no. 89233218CNA000001). M.S.M. acknowledges support from the National Science Foundation through award PHY-2108505. We thank J. Haack for insightful feedback on the paper. This document is LA-UR-20-24947.

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Authors and Affiliations

  1. Fusion Energy Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA

    A. Diaw

  2. Los Alamos National Laboratory, Los Alamos, NM, USA

    M. McKerns & I. Sagert

  3. Department of Mathematics and Statistics, San José State University, San José, CA, USA

    L. G. Stanton

  4. Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, USA

    M. S. Murillo

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Contributions

A.D., M.M. and M.S.M. conceived the project. M.M. developed the software. A.D., I.S. and M.M. performed simulations and prepared figures. All authors were responsible for the formal analysis.

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Correspondence to A. Diaw or M. McKerns.

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Nature Machine Intelligence thanks Martin Buhmann, André da Silva Schneider and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Table 1 Combined table of convergence and performance tests for benchmark functions
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Supplementary Information

Supplementary tests cases, molecular dynamics tests and method implementation.

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Diaw, A., McKerns, M., Sagert, I. et al. Efficient learning of accurate surrogates for simulations of complex systems. Nat Mach Intell 6, 568–577 (2024). https://doi.org/10.1038/s42256-024-00839-1

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