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A model-free shrinking-dimer saddle dynamics for finding saddle point and solution landscape

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Abstract

We propose a model-free shrinking-dimer saddle dynamics for finding any-index saddle points and constructing the solution landscapes, in which the force in the standard saddle dynamics is replaced by a surrogate model trained by the Gassian process learning. By this means, the exact form of the model is no longer necessary such that the saddle dynamics could be implemented based only on some observations of the force. This data-driven approach not only avoids the modeling procedure that could be difficult or inaccurate, but also significantly reduces the number of queries of the force that may be expensive or time-consuming. We accordingly develop a sequential learning saddle dynamics algorithm to perform a sequence of local saddle dynamics, in which the queries of the training samples and the update or retraining of the surrogate force are performed online and around the latent trajectory in order to improve the accuracy of the surrogate model and the value of each sampling. Numerical experiments are performed to demonstrate the effectiveness and efficiency of the proposed algorithm.

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Acknowledgements

We greatly appreciate Dr. Siwei Duo for providing the MATLAB code of computing the fractional Laplacian in [10].

Funding

This work was partially supported by the National Key R &D Program of China No. 2021YFF1200500, the Taishan Scholars Program of Shandong Province, and the National Natural Science Foundation of China No. 12225102, 12050002, 12288101.

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Authors and Affiliations

  1. Beijing International Center for Mathematical Research, Center for Machine Learning Research, Center for Quantitative Biology, Peking University, Beijing, 100871, China

    Lei Zhang

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Pingwen Zhang

  3. School of Mathematical Sciences, Laboratory of Mathematics and Applied Mathematics, Peking University, Beijing, 100871, China

    Pingwen Zhang

  4. School of Mathematics, Shandong University, Jinan, 250100, China

    Xiangcheng Zheng

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  1. Lei Zhang
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  2. Pingwen Zhang
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  3. Xiangcheng Zheng
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Contributions

LZ: Conceptualization, Funding acquisition, Methodology, Supervision, Writing – review & editing. PZ: Funding acquisition, Supervision. XZ: Formal analysis, Funding acquisition, Investigation, Methodology, Visualization, Writing – original draft.

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Correspondence to Lei Zhang.

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Zhang, L., Zhang, P. & Zheng, X. A model-free shrinking-dimer saddle dynamics for finding saddle point and solution landscape. Japan J. Indust. Appl. Math. 40, 1677–1693 (2023). https://doi.org/10.1007/s13160-023-00604-8

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  • DOI: https://doi.org/10.1007/s13160-023-00604-8

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