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We propose a model-free shrinking-dimer saddle dynamics for finding any-indexsaddle points and constructing the solution landscapes, in which the force inthe standard saddle dynamics is replaced by a surrogate model trained by theGassian process learning. By this means, the exact form of the model is nolonger necessary such that the saddle dynamics could be implemented based onlyon some observations of the force. This data-driven approach not only avoidsthe modeling procedure that could be difficult or inaccurate, but alsosignificantly reduces the number of queries of the force that may be expensiveor time-consuming. We accordingly develop a sequential learning saddle dynamicsalgorithm to perform a sequence of local saddle dynamics, in which the queriesof the training samples and the update or retraining of the surrogate force areperformed online and around the latent trajectory in order to improve theaccuracy of the surrogate model and the value of each sampling. Numericalexperiments are performed to demonstrate the effectiveness and efficiency ofthe proposed algorithm.

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