  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.
