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

TL;DR

This paper introduces a data-driven, model-free saddle dynamics method using Gaussian process learning to efficiently find saddle points and construct solution landscapes without requiring explicit models.

Contribution

It develops a novel surrogate model-based saddle dynamics approach that reduces modeling complexity and query costs, enabling efficient exploration of solution landscapes.

Findings

Effective in finding saddle points across various problems

Reduces number of force queries significantly

Demonstrates high accuracy and efficiency in numerical tests

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.