Neural Network–Based High-Index Saddle Dynamics Method for Searching Saddle Points and Solution Landscape

Abstract. The high-index saddle dynamics (HiSD) method is a powerful approach for computing saddle points and solution landscapes. However, its practical applicability is constrained by the requirement of an explicit expression of the energy function. To overcome this challenge, we propose a neural network–based high-index saddle dynamics (NN-HiSD) method. It utilizes a neural network-based surrogate model to approximate the energy function, allowing the use of the HiSD method in cases where the energy function is either unavailable or computationally expensive. We further enhance the efficiency of the NN-HiSD method by incorporating momentum acceleration techniques, specifically Nesterov’s acceleration and the heavy-ball method. We also provide approximation error estimates and a convergence analysis for the NN-HiSD method. We conduct numerical experiments on systems with and without explicit energy functions, specifically including the alanine dipeptide model and the pseudo-energy landscape of bacterial ribosomal assembly intermediates for the latter, demonstrating the effectiveness and reliability of the proposed NN-HiSD method. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/Sneakyturtle1234/NN-HiSD and in the supplementary materials ( NN-HiSD-main.zip [3.64MB]). [Formula: see text]