Neural Network-based High-index Saddle Dynamics Method for Searching Saddle Points and Solution Landscape

TL;DR

This paper introduces a neural network-based surrogate model for high-index saddle dynamics, enabling efficient computation of saddle points and solution landscapes even when the energy function is unavailable or costly to evaluate.

Contribution

It develops a neural network-enhanced HiSD method that approximates the energy function and incorporates momentum acceleration, with proven convergence and successful numerical tests.

Findings

Effective in systems without explicit energy functions

Enhanced efficiency with Nesterov's and heavy-ball acceleration

Demonstrated reliability on complex molecular models

Abstract

The high-index saddle dynamics (HiSD) method is a powerful approach for computing saddle points and solution landscape. However, its practical applicability is constrained by the need for the explicit energy function expression. To overcome this challenge, we propose a neural network-based high-index saddle dynamics (NN-HiSD) method. It utilizes neural network-based surrogate model to approximates the energy function, allowing the use of the HiSD method in the 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 a rigorous convergence analysis of the NN-HiSD method. We conduct numerical experiments on systems with and without explicit energy functions, specifically including the alanine dipeptide model and bacterial ribosomal assembly intermediates for the latter, demonstrating the effectiveness and reliability of the proposed method.