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

TL;DR

A neural network-based high-index saddle dynamics method that 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.

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.

Figures (12)

• Figure 1: Overview of the framework of the NN-HiSD method. (a) Function values at only a limited set of specific points available. (b) Construct neural network-based surrogate model. (c) Implement the HiSD method with the momentum acceleration. (d) Construct the solution landscape using downward and upward search.
• Figure 1: Trajectory of true potential and learned potential. (a) true energy potential $E(x)$. (b) learned energy potential. The initial point is represented by a brown solid circle, and the saddle point is represented by a red solid pentagram.
• Figure 2: $\|x^{(n)}-x_{NN}^*\|_2$ of the surrogate model-based HiSD method with respect to the iteration number. (a) dimer method to calculate $G(x)v_i$. (b) ADAD method to calculate $G(x)$.
• Figure 3: Solution landscape for both true model and surrogate model. (a) $\alpha=6$ (b) $\alpha=3$
• Figure 4: Trajectory of true potential and learned potential for the MB function. (a) True energy potential $E(x)$. (b) Learned energy potential $E_{NN}(x)$.

Theorems & Definitions (8)

• Remark 3.4
• Corollary 3.5
• Lemma 3.6
• Theorem 3.7
• Proof
• Theorem 3.8
• Proof
• Corollary 3.9