The high-index saddle dynamics (HiSD) method [J. Yin, L. Zhang, and P. Zhang, {\it SIAM J. Sci. Comput., }41 (2019), pp.A3576-A3595] serves as an efficient tool for computing index-$k$ saddle points and constructing solution landscapes. Nevertheless, the conventional HiSD method often encounters slow convergence rates on ill-conditioned problems. To address this challenge, we propose an accelerated high-index saddle dynamics (A-HiSD) by incorporating the heavy ball method. We prove the linear stability theory of the continuous A-HiSD, and subsequently estimate the local convergence rate for the discrete A-HiSD. Our analysis demonstrates that the A-HiSD method exhibits a faster convergence rate compared to the conventional HiSD method, especially when dealing with ill-conditioned problems. We also perform various numerical experiments including the loss function of neural network to substantiate the effectiveness and acceleration of the A-HiSD method.

翻译：高指标鞍点动力学（HiSD）方法[J. Yin, L. Zhang, and P. Zhang, {\it SIAM J. Sci. Comput., }41 (2019), pp.A3576-A3595]是计算指标$k$鞍点并构建解景观的有效工具。然而，传统HiSD方法在处理病态问题时常面临收敛速度慢的挑战。为解决此问题，我们通过引入heavy ball方法提出了加速高指标鞍点动力学（A-HiSD）。我们证明了连续A-HiSD的线性稳定性理论，并随后估计了离散A-HiSD的局部收敛率。分析表明，与常规HiSD方法相比，A-HiSD方法在病态问题场景下表现出更快的收敛速度。我们还通过包括神经网络损失函数在内的多种数值实验，验证了A-HiSD方法的有效性与加速性能。