We introduce a new stochastic algorithm to locate the index-1 saddle points of a function $V:\mathbb R^d \to \mathbb R$, with $d$ possibly large. This algorithm can be seen as an equivalent of the stochastic gradient descent which is a natural stochastic process to locate local minima. It relies on two ingredients: (i) the concentration properties on index-1 saddle points of the first eigenmodes of the Witten Laplacian (associated with $V$) on $1$-forms and (ii) a probabilistic representation of a partial differential equation involving this differential operator. Numerical examples on simple molecular systems illustrate the efficacy of the proposed approach.

翻译：我们提出了一种新的随机算法，用于定位函数$V:\mathbb R^d \to \mathbb R$（其中$d$可能较大）的1型鞍点。该算法可视为随机梯度下降的等价形式，而随机梯度下降是定位局部极小值的自然随机过程。它依赖于两个关键要素：(i) 关于$1$-形式上的Witten拉普拉斯算子（与$V$相关联）的第一本征模在1型鞍点处的浓度性质，以及(ii) 涉及该微分算子的偏微分方程的概率表示。对简单分子系统的数值算例验证了所提方法的有效性。