论文标题
使用Witten Laplacians定位Index-1鞍点
Using Witten Laplacians to locate index-1 saddle points
论文作者
论文摘要
我们引入了一种新的随机算法，以找到函数$ v的索引1鞍点：\ mathbb r^d \ to \ mathbb r $，其中$ d $可能很大。该算法可以看作是随机梯度下降的等效物，这是定位局部最小值的自然随机过程。它依赖于两种成分：（i）Witten laplacian的第一个特征模式（与$ v $相关的索引-1鞍点上的浓度属性）在$ 1 $ - forms上，以及（ii）（ii）涉及这种差异操作员的部分微分方程的概率表示。简单分子系统的数值示例说明了所提出的方法的功效。
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.
