The Ricci flow is a partial differential equation for evolving the metric in a Riemannian manifold to make it more regular. However, in most cases, the Ricci flow tends to develop singularities and lead to divergence of the solution. In this paper, we propose the linearly nearly Euclidean metric to assist manifold micro-surgery, which means that we prove the dynamical stability and convergence of the metrics close to the linearly nearly Euclidean metric under the Ricci-DeTurck flow. In practice, from the information geometry and mirror descent points of view, we give the steepest descent gradient flow for neural networks on the linearly nearly Euclidean manifold. During the training process of the neural network, we observe that its metric will also regularly converge to the linearly nearly Euclidean metric, which is consistent with the convergent behavior of linearly nearly Euclidean manifolds under Ricci-DeTurck flow.

翻译：里卡西流是一个局部的差别方程式,用来在里卡尼多管上演进量度,使其更加正常。然而,在多数情况下,里卡西流动倾向于形成独一性,导致解决方案的分歧。在本文中,我们提议了线性接近欧几里德的度量度,以协助多重微外科,这意味着我们证明,在里卡-德图尔克流下,接近线性近欧几里德多管的度量度值具有动态稳定性和趋同性。在实践中,从信息几何和镜面下沉点来看,我们给线性近欧几里德多管线性神经网络提供了最陡峭的下降梯度流。在神经网络的培训过程中,我们观察到,其度量度也将经常与线性近欧几里德多管流下的线性近欧几里德多管的趋同。