We revisit the relation between the gradient-flow equations and Hamilton's equations in information geometry. By regarding the gradient-flow equations as Huygens' equations in geometric optics, we have related the gradient flows to the geodesic flows induced by the geodesic Hamiltonian in an appropriate Riemannian geometry. The original evolution parameter $\textit{t}$ in the gradient-flow equations is related to the arc-length parameter in the associated Riemannian manifold by Jacobi-Maupertuis transformation. As a by-product, it is found the relation between the gradient-flow equation and replicator equations.

翻译：本文重新审视了信息几何中梯度流方程与哈密顿方程之间的关系。通过将梯度流方程视为几何光学中的惠更斯方程，我们将梯度流与适当黎曼几何中由测地哈密顿量诱导的测地流联系起来。梯度流方程中的原始演化参数$\textit{t}$通过雅可比-莫佩尔蒂变换与相关黎曼流形中的弧长参数建立关联。作为副产品，我们发现了梯度流方程与复制方程之间的关系。