In this paper, we establish the connection between exponential representation of curve (surface) and the classical linear differential system. That is, a curve (surface) can be described by a vector-valued function $X\left(t\right)=e^{At}{X}_0\left(t\right)$ ($X(s,t)=e^{At}{X}_0(s,t)$), which can be viewed as a solution of the ordinary (partial) linear differential system $\dot{X}\left(t\right)=AX\left(t\right)+f\left(t\right)$ ($\frac{\partial X(s,t)}{\partial t}=AX(s,t)+f(s,t)$). Further, we propose two algorithms to fit the given discrete data points by the solution curve (surface) of the ordinary (partial) linear differential system with nonhomogeneous term. We also show that the solution of the linear differential system described by $\dot{X}\left(t\right)=AX\left(t\right)+f\left(t\right)$ ($\frac{\partial X(s,t)}{\partial t}=AX(s,t)+f(s,t)$) is not unique, so we find a way to determine an optimal A based on the flow field information from the given data points. Numerical examples illustrate the effectiveness of the proposed algorithms.

在本文中，我们建立了曲线（表面）的指数表示与经典线性微分系统之间的联系。即，可以通过矢量值函数来描述曲线（曲面）$X（Ť）={Ë}^{\mathrm{一种}Ť}{X}_0（Ť）$ （$X（s，Ť）={Ë}^{\mathrm{一种}Ť}{X}_0（s，Ť）$），可以看作是普通（部分）线性微分系统的解决方案 $\dot{X}（Ť）=\mathrm{一种}X（Ť）+F（Ť）$ （$\frac{\partial X（s，Ť）}{\partial Ť}=\mathrm{一种}X（s，Ť）+F（s，Ť）$）。此外，我们提出了两种算法，通过具有非齐次项的普通（部分）线性微分系统的解曲线（表面）拟合给定的离散数据点。我们还表明，线性差分系统的解由$\dot{X}（Ť）=\mathrm{一种}X（Ť）+F（Ť）$ （$\frac{\partial X（s，Ť）}{\partial Ť}=\mathrm{一种}X（s，Ť）+F（s，Ť）$）不是唯一的，因此我们找到了一种基于给定数据点的流场信息确定最佳A的方法。数值算例说明了所提出算法的有效性。