We consider the following inverse problem for an ordinary differential equation (ODE): given a set of data points $P=\{(t_i,x_i),\; i=1,\dots,N\}$, find an ODE $x^\prime(t) = v (x)$ that admits a solution $x(t)$ such that $x_i \approx x(t_i)$ as closely as possible. The key to the proposed method is to find approximations of the recursive or discrete propagation function $D(x)$ from the given data set. Afterwards, we determine the field $v(x)$, using the conjugate map defined by Schr\"{o}der's equation and the solution of a related Julia's equation. Moreover, our approach also works for the inverse problems where one has to determine an ODE from multiple sets of data points. We also study existence, uniqueness, stability and other properties of the recovered field $v(x)$. Finally, we present several numerical methods for the approximation of the field $v(x)$ and provide some illustrative examples of the application of these methods.

中文翻译：

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使用共轭法求解常微分方程的逆问题

我们考虑以下常微分方程 (ODE) 的逆问题：给定一组数据点 $P=\{(t_i,x_i),\; i=1,\dots,N\}$，找到一个 ODE $x^\prime(t) = v (x)$ ，它承认一个解 $x(t)$ 使得 $x_i \approx x(t_i)$尽可能接近。所提出方法的关键是从给定的数据集中找到递归或离散传播函数 $D(x)$ 的近似值。之后，我们使用 Schr\"{o}der 方程定义的共轭映射和相关的 Julia 方程的解来确定场 $v(x)$。此外，我们的方法也适用于必须解决的逆问题从多组数据点确定 ODE。我们还研究了恢复字段 $v(x)$ 的存在性、唯一性、稳定性和其他属性。最后，