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How to Reconstruct the System's Dynamics by Differentiating Interval-Valued and Set-Valued Functions

机译:如何通过区分区间值函数和集合值函数来重建系统动力学

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To predict the future state of a physical system, we must know the differential equations x = f(x) that describe how this state changes with time. In many practical situations, we can observe individual trajectories x(t). By differentiating these trajectories with respect to time, we can determine the values of f(x) for different states x; if we observe many such trajectories, we can reconstruct the function f{x). However, in many other cases, we do not observe individual systems, we observe a set X of such systems. We can observe how this set X changes, but not how individual states change. In such situations, we need to reconstruct the function f(x) based on the observations of such "set trajectories" X(t). In this paper, we show how to extend the standard differentiation techniques of reconstructing f(x) from vector-valued trajectories x(t) to general set-valued trajectories X(t).
机译:要预测物理系统的未来状态,我们必须知道描述该状态如何随时间变化的微分方程x = f(x)。在许多实际情况下,我们可以观察到单个轨迹x(t)。通过相对于时间区分这些轨迹,我们可以确定不同状态x的f(x)值;如果我们观察到许多这样的轨迹,我们可以重建函数f {x)。但是,在许多其他情况下,我们没有观察到单个系统,而是观察到此类系统的集合X。我们可以观察到这组X的变化,但是观察不到各个状态的变化。在这种情况下,我们需要基于对这种“设定轨迹” X(t)的观察来重建函数f(x)。在本文中,我们展示了如何将重构f(x)的标准微分技术从矢量值轨迹x(t)扩展到通用集值轨迹X(t)。

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