An important mathematical tool in the analysis of dynamical systems is the approximation of the reach set, i.e., the set of states reachable after a given time from a given initial state. This set is difficult to compute for complex systems even if the system dynamics are known and given by a system of ordinary differential equations with known coefficients. In practice, parameters are often unknown and mathematical models difficult to obtain. Data-based approaches are promised to avoid these difficulties by estimating the reach set based on a sample of states. If a model is available, this training set can be obtained through numerical simulation. In the absence of a model, real-life observations can be used instead. A recently proposed approach for data-based reach set approximation uses Christoffel functions to approximate the reach set. Under certain assumptions, the approximation is guaranteed to converge to the true solution. In this paper, we improve upon these results by notably improving the sample efficiency and relaxing some of the assumptions by exploiting statistical guarantees from conformal prediction with training and calibration sets. In addition, we exploit an incremental way to compute the Christoffel function to avoid the calibration set while maintaining the statistical convergence guarantees. Furthermore, our approach is robust to outliers in the training and calibration set.

通过利用基于数据的方法中的克里斯托费尔函数进行接近集的近似计算，我们改进了现有结果，提高了样本效率，并通过利用训练和校准集的统计保证，放宽了一些假设。此外，我们采用了增量计算克里斯托费尔函数的方法，避免使用校准集，同时保持统计收敛保证，并且我们的方法对训练集和校准集中的异常值具有鲁棒性。

基于数据驱动的可达性使用Christoffel函数和共形预测