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# Data-driven Reachability using Christoffel Functions and Conformal Prediction

*Proceedings of the Twelfth Symposium on Conformal and Probabilistic Prediction with Applications*, PMLR 204:194-213, 2023.

#### Abstract

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.