Data-Driven Reachability Analysis with Christoffel Functions

We present an algorithm for data-driven reachability analysis that estimates finite-horizon forward reachable sets for general nonlinear systems using sub-level sets of a certain class of polynomials known as empirical inverse Christoffel functions, which are known empirically to provide good approximations to the support of probability distributions. The algorithm uses this property for reachability analysis by solving a probabilistic relaxation of the reachable set computation problem. We also provide a guarantee that the output of the algorithm is an accurate reachable set approximation in a probabilistic sense, provided that a certain sample size is attained. We also investigate three numerical examples to demonstrate the algorithm's capabilities, such as providing non-convex reachable set approximations and detecting holes in the reachable set.