Multivariate AAA-style near-best rational approximation

Develop an algorithm for multivariate near-best rational approximation analogous to AAA in one variable, including a robust multivariate barycentric-like framework and stability guarantees.

Background

The authors demonstrate the effectiveness of AAA in univariate settings and illustrate multivariate rational approximations by reductions or line-wise fits, but they explicitly identify the absence of a true multivariate AAA algorithm. They regard this as an important open problem with significant implications for high-dimensional function approximation and PDE modeling.