The theoretical guarantees on the approximation accuracy and stability of the constructed multivariate models can be established through rigorous analysis of the interpolation error and the properties of the chosen basis functions. By ensuring that the chosen basis functions have good approximation properties and that the interpolation points are strategically selected, the accuracy of the model can be controlled. Additionally, stability analysis can be performed by examining the behavior of the system under perturbations in the data or parameters, ensuring that the model remains stable and reliable in practical applications.