Computing numerical approximations of high-dimensional PDEs is affected by the so-called curse of dimensionality. In other words, given a fixed grid size, the number of required spatial discretization points to realize that size increases exponentially as the dimension of the domain grows. As a result, scientists are constantly developing efficient algorithms in order to reduce the computational cost involved in computing approximate solutions of high-dimensional problems. However, when done at the frontiers of computability, the numerical outputs resulting from these algorithms may be extremely hard to validate. In order to address this fundamental issue in the context of high-dimensional PDEs, we introduce in this talk a computational method that allows on the one hand producing rigorous mathematical theorems regarding the validity of the numerical outputs, while on the other hand having the important property that the extra computational cost involved in the proof is not affected by the curse of dimensionality. This is joint work with Marcio Gameiro (University of São Paulo at São Carlos).