Abstract: Accurate molecular simulation requires computationally expensive quantum chemistry models that makes simulating complex material phenomena or large molecules intractable. However, if no chemistry is required, but only interatomic forces then it should in principle be possible to construct much cheaper surrogate models, interatomic potentials, that capture full QM accuracy. This talk will review recent attempts, with focus on my personal analysis and numerical analysis perspectives, to achieve this. Specifically, I will explore whether we can rigorously justify the extremely low-dimensional functional forms proposed for interatomic potentials, and whether we can construct practical approximation schemes that can, in principle at least, close the complexity gap between density functional theory and interatomic potentials.