Theoretical Limits and Sampling Guarantees for Machine-Learned Interatomic Potentials

Machine-learned interatomic potentials (MLIPs) accelerate molecular dynamics (MD) simulations by learning the energies and forces that govern atomic motion. Recent work shows that relaxing physical inductive biases can improve computational efficiency and predictive accuracy, raising the question of whether energy-and-force matching alone ensures accurate sampling. We show that the relationship between prediction error and sampling accuracy is exponentially ill-conditioned in dimension. In particular, prediction errors that decay exponentially with dimension can still coexist with nearly maximal total variation distance between the learned and target distributions. We identify two distinct failure mechanisms. The first is kinetic trapping: relaxation to an accurate learned equilibrium requires exponentially long times, even when simulations start from training configurations whose energies and forces are matched exactly. The second is equilibrium mismatch: the learned stationary distribution itself is far from the target. Under explicit assumptions, we show that equilibrium mismatch can persist in local, symmetry-preserving models, where small per-atom errors coexist with a distributional discrepancy that approaches its maximum as system size increases. Finally, we establish global energy-control conditions that ensure equilibrium accuracy and identify additional conditions needed for rapid relaxation.