From Force Accuracy to Reliable Molecular Dynamics: Limits and Guarantees for Unconstrained MLIPs

Machine-learned interatomic potentials (MLIPs) are commonly trained by energy-and-force matching and deployed in molecular dynamics (MD) simulations. While many MLIPs incorporate physical inductive biases into their architectures, recent work has shown that relaxing these constraints can improve both computational efficiency and predictive accuracy. Motivated by this trend, we investigate whether energy-and-force matching is sufficient for reliable dynamics when the hypothesis class imposes no additional structural constraints. We prove that, in general, it is not. We show that smooth, conservative surrogates with uniformly bounded Hessians can achieve arbitrarily small population energy and force errors with respect to the target distribution in the unconstrained hypothesis class. However, their induced dynamics remain far from the target distribution in total variation distance, and this discrepancy grows with dimension. We isolate two failure mechanisms: kinetic trapping, where the learned stationary distribution is accurate but cannot be reached within any sub-exponential time horizon, and equilibrium mismatch, where the learned stationary distribution differs from the target. We further establish sufficient conditions under which energy-and-force matching provides sampling guarantees and discuss their implications for future MLIP design.