Abstract

Atomic simulations employ interatomic potentials as a surrogate model for electronic structure calculations of atomic energies and forces. In practice, interatomic potentials are always misspecified – no one choice of parameters can exactly match training data. Model parameters are thus intrinsically uncertain, and any scheme to propagate uncertainty to simulation results of interest should capture this uncertainty.

The posterior distribution from Bayesian inference would appear to be ideal for this purpose, but in fact the posterior is completely blind to misspecification uncertainty. This is most problematic when the underlying data is deterministic- the same input gives the same output, as is the case for electronic structure calculations and many other settings in computational science. As a result, Bayesian parameter uncertainties are severe underestimates, rapidly decaying to zero as the number of training points increases.

With analogy to the Gibbs-Bogoliubov bound in free energy estimation, I will discuss how the loss is only an upper bound to the true generalisation error. I will derive a condition any minimiser of the generalisation error must obey, and design a simple ansatz that can be variationally minimised¹. The variational minimum can be found analytically for high dimensional linear models, giving efficient estimation and very useful bounding of worse-case errors. Importantly, model prediction errors are now directly related to uncertainties in model parameters, essential to capture correlations present when propagating uncertainty through multi-scale simulations. If time allows, I will discuss how parameter uncertainty can be efficiently propagated when simulation results are stationary points on the atomic energy landscape².

[1] https://arxiv.org/abs/2402.01810v3 (with Danny Perez, Los Alamos National Laboratory) [2] https://arxiv.org/abs/2407.02414 (with Ivan Maliyov and Petr Grigorev, CNRS /Aix-Marseille U)