Efficient Crystalline Anharmonic Potential Energy Surfaces by Taking the Derivative of a Gaussian Process'

Machine learning (ML) a surrogate model is increasingly demonstrated as a useful tool for exploring potential energy surfaces (PESs). One such method is Gaussian process (GP) regression, which is particularly well suited to Bayesian (probabilistic) model construction. To obtain highly accurate PESs with a minimal number of electronic structure calculations, the GP model can be differentiated, and conditioned (trained) directly on the available forces (energy derivative). This exact transformation requires differentiating the GP kernel for the problem at hand. This could reduce the number of data points (from expensive electronic structure calculations) required for a given accuracy (at the cost of a more complex GP model).  Once you have a trained GP model, force-constants (to arbitrary order) can be calculated by taking the derivative of the GP model, which as differentiation is a linear operator, yields another GP model. We apply these new techniques to inferring the anharmonic properties of crystalline materials and compare it to the more standard approach of using finite displacements.