Estimating theoretical uncertainties of the two-nucleon observables by using backpropagation

Abstract We present a novel approach to calculating theoretical uncertainties in few-nucleon calculations, making use of automatic differentiation via backpropagation, which is particularly efficient when there are many input variables but only a few outputs. The methods described in this paper constitute tools that can be used to investigate the properties of scalar functions used to define nuclear potentials and quantify their contribution to the uncertainty of few nucleon calculations. We demonstrate these methods in deuteron bound state and nucleon–nucleon scattering calculations. Backpropagation, implemented in the Python pytorch library, is used to calculate the gradients with respect to model parameters and propagate errors from these parameters to the deuteron binding energy and selected phase-shift parameters. The uncertainty values obtained using this approach are validated by directly sampling from the potential parameters. We find very good agreement between two ways of estimating that uncertainty.