Predicting goal error evolution from near-initial-information: A learning algorithm

We estimate the discretization error of time-dependent goals that are calculated from a numerical model of the spherical shallow-water equations. The goal errors are described as a weighted sum of local model errors. Our algorithm divides goal error estimation into three phases. In phase one, we select deterministic functionals of the flow as a mathematical description of local model error estimators. In phase two, a learning algorithm adapts the selected functionals to the numerical experiment under consideration by determining the free parameters of the functionals. To do this, the learning algorithm analyzes a short numerical simulation at two different resolutions. In phase three, goal errors are estimated using the local error estimators with the parameters learned in phase two. The required weights are the sensitivities of the goal with respect to local model errors; these sensitivities are calculated automatically with an Algorithmic Differentiation tool applied to the model’s source code.We apply this new error estimation algorithm to two different shallow water test cases: solid-body rotation and zonal flow against a mountain. For the solid-body rotation we successfully estimate the error of simulated regional potential energy and can track its evolution for up to 24 h. For the zonal flow against a mountain we also successfully estimate the error of simulated regional potential energy. From the comparison of the two test cases we see that the learning period must incorporate a similar flow state as the prediction period to enable useful goal error estimators.Our algorithm produces goal error estimates without detailed knowledge of the employed discretization. We believe that this learning approach can be useful in adapting error estimation techniques to complex models.

► We show how to estimate numerical errors in key physical quantities for a shallow water model.
► We reformulate the dual-weight residual method to be able to “learn” local model error characteristics from the flow state.
► The use of initial local information leads to goal error estimates for two test cases.