Error estimation in geophysical fluid dynamics through learning

Abstract

Current models of Geophysical Fluid Dynamics (GFD) lack the capability to quantify computationally induced errors. To address this issue, we present a new approach for numerical uncertainty quantification in GFD models: goal error estimation through learning. We estimate the error in important physical quantities – so-called goals – as a weighted sum of local model errors. Our algorithm divides this goal error estimation into three phases. In phase one, we select a mathematical description of local model errors, either a deterministic functional of the solution or a stochastic process. In phase two, a learning algorithm adapts the selected mathematical description to the numerical experiment under consideration by determining the free parameters of the mathematical description. The learning algorithm analyzes a series of short numerical simulations on different resolutions. In phase three, goal errors are estimated using the learned parameters of the local error description. The deterministic description produces a goal error estimate that can be used to correct the original goal approximation. The stochastic description produces a goal error estimate ensemble that can be used to construct error bounds for the original goal approximation. The goal error ensemble is generated from a single model forward evaluation. The weights that are required for both approaches 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 evaluate both algorithms within ICOSWM, a numerical model for the shallow water equations on the sphere, and implement an Algorithmic Differentiation framework that calculates any required goal sensitivity. With our deterministic approach, we are the first to estimate time-dependent goal approximation errors for the spherical shallow water equations. With our stochastic approach, we are the first to estimate an ensemble of goal approximation errors from only one forward solution of the model. We combine our local error learning algorithm with stochastic physics and initial condition ensemble techniques and compare the results of both forward ensembles and our a posteriori ensemble. For our test cases, we see that an a posteriori ensemble – derived from a single model solution – delivers comparable results as a stochastic physics ensemble that requires multiple model solutions. We suggest the extension of our method to total model error and discuss the general nature of local model errors. The algorithm proposed in this thesis bridges the gap between deterministic numerical methods and stochastic ensemble methods. It is generally applicable, easy to use, and simple compared to classical goal error estimation methods. Goal error estimation through learning is a first step towards automatic error bars for GFD models.