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Ulrich Friedrich

Publications and source records attributed to Ulrich Friedrich.

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Inductive biases in deep learning models for weather prediction

Deep learning has gained immense popularity in the Earth sciences as it enables us to formulate purely data-driven models of complex Earth system processes. Deep learning-based weather prediction (DLWP) models have made significant progress in the last few years, achieving forecast skills comparable to established numerical weather prediction models with comparatively lesser computational costs. In order to train accurate, reliable, and tractable DLWP models with several millions of parameters, the model design needs to incorporate suitable inductive biases that encode structural assumptions about the data and the modelled processes. When chosen appropriately, these biases enable faster learning and better generalisation to unseen data. Although inductive biases play a crucial role in successful DLWP models, they are often not stated explicitly and their contribution to model performance remains unclear. Here, we review and analyse the inductive biases of state-of-the-art DLWP models with respect to five key design elements: data selection, learning objective, loss function, architecture, and optimisation method. We identify the most important inductive biases and highlight potential avenues towards more efficient and probabilistic DLWP models.

physics.ao-ph

On the convergence analysis of the inexact linearly implicit Euler scheme for a class of SPDEs

This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an elliptic equation with random right-hand side has to be solved. In practice, this cannot be performed exactly, so that efficient numerical methods are needed. Well-established adaptive wavelet or finite-element schemes, which are guaranteed to converge with optimal order, suggest themselves. We investigate how the errors corresponding to the adaptive spatial discretization propagate in time, and we show how in each time step the tolerances have to be chosen such that the resulting perturbed discretization scheme realizes the same order of convergence as the one with exact evaluations of the elliptic subproblems.

math.PR