Steffen Maaß | Searcharxiv

arXiv2026

Effects of Horizontal Discretization on Triangular and Hexagonal Grids on Linear Baroclinic and Symmetric Instabilities

As global ocean general circulation models are run at eddy-permitting resolutions, reproducing accurate growth rates of baroclinic instabilities is a major concern when choosing a discretization of the equations of motion. From this viewpoint, we analyze discretizations on triangular and hexagonal grids with different types of variable staggering used in several ocean circulation models. By extending the linear baroclinic instability analysis in the Eady configuration to discretizations on more complex grids, several numerical subtleties are revealed. In comparison to discretizations on quadrilateral grids, the analyzed discretizations are less robust against unstable spurious modes, partly created by the mesh geometry. Some of the subtleties arise because spurious modes on staggered triangular and hexagonal grids do not adhere to Galilean invariance. As a consequence, their growth rates demonstrate a dependence on the alignment between the background flow and the grid, as well as the strength of a uniform background flow. The interactions with spurious modes become more significant on the axis of symmetric instabilities where the physical and spurious branches of instability are more difficult to separate in wavenumber space. Our analysis shows that in most cases moderate biharmonic viscosity and diffusion suppress spurious branches. However, one needs to carefully calibrate the viscosity and diffusivity parameters for each of the considered discretizations in order to achieve this.

physics.ao-ph↗

arXiv2023

A differentiable Gaussian Prototype Layer for explainable Segmentation

We introduce a Gaussian Prototype Layer for gradient-based prototype learning and demonstrate two novel network architectures for explainable segmentation one of which relies on region proposals. Both models are evaluated on agricultural datasets. While Gaussian Mixture Models (GMMs) have been used to model latent distributions of neural networks before, they are typically fitted using the EM algorithm. Instead, the proposed prototype layer relies on gradient-based optimization and hence allows for end-to-end training. This facilitates development and allows to use the full potential of a trainable deep feature extractor. We show that it can be used as a novel building block for explainable neural networks. We employ our Gaussian Prototype Layer in (1) a model where prototypes are detected in the latent grid and (2) a model inspired by Fast-RCNN with SLIC superpixels as region proposals. The earlier achieves a similar performance as compared to the state-of-the art while the latter has the benefit of a more precise prototype localization that comes at the cost of slightly lower accuracies. By introducing a gradient-based GMM layer we combine the benefits of end-to-end training with the simplicity and theoretical foundation of GMMs which will allow to adapt existing semi-supervised learning strategies for prototypical part models in future.

cs.CV↗