Symmetry group of the equiangular cubed sphere

The equiangular cubed sphere is a spherical grid, widely used in computational physics. This paper deals with mathematical properties of this grid. We identify the symmetry group, i.e. the group of the orthogonal transformations that leave the cubed sphere invariant. The main result is that it coincides with the symmetry group of a cube. The proposed proof emphasizes metric properties of the cubed sphere. We study the geodesic distance on the grid, which reveals that the shortest geodesic arcs match with the vertices of a cuboctahedron. The results of this paper lay the foundation for future numerical schemes, based on rotational invariance of the cubed sphere.

Three dimensional visualization of simulations of liquids and solids

Visualization in three dimensions is invaluable for understanding the nature of condensed and fluid systems, but it is not always easy. In nature it is hard to view sample interiors, but on computers it is possible. We describe and contrast two opposite approaches - “smoke” visualization for viewing interiors of liquid samples and interactive WebGL for solids and molecules. Both are extensions of earlier Technion Computational Physics group projects and complement and are interoperable with the recent SimPhoNy Fp7 project. They require only desktop hardware and software accessible to students. Examples and standalone instructions for both are presented, starting with sample creation and concluding with image galleries.

Parameterized neural ordinary differential equations: applications to computational physics problems

This work proposes an extension of neural ordinary differential equations (NODEs) by introducing an additional set of ODE input parameters to NODEs. This extension allows NODEs to learn multiple dynamics specified by the input parameter instances. Our extension is inspired by the concept of parameterized ODEs, which are widely investigated in computational science and engineering contexts, where characteristics of the governing equations vary over the input parameters. We apply the proposed parameterized NODEs (PNODEs) for learning latent dynamics of complex dynamical processes that arise in computational physics, which is an essential component for enabling rapid numerical simulations for time-critical physics applications. For this, we propose an encoder–decoder-type framework, which models latent dynamics as PNODEs. We demonstrate the effectiveness of PNODEs on benchmark problems from computational physics.