Boundary Correspondence for Iso-Geometric Analysis Based on Deep Learning

One of the key problems in isogeometric analysis(IGA)is domain parameterization,i.e.,constructing a map between a parametric domain and a computational domain.As a preliminary step of domain parameterization,the mapping between the boundaries of the parametric domain and the computational domain should be established.The boundary correspondence strongly affects the quality of domain parameterization and thus subsequent numerical analysis.Currently,boundary correspondence is generally determined manually and only one approach based on optimal mass transport discusses automatic generation of boundary correspondence.In this article,we propose a deep neural network based approach to generate boundary correspondence for 2D simply connected computational domains.Given the boundary polygon of a planar computational domain,the main problem is to pick four corner vertices on the input boundary in order to subdivide the boundary into four segments which correspond to the four sides of the parametric domain.We synthesize a dataset with corner correspondence and train a fully convolutional network to predict the likelihood of each boundary vertex to be one of the corner vertices,and thus to locate four corner vertices with locally maximum likelihood.We evaluate our method on two types of datasets:MPEG-7 dataset and CAD model dataset.The experiment results demonstrate that our algorithm is faster by several orders of magnitude,and at the same time achieves smaller average angular distortion,more uniform area distortion and higher success rate,compared to the traditional optimization-based method.Furthermore,our neural network exhibits good generalization ability on new datasets.

A new way to construct topological invariants of non-Hermitian systems with the non-Hermitian skin effect

The non-Hermitian skin effect breaks the conventional bulk–boundary correspondence and leads to non-Bloch topological invariants.Inspired by the fact that the topological protected zero modes are immune to perturbations,we construct a partner of a non-Hermitian system by getting rid of the non-Hermitian skin effect.Through adjusting the imbalance hopping,we find that the existence of zero-energy boundary states still dictate the bulk topological invariants based on the band-theory framework.Two non-Hermitian Su–Schrieffer–Heeger(SSH)models are used to illuminate the ideas.Specially,we obtain the winding numbers in analytical form without the introduction of the generalized Brillouin zone.The work gives an alternative method to calculate the topological invariants of non-Hermitian systems.