Laplacian networks: Growth, local symmetry, and shape optimization

OPTIMUM SHAPE DESIGN OF DOUBLE-LAYER GRIDS BY QUANTUM BEHAVED PARTICLE SWARM OPTIMIZATION AND NEURAL NETWORKS

In this paper, a methodology is presented for optimum shape design of double-layer grids subject to gravity and earthquake loadings. The design variables are the number of divisions in two directions, the height between two layers and the cross-sectional areas of the structural elements. The objective function is the weight of the structure and the design constraints are some limitations on str...

Laplacian shape modeling

Symmetry-invariant optimization in deep networks

Recent works have highlighted scale invariance or symmetry that is present in the weight space of a typical deep network and the adverse effect that it has on the Euclidean gradient based stochastic gradient descent optimization. In this work, we show that these and other commonly used deep networks, such as those which use a max-pooling and sub-sampling layer, possess more complex forms of sym...

Modal shape analysis beyond Laplacian

In recent years, substantial progress in shape analysis has been achieved through methods that use the spectra and eigenfunctions of discrete Laplace operators. In this work, we study spectra and eigenfunctions of discrete differential operators that can serve as an alternative to the discrete Laplacians for applications in shape analysis. We construct such operators as the Hessians of surface ...

Efficient Shape Optimization Using Polynomial Chaos Expansion and Local Sensitivities

This paper presents an efficient shape optimization technique based on stochastic response surfaces (polynomial chaos expansion) constructed using performance and local sensitivity data at heuristically selected collocation points. The cited expansion uses Hermite polynomial bases for the space of square-integrable probability density functions and provides a closed form solution of the perform...