In its simplest form, convolution neural networks (CNNs) consist of a fully connected layer g composed with a sequence of convolution layers T . Although g is known to have the universal approximation property, it is not known if CNNs, which has the form g ◦ T inherits this property, especially when the kernel size in T is small. In this paper, we show that under suitable conditions, CNNs does inherit the universal approximation property and its sample complexity can be characterized. In addition, we discuss concretely how the nonlinearity of T can improve the approximation power. Finally, we show that when the target function class has a certain compositional form, convolutional networks are far more advantageous compared with fully connected networks, in terms of number of parameters needed to achieve a desired accuracy.

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

Accurate simulation of seismic waves is of critical importance in a variety of geophysical applications. Based on recent works on staggered discontinuous Galerkin methods, we have developed a new method for the simulations of seismic waves, which has energy conservation and extremely low grid dispersion, so that it naturally provided accurate numerical simulations of wave propagation useful for geophysical applications and was a generalization of classical staggered-grid finite-difference methods. Moreover, it could handle with ease irregular surface topography and discontinuities in the subsurface models. Our new method discretized the velocity and the stress tensor on this staggered grid, with continuity imposed on different parts of the mesh. The symmetry of the stress tensor was enforced by the Lagrange multiplier technique. The resulting method was an explicit scheme, requiring the solutions of a