Executive Summary : | The availability of rich computational resources has immensely boosted entangled advances in numerical differential equations and deep learning. In one hand there are instances where numerical analysis and PDE's are applied in Deep learning. On the other hand data science and neural networks are applied to learn underlying differential equation model and to compute the solution of underlying PDE's. We consider the high order polynomial interpolation and their applications in deep learning. It is well known that they are core of numerical techniques like, finite difference formulas, ODE and PDE solvers. They also play crucial role in many area of engineering such as computer vision problems like image resizing, image resolution enhancement etc. For example, the traditional image to reshaping algorithms include nearest neighbor interpolation, bi-linear interpolation and high-order interpolation techniques. Recently from finite difference interpolations approximations, 3x3 and 5x5 sparse convolution filters are proposed which leads to memory efficient Resnet blocks. The correspondence between first order ODE and its numerical solvers are utilized to train deep learning networks having skip connections and hidden states can be visualized analogous to Euler method. Such residual networks are shown to be related with ODEs/PDEs. To achieve super-resolution reconstruction of images, a composite algorithm using traditional interpolation algorithms and deep learning is proposed. On the other hand, one can see applications of deep leaning in scientific computing. In particular, in one of the core area of scientific computing i.e., is to devise high order accurate new numerical methods for convection dominated or hyperbolic conservation laws type PDE models of fluid flow phenomena. Recently physics informed deep regression neural network are proposed compute the solution of such PDEs. Also deep regression networks are used to improve numerical schemes for PDEs'. Such improvement is done by learning some essential feature in underlying scheme e.g., learning viscosity, learning shock detection, learning WENO weights. The eccentricity of this study is entangled around polynomial interpolations and deep neural networks: their constructions, analysis, applications. |