  In this paper we review the mathematical foundations of convolutional neural
nets (CNNs) with the goals of: i) highlighting connections with techniques from
statistics, signal processing, linear algebra, differential equations, and
optimization, ii) demystifying underlying computations, and iii) identifying
new types of applications. CNNs are powerful machine learning models that
highlight features from grid data to make predictions (regression and
classification). The grid data object can be represented as vectors (in 1D),
matrices (in 2D), or tensors (in 3D or higher dimensions) and can incorporate
multiple channels (thus providing high flexibility in the input data
representation). CNNs highlight features from the grid data by performing
convolution operations with different types of operators. The operators
highlight different types of features (e.g., patterns, gradients, geometrical
features) and are learned by using optimization techniques. In other words,
CNNs seek to identify optimal operators that best map the input data to the
output data. A common misconception is that CNNs are only capable of processing
image or video data but their application scope is much wider; specifically,
datasets encountered in diverse applications can be expressed as grid data.
Here, we show how to apply CNNs to new types of applications such as optimal
control, flow cytometry, multivariate process monitoring, and molecular
simulations.
