Mach. Learn. Knowl. Extr., Volume 2, Issue 1 (March 2020) – 4 articles

This article presents a new methodology called Deep Theory of Functional Connections (TFC) that estimates the solutions of partial differential equations (PDEs) by combining neural networks with the TFC. The TFC is used to transform PDEs into unconstrained optimization problems by analytically embedding the PDE’s constraints into a “constrained expression” containing a free function. In this research, the free function is chosen to be a neural network, which is used to solve the now unconstrained optimization problem. This optimization problem consists of minimizing a loss function that is chosen to be the square of the residuals of the PDE. The neural network is trained in an unsupervised manner to minimize this loss function. This methodology has two major differences when compared with popular methods used to estimate the solutions of PDEs. First, this methodology does not need to discretize the domain into a grid, rather, this methodology can randomly sample points from the domain during the training phase. Second, after training, this methodology produces an accurate analytical approximation of the solution throughout the entire training domain. Because the methodology produces an analytical solution, it is straightforward to obtain the solution at any point within the domain and to perform further manipulation if needed, such as differentiation. In contrast, other popular methods require extra numerical techniques if the estimated solution is desired at points that do not lie on the discretized grid, or if further manipulation to the estimated solution must be performed. Full article

Forest structure estimation is very important in geological, ecological and environmental studies. It provides the basis for the carbon stock estimation and effective means of sequestration of carbon sources and sinks. Multiple parameters are used to estimate the forest structure like above ground biomass, leaf area index and diameter at breast height. Among all these parameters, vegetation height has unique standing. In addition to forest structure estimation it provides the insight into long term historical changes and the estimates of stand age of the forests as well. There are multiple techniques available to estimate the canopy height. Light detection and ranging (LiDAR) based methods, being the accurate and useful ones, are very expensive to obtain and have no global coverage. There is a need to establish a mechanism to estimate the canopy height using freely available satellite imagery like Landsat images. Multiple studies are available which contribute in this area. The majority use Landsat images with random forest models. Although random forest based models are widely used in remote sensing applications, they lack the ability to utilize the spatial association of neighboring pixels in modeling process. In this research work, we define Convolutional Neural Network based model and analyze that model for three test configurations. We replicate the random forest based setup of Grant et al., which is a similar state-of-the-art study, and compare our results and show that the convolutional neural networks (CNN) based models not only capture the spatial association of neighboring pixels but also outperform the state-of-the-art. Full article

An artificial neural network (ANN) is an automatic way of capturing linear and nonlinear correlations, spatial and other structural dependence among features. This machine performs well in many application areas such as classification and prediction from magnetic resonance imaging, spatial data and computer vision tasks. Most commonly used ANNs assume the availability of large training data compared to the dimension of feature vector. However, in modern applications, as mentioned above, the training sample sizes are often low, and may be even lower than the dimension of feature vector. In this paper, we consider a single layer ANN classification model that is suitable for analyzing high-dimensional low sample-size (HDLSS) data. We investigate the theoretical properties of the sparse group lasso regularized neural network and show that under mild conditions, the classification risk converges to the optimal Bayes classifier’s risk (universal consistency). Moreover, we proposed a variation on the regularization term. A few examples in popular research fields are also provided to illustrate the theory and methods. Full article