Abstract

Motivated by a growing need for explainable artificial intelligence, this thesis explores quadratic neural networks (QNNs) as a solution to many issues that limit the application of neural networks to safety-critical systems. Specifically, this thesis shows that the training of both a two-layer convolutional quadratic neural network (CQNN) and a two-layer quadratic physics-informed neural network (QPINN), with no regularization and constraints added on their activation function parameters and the norm of their weights, can be formulated as a least squares problem. Using this method, an analytic expression for the globally optimal weights is obtained, alongside a quadratic input-output mapping for the network. These properties make the network a viable tool in system theory by enabling formal analysis. Furthermore, compared to backpropagation-trained networks, the least squares training significantly reduces both training time and the number of hyperparameters that the designer must select. Additionally, it is proven that two-layer QNNs become universal approximators when a monomial lifted input of sufficiently high degree is used. Finally, ensemble QNNs (EQNNs) are proposed to train multiple QNNs across the training space to increase the representational capacity of QNNs.