Selected Topics in Mathematical Modelling: Machine Learning and Tugs-of-War

Abstract: This thesis concerns selected topics in mathematical modelling, namely in machine learning and stochastic games called tugs-of-war. It consists of four scientific articles. The first and second are about machine learning topics, while the third and fourth articles are about tug-of-war games.Article I presents an ensemble of independent and parallel long short-term memory (LSTM) neural networks for the prediction of financial time series. The model is applied to the constituents of the OMX30 index, and the resulting portfolio, when compared to benchmark portfolios, exhibits higher returns and lower volatility.Article II delves into the high-dimensional parameter space in which the optimization of deep neural networks occurs. The landscape traced out by stochastic gradient descent (SGD) is analyzed using diffusion maps to gain a deeper understanding of the landscape and possibly discover (local) low-dimensional representations. Empirical results show that SGD does indeed tend to move in a much lower-dimensional subspace, the dimension of which is robust to noise, depth, and weight initializations.Article III proves asymptotic mean value representation formulas for functions with respect to the fractional heat operator. Moreover, new nonlocal, nonlinear parabolic operators called the forward and backward infinity fractional heat operators are introduced. These operators are related to tug-of-war games that account for waiting times and space-time couplings.Article IV introduces a new class of strongly degenerate nonlinear parabolic partial differential equations (PDEs). The solutions are characterized in terms of an asymptotic mean value property and the results are connected to the analysis of modified tug-of-war games with noise. Existence and uniqueness of viscosity solutions to the Dirichlet problem are also established.