Approximating Quasistationary Distributions Using Deep Learning

Sammanfattning: We study a class of It\={o} diffusion processes on domains with smooth boundary, at which the process is killed. Such a process, when conditioned on non-extinction, gives rise to a stationary state known as a \emph{quasistationary distribution} (QSD). Let $\mathcal{L}$ be the \emph{infinitesimal generator} of the \emph{Markov semigroup} of the process and let $\mathcal{L}^'$ denote the formal adjoint of $\mathcal{L}$, given by $\langle \mathcal{L} f, g \rangle=\langle f, \mathcal{L}^' g \rangle$ for suitable functions $f$ and $g$. For a killed It\={o} diffusion on a domain $G$, $\mathcal{L}^'$ is a differential operator that can be expressed analytically in terms of the drift and diffusion coefficients of the process. It is well-known that an eigenfunction of the Dirichlet eigenvalue problem\begin{align'}\mathcal{L}^' u(x) &= \lambda u(x), ~~~~~~~~~~~ x \in G \\u(x) &= 0, ~~~~~~~~~~~~~~~~~ x \in \partial G\end{align'}gives the unique (up to normalization) QSD of the process. The goal of this thesis is to develop a Deep Learning-based numerical scheme to approx\hyp{}imate QSDs by exploiting the connection to the aforementioned eigenvalue problem. To this end we consider both methods based on minimizing a residual that takes account of the PDE dynamics---so called physics based methods---and schemes that use a stochastic representation of the PDE, with emphasis on the former. As a basis for comparison, we derive analytical expressions for the QSD of a diffusion process with constant coefficients and for the Ornstein-Uhlenbeck (OU) process. In the constant coefficient case, we highlight a general problem with penalty-based methods that gives rise to \emph{spurious solutions} due to the numerical method solving a relaxed problem in practice. Finally, we show empirically that, at least in the OU case, it is possible to obtain a good approximation of the QSD using Deep Learning.