Time-Reversible Bridges of Data with Machine Learning (2412.13665v1)

Abstract: The analysis of dynamical systems is a fundamental tool in the natural sciences and engineering. It is used to understand the evolution of systems as large as entire galaxies and as small as individual molecules. With predefined conditions on the evolution of dy-namical systems, the underlying differential equations have to fulfill specific constraints in time and space. This class of problems is known as boundary value problems. This thesis presents novel approaches to learn time-reversible deterministic and stochastic dynamics constrained by initial and final conditions. The dynamics are inferred by machine learning algorithms from observed data, which is in contrast to the traditional approach of solving differential equations by numerical integration. The work in this thesis examines a set of problems of increasing difficulty each of which is concerned with learning a different aspect of the dynamics. Initially, we consider learning deterministic dynamics from ground truth solutions which are constrained by deterministic boundary conditions. Secondly, we study a boundary value problem in discrete state spaces, where the forward dynamics follow a stochastic jump process and the boundary conditions are discrete probability distributions. In particular, the stochastic dynamics of a specific jump process, the Ehrenfest process, is considered and the reverse time dynamics are inferred with machine learning. Finally, we investigate the problem of inferring the dynamics of a continuous-time stochastic process between two probability distributions without any reference information. Here, we propose a novel criterion to learn time-reversible dynamics of two stochastic processes to solve the Schr\"odinger Bridge Problem.