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.

翻译：动力学系统分析是自然科学与工程领域的基础工具，其应用范围涵盖从星系演化到分子运动的各类系统。当对动力学系统的演化过程施加预定义条件时，底层微分方程必须在时空维度满足特定约束，此类问题被称为边值问题。本论文提出了在初末条件约束下学习时间可逆确定性及随机动力学的新方法。与传统通过数值积分求解微分方程的途径不同，本文通过机器学习算法从观测数据中推断动力学规律。研究按难度递增序列探讨了动力学学习的不同维度：首先基于确定性边界条件约束的真实解学习确定性动力学；其次研究离散状态空间的边值问题，其中正向动力学遵循随机跳跃过程而边界条件为离散概率分布，特别针对Ehrenfest过程这一特定跳跃过程，利用机器学习推断其逆时间动力学；最终探索在无参考信息条件下推断连续时间随机过程在两个概率分布间的动力学规律，为此提出学习两个随机过程时间可逆动力学的新判据以求解薛定谔桥问题。