DEEP LEARNING BASED METHODS FOR THE STUDY OF DYNAMICAL SYSTEMS

Abstract

Dynamical systems under the influence of random perturbations often arise in scientific modeling. The committor function plays an important role in understanding the rare but important transition events between metastable states. Quasipotential is a natural generalization of the concept of energy functions to non-equilibrium dynamical systems. The invariant distribution is another important object in the study of randomly perturbed dynamical systems. However, computing these objects for high-dimensional dynamical systems is a challenging task, due to the curse of dimensionality and other issues such as the scarcity of transition data at low temperatures. In this thesis, we develop effective and robust methods based on deep learning for computing the committor function at low temperatures, mapping the quasipotential landscape and solving the Fokker-Planck equation for the invariant distribution that overcome these issues. The ability and effectiveness of the proposed methods are demonstrated by complex and high-dimensional examples.