Time Changed Normalizing Flows for Accurate SDE Modeling

Abstract

The generative paradigm has become increasingly important in machine learning and deep slearning models. Among popular generative models are normalizing flows, which enable exact likelihood estimation by transforming a base distribution through diffeomorphic transformations. Extending the normalizing flow framework to handle time-indexed flows provided dynamic normalizing flows, a powerful tool to model time series, stochastic processes, and neural stochastic differential equations (SDEs). In this work, we propose a novel variant of dynamic normalizing flows, a Time-Changed Normalizing Flow (TCNF), based on time deformation of a Brownian motion which constitutes a versatile and extensive family of Gaussian processes. This approach enables us to effectively model some SDEs that cannot be modeled otherwise, including standard ones such as the well-known Ornstein-Uhlenbeck process, generalizes prior methodologies, and leads to improved results and better inference and prediction capability.