Construction of Predictive Dynamical Systems from Observed Data Through Data Driven Forecasting

Book excerpt: The evolution of particles in space, flows on an ocean surface, or orbits of the planets can all be thought of as their own dynamical systems who's [whose] forecasts and models are crucial to many scientific disciplines. These dynamical system models depict the physics of what is going on by mathematically describing how each state variable of the system evolves in time. It is our role as computational physicists to find solutions to these complex and often analytically unsolvable dynamical system models to aid in the study of interesting and important physics. In this dissertation we will go through the development and deployment of a melding of methods in applied mathematics and machine learning to construct approximate forms to dynamical systems equations for forecasting from data alone in a method known as Data Driven Forecasting (DDF). A theoretical background for the method is first discussed along with a sampling of the different variations of DDF. The utilization of Radial Basis Functions (RBF) to interpolate the behavior of dynamical systems plays a major role approximating the flow of the model dynamics. A breakdown of what dynamical properties like chaos, fractal dimension, Lyapunov exponent, and Jacobian are preserved and under what conditions in reconstructing the model from data. As DDF builds models from observed data alone, it will contend with the challenge of construction model approximations when fewer than the total dimensions are observed. Through the use of Taken's Embedding Theorem and time delay embedding techniques, the attractor can be reconstructed and forecasting made possible. This dissertation concludes with a thorough exploration of the method on a Neuro Dynamical system and Fluid Dynamical system where reduced dimensional observations are made and time delay embedding techniques must be used. The results shown in these sections are indicative of the potential for this method to both be expanded upon and applied for modern scientific pursuits.