Weak-form Sparse Identification of Differential Equations from Noisy Measurements

Messenger, Daniel Ames

Graduate Thesis Or Dissertation

Weak-form Sparse Identification of Differential Equations from Noisy Measurements Público Deposited

Analytics

Citations

Citeable URL: https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/7d278v276

Abstract

Data-driven modeling refers to the use of measurement data to infer the parameters and structure ofa mathematical model. While currently an active area research, data-driven modeling is characterized by the cycle of hypothesis, observation, and conclusion, which is none other than the scientific method, and can be traced back to Aristotle. For millenia, attempts have been made to distill governing laws from the observations made on a given system, with the hope of both explaining the observations and predicting future observations. Major accomplishments in this vein include Archimedes’ principle of buoyancy, Newtonian physics, and Röntgen’s X-rays. In each of these cases, the observations of a peculiar phenomena, often accidental, compelled the researcher to develop models. It is this map from observations to models that is at the heart of this thesis.The paradigm shift in recent years, driven by increased computing power, availability of large quan-tities of data, and the development of advanced mathematical techniques and algorithms, has been to automate the process of data-driven modeling. In the terminology of hypothesis, observation, and conclu- sion, automation can occur at the level of developing hypotheses about possible mathematical models, or the design of experiments which differentiate between the many possible models, or the map which takes experimental data and returns a mathematical model. To complete the cycle, one could also consider algorithms which generate potential nearby models given the model that is found to best fit the data. In this dissertation, we explore algorithms which automate the map from observations to governing equations, specifically differential equations. Our key contribution is the development of algorithms which identify differential equations in a weak form, which loosely refers to integrating the differential equation against arbitrary functions. We will show that the weak form is an ideal framework for identifying models from data if the criteria are robustness to data corruptions, highly accurate model recovery when corruption levels are low, and computational efficiency.We will demonstrate the superiority of our weak-form sparse identification for nonlinear dynamicsalgorithm (WSINDy) in the discovery of correct underlying model equations in a variety of differential equation and data corruption scenarios. We start with the simplest case, of ordinary differential equations (ODEs) depending only on time. We then move to partial differential equations (PDEs), where state variables change in both time and space. We then bridge the two previous regimes by considering interacting particle systems (IPSs), where the weak form is used to identify a mean-field PDE using data that can also be described as a large system of ODEs. Finally, we establish feasibility of weak- form identification of PDEs in an online context, where data is streamed in. In the online setting, we demonstrate the possibility of identifying time-varying models as well as models from data in four dimensions (3 space + 1 time).

Creator