Dissertation

Discovering Governing Equations from Noisy and Incomplete Data

Reinbold, Patrick A. K

2020

Overview

Partial differential equations (PDEs) provide macroscopic descriptions of systems in numerous fields, such as physics, biology, and chemistry. Additionally, with increasingly vast amounts of data becoming available with the advancement of technology, machine learning is now offering an alternative to traditional model construction (eg. from first principles). This alternative is particularly attractive for systems that are too complex for derivation from first principles to be tractable, or worse, where no first principles are known at all. The research presented in this thesis advances the current state of data-driven PDE modeling. The fundamental approach involves converting a candidate PDE into a system of algebraic equations linear in model parameters via a carefully designed evaluation procedure, and then using sparse regression to narrow down to the model that best fits the data. Existing regression methods, when applied to PDEs, rely on linear systems that are sensitive to noise corruption in the observations. However, regression methods can quickly explore the fitness of many different model options, and so are the default choice for systems that could be described by a large set of potential models.Several data-driven approaches to model discovery have been designed recently, but most have been shown to be inadequate for application to high-dimensional data described by PDEs. Some approaches (like training a neural network to ’learn’ the dynamics, from which the model can be backed out of the network parameters) are less susceptible to noise corruption, but are currently ill-suited to finding the best model out of a large set of candidates, since it does not directly accomodate changes to the model structure. On the other hand, symbolic regression methods of the type mentioned above can discern between different potential models efficiently. They do well for low-dimensional systems described by ordinary differential equations (ODEs), but have yet to see successful application to PDEs for experimental data. The main problem is that constructing the linear system prior to performing the regression has traditionally required locally evaluating derivatives of discrete and noisy data. Derivatives (especially of higher order) computed from discrete and noisy data is notoriously inaccurate and has hindered many attempts at PDE model identification.Besides relying on local (inaccurate) derivative information, existing regression methods rarely constrain the library of candidate model terms. However, it is found here that utilizing knowledge of relevant physics and symmetry can ensure that the candidate terms are both interpretable and not un-physical. The main breakthrough of this thesis is developing an alternative approach to building the linear system fed to the regression algorithm. By considering the weak form of the candidate model, the derivatives on many model terms can be moved onto a weight function whose derivatives are analytically known. The linear system is then filled with integral values that are far less sensitive to noise corruption. Furthermore, the weight function can be carefully designed to remove dependence on certain latent variables from the weak form, which enables identification of PDEs that would otherwise be impossible to work with due to the missing information. These latent variables can later be reconstructed using the available data, domain knowledge, and (crucially) the model identified using the weak formulation. The preceding approach was tested on a number of synthetic examples, and then applied to experimental turbulent fluid flow data to obtain a 2D model consistent over a range of driving values. In summary, the research presented here develops and validates a methodology for data-driven discovery of PDEs that is robust to noise and latent variables, and it demonstrates the ability to do so on real-world data for the first time.

• Publisher: ProQuest Dissertations & Theses
• Subject: Corruption / Electromagnetics / Fluid mechanics / Libraries / Magnetic fields / Mathematics / Ordinary differential equations / Parameter estimation / Partial differential equations / Physics / Reynolds number
• ISBN: 9798263382919