Organizational Unit:
School of Physics
School of Physics
Permanent Link
Research Organization Registry ID
Description
Previous Names
Parent Organization
Parent Organization
Organizational Unit
Includes Organization(s)
ArchiveSpace Name Record
3 results
Publication Search Results
Now showing
1 - 3 of 3
-
ItemOn the Generalization of Shadowing to Fluid Turbulence: Practical Methods For Quantifying Dynamical Similarity(Georgia Institute of Technology, 2023-07-31) Pughe-Sanford, Joshua L.Chaos is an intrinsic property of many real world systems, impacting a number of today's open research questions. While many chaotic systems have known governing equations and are deterministically “solved,” we still lack a comprehensive framework for predicting, controlling, and simply making sense of such systems. And while recent advances in technology allow us to explore these systems through direct numerical simulation better than ever before, the need for an insightful theoretical framework is still very much alive. Such a framework exists in a subset of chaotic systems, known as Axiom A chaotic systems. As a result, Axiom A systems are understood quite well. However, the requirements for a system to be Axiom A are quite strict, and the overlap between systems that are Axiom A and those that are physically significant is quite small. A very important concept in Axiom A systems is the notion of shadowing, which allows the chaotic dynamics to be decomposed piecewise-in-time in terms of much easier to analyze solutions known as periodic orbits. Periodic orbits are solutions to the governing equations that, unlike chaos, repeat in time. Their compactness make periodic orbits very simple objects to manipulate, both numerically and theoretically. This decomposition ultimately results in a predictive theory of Axiom A systems both deterministically and statistically. In this dissertation, we seek to generalize the concept of shadowing to a broader class of (non Axiom A) chaotic systems, specifically, fluid turbulence. Although recent studies suggest that Exact Coherent Structures—e.g., repeating solutions to the Navier-Stokes equation—are descriptive of turbulence, it is an open question whether they are to turbulence what periodic orbits are to Axiom A chaos. Here, we propose a generalized method for quantifying shadowing and discuss the generalized nature of shadowing in turbulence. Our results suggest that an axiom A framework for chaos may be more generalizable than previously thought.
-
ItemPhysics-Inspired Machine Learning of Partial Differential Equations(Georgia Institute of Technology, 2023-07-30) Golden, Matthew RyanThis dissertation discusses the Sparse Physics-Informed Discovery of Empirical Relations (SPIDER) algorithm, which is a technique for data-driven discovery of governing equations of physical systems. SPIDER combines knowledge of symmetries, physical constraints like locality, the weak formulation of differential equations, and sparse regression to construct mathematical models of spatially extended physical systems. SPIDER is a valuable tool in synthesizing scientific knowledge as demonstrated by its applications. First, libraries of terms are constructed using available physical fields. The symmetries of a system allow libraries to be projected into independently transforming spaces, known as irreducible representations. This breaks relations down into their indivisible parts; each minimal physical relation is learned independently to reduce implicit bias. A library of nonlinear functions is constructed for each irreducible representation of interest. Second, each library term is evaluated in the weak formulation. SPIDER is aimed at experimental systems with inherently noisy data making accurate estimation of derivatives difficult. The weak formulation solves this problem: library terms are integrated over spacetime domains with flexible weight functions. Integration by parts can avoid numerical differentiation in many situations and increases robustness to noise by orders of magnitude. Clever weight functions can remove discontinuities and even entirely remove unobserved fields from analysis. Third, a sparse regression algorithm can find parsimonious relations ranging from dominant balances to multi-scale quantitatively accurate relations. Applications to direct numerical simulation of 3D fluid turbulence and experimental 2D active nematic turbulence are presented. SPIDER recovered complete mathematical models of both systems. The active nematic system is of particular interest; SPIDER identified a 2D description contradicting widely accepted theoretical descriptions used for over a decade. SPIDER facilitated the discovery of a new physical constraint on the fluid flow.
-
ItemDiscovering Governing Equations from Noisy and Incomplete Data(Georgia Institute of Technology, 2020-12-07) Reinbold, Patrick K.Partial differential equations (PDEs) provide macroscopic descriptions of systems in many disciplinesthroughout physical science, such as for fluid flows. With increasingly vast amounts of data becoming avail-able with the advancement of technology, machine learning is now offering an alternative to traditional modelconstruction, which may not feasible if the systems are too complex or have no known first principles. Theresearch presented in my thesis advances the current state of data-driven PDE modeling. The fundamentalapproach is to use symbolic regression to select a model that best fits the data out of a ”library” of can-didates. Existing symbolic regressions methods have done well for low-dimensional systems described byordinary differential equations (ODEs), but have yet to see successful application to PDEs for experimentaldata. This is due to three main issues common to measurements of extended systems: discretization of con-tinuous fields, noise corruption, and incomplete state measurements. To address these concerns, I treat thegoverning equations in their weak form. The weak model terms can be manipulated to remove completely orreduce the number of necessary derivative estimates, which are notoriously inaccurate for discrete and noisydata. Additionally, the missing (latent) variables can be removed from the weak form by carefully designinga weight function. These latent variables can later be reconstructed using the available data, domain knowl-edge, and (crucially) the model identified using the weak formulation. I tested this approach on a number ofsynthetic examples, with experimentally realistic conditions, and found the method to be significantly lesssensitive to noise and discretization spacing than alternative approaches. The method was then applied toexperimental turbulent fluid flow data to obtain a 2D model consistent over a range of driving values, usingonly measurements of the horizontal velocity. In summary, this work develops and validates a methodologyfor data-driven discovery of PDEs that is robust to noise and latent variables, and it demonstrates the abilityto do so on real-world data for the first time.1