Discovering Governing Equations from Noisy and Incomplete Data
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Reinbold, Patrick K.
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Abstract
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
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2020-12-07
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Dissertation