Data Driven Approximations Of PDEs

On Robustness of Reduced Order Mappings between Function Spaces Against Noise

Bachelor thesis (2024)

Authors

P.O. Lacombe Electrical Engineering, Mathematics and Computer Science

Contributors

D.M.J. Tax Pattern Recognition and Bioinformatics - EEMCS (supervisor 1)
M. Naderibeni Pattern Recognition and Bioinformatics - EEMCS (supervisor 1)
N. Tömen Pattern Recognition and Bioinformatics - EEMCS (supervisor 2)
Faculty
Electrical Engineering, Mathematics and Computer Science
Copyright: © 2024 Pablo Lacombe
More Info
expand_more

Published Date

02-02-2024

Language

English

Reuse Rights

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.

Faculty

Electrical Engineering, Mathematics and Computer Science

Abstract

This paper presents a comprehensive exploration of a novel method combining Principal Component Analysis (PCA) and Neural Networks (NN) to efficiently solve Partial Differential Equations (PDEs), a fundamental challenge in modeling a wide range of real-world phenomena. Our research extends the work of Bhattacharya et al. by focusing on PCA for effective dimensionality reduction and utilizing NN for mapping in the reduced dimension. This approach addresses the significant computational challenges and inaccuracies often encountered with classical numerical techniques in solving PDEs.

We specifically investigate the still-water equation, employing our PCA-NN method to learn a reduced order mapping of PDE solutions and evaluate its robustness in diverse noisy environments. Our findings reveal a notable relationship between noise intensity and error, indicating a linear trend for Gaussian and Salt and Pepper noise, and an exponential trend for Uniform noise. Furthermore, this study uncovers a critical weakness of the model in predicting points with a high rate of change.

Overall, our research significantly contributes to understanding the practical applicability and limitations of PCA-NN methods in real-world, noisy settings, offering valuable insights for future applications in this domain.