Data Driven Approximations Of PDEs

Data Driven Approximations Of PDEs: On Robustness of Reduced Order Mappings between Function Spaces Against Noise

Lacombe, Pablo (TU Delft Electrical Engineering, Mathematics and Computer Science)

Tax, D.M.J. (mentor)
Naderibeni, M. (mentor)
Tömen, N. (graduation committee)

Delft University of Technology

Computer Science and Engineering

CSE3000 Research Project

2024-02-02

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.

Partial Differential Equations
Machine Learning (ML)
Principal Component Analysis (PCA)
Neural Network

http://resolver.tudelft.nl/uuid:588a0a1a-5193-4371-9f14-88ae65cbfb6b

Student theses

bachelor thesis

© 2024 Pablo Lacombe