PhD Comprehensive Exam | Phuong Dong Le, Machine Learning Methods for Solving Partial Differential Equations

DC 2314

Candidate

Phuong Dong Le | Applied Mathematics, University of Waterloo

Title

Machine Learning Methods for Solving Partial Differential Equations

Abstract

One of the fundamental problems in fields of science, physical phenomena and engineering is partial differential equations (PDE). Those are used to formulate problems of propagation of sound or heat, electrostatics, fluid flow and elasticity. Neural network models have shown a great potential in solving partial differential equations (PDE). Once trained with numerical simulation data, these models can provide faster alternative to traditional simulators and be efficient. However, they suffer from the generalization problem. There have been previous works that address the issue by applying universal approximation theorem for operator (DeepONet) using two sub-neural-networks. This approach has been generalized to neural network models that can learn mappings between function spaces. A recent work of neural operator (FNO) has been formulated as a new method by parameterizing the integral kernel into a Fourier space. In the proposal, we review a recent literature in the field of deep learning methods to approximate solution for partial differential equations. We consider the proposed neural architecture in solving examples of elliptic and hyperbolic partial differential equations.an more responsibly administer antibiotics.