Physics-Informed Neural Networks: an AI Approach to Solve Direct and Inverse PDE Problems

Abstract

Partial differential equations (PDEs) arise naturally when modeling physical phenomena mathematically. The importance of PDE goes far beyond just solving a math problem; they are crucial for many physics and engineering problems; saying that the world is governed by PDEs is not an exaggeration. However, solving a PDE is not always straight- forward; despite their extreme importance, not all PDEs are solvable. Many numerical methods have been devised to solve PDEs over the years, like Finite Element Method, Finite Volume, etc.; nevertheless, these methods can suffer from numerical instability and being computationally expensive. Physics-Informed Neural Networks (PINNs) are a novel approach that uses the properties of deep learning to solve PDEs by combining AI and physics hand in hand without relying on traditional numerical discretization methods. In this thesis, we investigate the capabilities and limitations of PINNs in solving both di- rect and inverse PDE problems. We introduce a novel architecture that can be adjusted to any PDE with any boundary/initial conditions. We also attempt to solve the Firn PDE, a very important and complex PDE. Our findings provide practical guidelines on the imple-mentation of PINNs across different PDEs, highlighting where PINNs succeed and may potentially fail. These findings suggest that PINNs have the future potential to be the go-to PDE solving method.