Using Physics-informed Neural Networks to Find Soliton Solutions to Nonlinear Dispersive Equations
Abstract
Partial differential equations (PDEs) can model different physical phenomena. They can be especially challenging to solve analytically or numerically if they contain nonlinear or dispersive (i.e. high order derivative) terms, which is the case with PDEs governing water waves. Recently, physics-informed neural networks (PINNs) have gained popularity as an alternative way of approximating the solution function. This deep learning method utilizes the governing PDE in the neural network's loss function. In this thesis, we explore the application of PINNs to nonlinear dispersive equations to find soliton solutions. Solitons are a type of solitary wave with special properties, such as being able to maintain their shape after nonlinearly interacting with each other. We use PINNs to find one-, two- and three-soliton solutions. We then generalize the network to two spatial dimensions. We also explore an application of solitons in modelling blood pressure profiles using PINNs. We comment on the effectiveness of PINNs, their shortcomings and provide insight for future research.