Thesis Format

Integrated Article

Using Physics-informed Neural Networks to Find Soliton Solutions to Nonlinear Dispersive Equations

Author

Nadia AiasehFollow

Degree

Master of Science

Program

Physics

Collaborative Specialization

Machine Learning in Health and Biomedical Sciences

Supervisor

Trichtchenko, Olga

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.

Summary for Lay Audience

Water waves on shallow beaches or the evolution of a pulsatile blood pressure wave as it travels down the arterial network can both be modelled using similar partial differential equations (PDEs). A PDE is an equation comprised of a function and its derivatives, wherein the function depends on more than one variable. They serve as powerful modelling tools because they provide information about how different rates of change (i.e. derivatives) relate to each other. By finding the function that satisfies this equation, the physical quantity under consideration - water wave amplitude or blood pressure for example - can be predicted. As such, solving PDEs is a rich mathematical research area. PDEs can be solved exactly (analytical methods) or approximated using computers (numerical methods). These methods are not always easy to implement. For instance, sometimes numerical methods can require a lot of computational resources.

With recent advances in deep learning, physics-informed neural networks (PINNs) have attracted attention as an alternative way of finding solutions to PDEs. Every neural network learns by minimizing a loss function during training. PINNs are a type of neural network that contain the PDE in their loss, making them informed of the governing physics of the problem. In this work, we use PINNs to find soliton solutions to PDEs governing water waves. PDEs modelling water waves are particularly challenging to solve due to being nonlinear and dispersive. Nonlinearity complicates the behaviour of these waves, while dispersion causes waves of different frequencies to travel at different speeds.

In this thesis, we bring together two areas of research. We combine the new and rapidly evolving deep learning and the old and well-researched nonlinear waves. We use PINNs to find soliton solutions to these nonlinear dispersive PDEs. Solitons are a type of solitary wave, that are able to maintain their shape and propagate with constant speed. They do so by balancing nonlinearity with dispersion. Despite being mathematical abstractions, their special properties makes them an attractive choice for modelling physical phenomena. For example, due to their lossless nature, they are taken advantage of in fibre optics to carry information with minimal degradation. In this thesis, we show how well PINNs can find solutions that are in agreement with what we know analytically, using only the PDE along with minimum initial and boundary conditions that are necessary for uniquely determining a solution. In addition, we take advantage of PINNs to find solutions for cases without an analytical solution. We also explore the aforementioned applications of modelling shallow beach and pulsatile blood pressure waves using PINNs and solitary wave initial profiles.

Recommended Citation

Aiaseh, Nadia, "Using Physics-informed Neural Networks to Find Soliton Solutions to Nonlinear Dispersive Equations" (2024). Electronic Thesis and Dissertation Repository. 10405.
https://ir.lib.uwo.ca/etd/10405

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