Utilizing deep learning algorithms for the resolution ofpartial differential equations

Abstract

Partial differential equations (PDEs) are mathematical equations that are used tomodel physical phenomena around us, such as fluid dynamics, electrodynamics,general relativity,  electrostatics,  and diffusion.   However,  solving these equa-tions can be challenging due to the problem known as the dimensionality curse,which makes classical numerical methods less effective. To solve this problem,we propose a deep learning approach called deep Galerkin algorithm (DGA).This technique involves training a neural network to approximate a solution bysatisfying the difference operator, boundary conditions and an initial condition.DGA alleviates the curse of dimensionality through deep learning, a meshlessapproach, residue-based loss minimisation and efficient use of data. We will testthis approach for the transport equation,  the wave equation,  the Sine-Gordonequation and the Klein-Gordon equation.