Utilizing deep learning algorithms for the resolution of partial differential equations

Abstract

Partial differential equations (PDEs) are mathematical equations that are used to model physical phenomena around us, such as fluid dynamics, electrodynamics, general relativity, electrostatics, and diffusion. However, solving these equations 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 by satisfying the difference operator, boundary conditions and an initial condition. DGA alleviates the curse of dimensionality through deep learning, a meshless approach, residue-based loss minimisation and efficient use of data. We will test this approach for the transport equation, the wave equation, the Sine-Gordon equation and the Klein-Gordon equation.