A Novel Approach for Data-Free, Physics-Informed Neural Networks in Fluid Mechanics Using the Principle of Minimum Pressure Gradient
Ahmed M. Atallah, Abdelrahman A. Elmaradny, Haithem E. Taha
Abstract
Fluid dynamics traditionally relies on the solution of the Navier-Stokes equations -- a set of non-linear partial differential equations that represent Newton's second law for fluid motion. However, these equations can be challenging to solve numerically for complex flow scenarios. This paper introduces a novel approach, applying Physics-Informed Neural Networks (PINNs) to fluid mechanics problems by implementing the principle of minimum pressure gradient (PMPG), which asserts that an incompressible flow evolves from one instant to another such that the total magnitude of the pressure gradient in the domain is minimized. Leveraging Gauss' principle of least constraint, this method bypasses the need for a direct solution of the Navier-Stokes equations, and instead, translates the fluid mechanics problem into a minimization problem. We demonstrate the effectiveness of this data-free method by solving the problem of flow around a cylinder, resulting in a model that aligns well with the expected physics, and showing a negligible deviation from Euler's equation in a scenario of an inviscid, and incompressible flow.