Minimizing nature's cost: Exploring data-free physics-informed neural network solvers for fluid mechanics applications

In this paper, we present a novel approach for fluid dynamic simulations by leveraging the capabilities of Physics-Informed Neural Networks (PINNs) guided by the newly unveiled Principle of Minimum Pressure Gradient (PMPG). In a PINN formulation, the physics problem is converted into a minimization problem (typically least squares). The PMPG asserts that for incompressible flows, the total magnitude of the pressure gradient over the domain must be minimum at every time instant, turning fluid mechanics into minimization problems, making it an excellent choice for PINNs formulation. Following the PMPG, the proposed PINN formulation seeks to construct a neural network for the flow field that minimizes Nature's cost function for incompressible flows in contrast to traditional PINNs that minimize the residuals of the Navier–Stokes equations. This technique eliminates the need to train a separate pressure model, thereby reducing training time and computational costs. We demonstrate the effectiveness of this approach through a case study of inviscid flow around a cylinder. The proposed approach outperforms the traditional PINNs approach in terms of training time, convergence rate, and compliance with physical metrics. While demonstrated on a simple geometry, the methodology is extensible to more complex flow fields (e.g., three-dimensional, unsteady, and viscous flows) within the incompressible realm, which is the region of applicability of the PMPG.