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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.
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A minimization principle for incompressible fluid mechanics
Most variational principles in classical mechanics are based on the principle of least action, which is only a stationary principle. In contrast, Gauss' principle of least constraint is a true minimum principle. In this paper, we apply Gauss' principle to the mechanics of incompressible flows, thereby discovering the fundamental quantity that Nature minimizes in most flows encountered in everyday life. We show that the magnitude of the pressure gradient over the domain is minimum at every instant of time. We call it the principle of minimum pressure gradient (PMPG). It turns a fluid mechanics problem into a minimization one. We demonstrate this intriguing property by solving four classical problems in fluid mechanics using the PMPG without resorting to Navier–Stokes' equation. In some cases, the PMPG minimization approach is not any more efficient than solving Navier–Stokes'. However, in other cases, it is more insightful and efficient. In fact, the inviscid version of the PMPG allowed solving the long-standing problem of the aerohydrodynamic lift over smooth cylindrical shapes where Euler's equation fails to provide a unique answer. The PMPG transcends Navier–Stokes' equations in its applicability to non-Newtonian fluids with arbitrary constitutive relations and fluids subject to arbitrary forcing (e.g., electromagnetic).
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- Award ID(s):
- 2332556
- PAR ID:
- 10589109
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- Physics of Fluids
- Volume:
- 35
- Issue:
- 12
- ISSN:
- 1070-6631
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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