The Principle of Minimum Pressure Gradient: An Alternative Basis for Physics-Informed Learning of Incompressible Fluid Mechanics

TL;DR

This work addresses the computational cost of physics-informed learning for incompressible fluid flows by introducing PMPG-PINN, which combines Gauss' principle of least constraint with PINNs to minimize the pressure gradient rather than solving for pressure directly. The method relies on minimizing the action $S$ under the continuity constraint, with training stabilized via Augmented Lagrangian Methods, thereby removing pressure from the training loop. On a lid-driven cavity benchmark at $\text{Re}=100$, PMPG-PINN achieves results in line with conventional NSE-PINN while reducing the per-epoch training time by approximately 12%. The approach offers a promising variational alternative for efficient CFD learning and could extend to transient, turbulent, and fluid-structure interaction problems, with post-hoc recovery of the pressure field if needed.

Abstract

Recent advances in the application of physics-informed learning into the field of fluid mechanics have been predominantly grounded in the Newtonian framework, primarly leveraging Navier-Stokes Equation or one of its various derivative to train a neural network. Here, we propose an alternative approach based on variational methods. The proposed approach uses the principle of minimum pressure gradient combined with the continuity constraint to train a neural network and predict the flow field in incompressible fluids. We describe the underlying principles of the proposed approach, then use a demonstrative example to illustrate its implementation and show that it reduces the computational time per training epoch when compared to the conventional approach.

Figures (6)

• Figure 1: Illustration of the evolution of the flow field in the space-time configuration, highlighting the path traced by $\mathbf{u}$ with a stationary quantity $\nabla S = 0$.
• Figure 2: Schematic of a physics-informed neural network showing both conventional PINN, and PMPG-PINN schemes.
• Figure 3: Schematic diagram of the lid-driven cavity problem showing the domain and boundary conditions.
• Figure 4: A physics-informed solution of the Lid-driven cavity problem using conventional PINN. (a) Convergence of the residual loss functions, (dashed) $\mathcal{L}_{PDE}$, and (line) $\mathcal{L}_c$. (b) Heat map of the velocity magnitude: $\sqrt{u^2+v^2}$.
• Figure 5: Variation of the quantity $S$ with the penalty weight $\mu_c$.