Neural-Network-Based Computational Framework for the Variational Theory of Lift

We present a neural-network framework for modeling inviscid flows over airfoils, based on the philosophy of the Principle of Minimum Pressure Gradient (PMPG). In this setting, the fluid dynamics problem is recast as the minimization problem: among all kinematically admissible velocity fields, we seek the one that minimizes the L-2 norm of the pressure gradient. Neural networks serve as generic parametrizations of the admissible flow field, enabling a generic approximation of the infinite-dimensional function space over which the PMPG functional is minimized. The neural network’s inputs are the spatial coordinates and the output is the streamfunction to naturally generate a divergence-free velocity field. In this constrained optimization setup, boundary conditions such as no-penetration and far-field are imposed as well as equilibrium and irrotationality constraints. The resulting framework is pressure-free and data-free; relying solely on physical principles, the variational PMPG formulation, and the function-approximation capabilities of neural networks. We assess the accuracy of the proposed method on benchmark inviscid test cases and compare the neural-network solutions against classical potential-flow theory in the light of the variational theory of lift. Our results show that the framework recovers potential-flow velocity fields, captures circulation effects, and implicitly converges to Kutta-like conditions through the variational minimization. Although the present work focuses on steady inviscid airfoil problem, the formulation is, in principle, extensible to viscous flows, unsteady configurations, three-dimensional geometries, and more general rheological models within the same variational-neural framework.