Learning Discontinuous Flows: Towards a Neural-Network-Based Platform for Variational Simulators

This work presents a neural-network–based computational framework for approximating weak solutions of the incompressible Euler equations in the light of the variational Principle of Minimum Pressure Gradient (PMPG). Classical smooth solutions of Euler fail to capture discontinuities such as vortex sheets and separation lines. To access these weak, possibly discontinuous solutions, we leverage the universal approximation capability of neural networks together with activation functions that admit sharp gradients, such as the Gaussian Error Linear Unit (GELU), while retaining the differentiability needed for automatic differentiation. The PMPG, derived from Gauss’ principle of least constraint, yields a pressure-free cost functional that selects the dynamically-correct flow field from the set of kinematically admissible ones by minimizing the L-2 norm of the pressure-gradient. We validate the framework on canonical discontinuous configurations. First, we approximate a vortex sheet using a regularized Gaussian representation of the Dirac delta, and show that the neural network recovers the expected vorticity distribution and circulation while enabling stable evaluation of convective acceleration and the PMPG cost. Second, we reconstruct the Parkinson-Jandali family of separating flows over a circular cylinder, demonstrating accurate representation of the separated flow and the associated shear layer