Implicit Branch Selection in Physics-Informed Neural Networks for an Underdetermined Exterior Laplace Problem: Potential Flow Around a Circular Cylinder with Weak Far-Field Regularization

Abstract

Abstract

Physics-informed neural networks (PINNs) have emerged as a promising framework for solving partial differential equations by embedding physical laws directly into the learning process. This study investigates the behaviour of a PINN applied to the classical exterior potential-flow problem around a circular cylinder, focusing on convergence under weak soft-penalty regularization rather than on explicit far-field boundary conditions. The no-penetration condition on the cylinder surface is the primary boundary constraint, while the physics-informed loss enforces the Laplace equation for the stream function. A weak far-field soft penalty provides qualitative free-stream information without constituting a hard far-field boundary condition when applied at radial distances greater than five-cylinder radii and weighted one-tenth of the solid boundary loss. It is important to establish that the exterior Laplace problem with only a solid wall boundary condition is mathematically non-unique; many harmonic functions, including the trivial zero solution, satisfy both the governing equation and the no-penetration condition. This work does not seek to resolve that non-uniqueness analytically or enforce mathematical uniqueness. Instead, it examines the empirical behaviour of the PINN under the given formulation. The results show convergence toward the classical uniform-flow solution branch, which is interpreted as an implicit branch selection influenced by the combined effects of global residual minimization, boundary anchoring, the soft far-field penalty, and the network's bias toward smooth, low-frequency functions. A parametric study spanning 33 training runs across 2000 to 10000 epochs, and 5000 to 60000 collocation points, reveals a highly non-monotonic optimization landscape. A recurrent failure mode is identified in which the absolute error converges to approximately 5.4, confirmed to equal the maximum magnitude of the analytical stream function over the evaluation domain, indicating collapse toward the trivial zero solution. A stability-optimal epoch regime near 8000 epochs is identified, producing zero failures across all tested collocation densities. The best-performing configuration achieves close agreement with the analytical solution for the stream function, velocity field, and pressure coefficient distribution.