Physics-informed neural networks for viscoelastic fluid flows around a cylinder in a two-dimensional channel
Grid-based fluid dynamics solvers routinely struggle with exhaustive meshing demands and ill-posed inverse problems. A practical mesh-free alternative is given by Physics-informed neural networks (PINNs) but, applying them to highly elastic Oldroyd-B fluids results in many training failures. The High Weissenberg Number Problem (HWNP), driven by the exponential stress growth near stagnation points is a major issue, which causes standard PINN optimizers to diverge. To prevent the network from crashing, we apply a Cholesky decomposition to the conformation tensor. This mathematical constraint stabilizes the gradients by guaranteeing a positive-definite stress field. Beyond mathematical stability, the inherent spectral bias of deep learning models can hinder the network from accurately capturing the highly elastic wake structures. Therefore, we used sparse data assimilation to force the model toward the actual physical solution. By anchoring the physics loss with targeted CFD data points and accelerating training via transfer learning, we successfully pushed the network past non-physical local minima. We validated this Cholesky-PINN approach on flow past cylindrical geometries for Reynolds numbers (Re) between 5-25, for the single-cylinder setup. In addition, the relaxation time ({\lambda}) is increased from 0.1 to 0.5 to test the stability of the network. Finally, we scale the framework to a complex 3-cylinder array which proved our constructive solid geometry approach completely bypasses the tedious re-meshing steps of traditional CFD. The combined framework accurately captured sharp viscoelastic wakes, providing a stable computational tool for complex rheological modeling.