Separating Artificial-Viscosity Annealing from the L-BFGS Viscosity in Physics-Informed Neural Networks for Two-Phase Flow

Abstract

Physics-informed neural networks (PINNs) remain difficult to train for hyperbolic conservation laws. Artificial viscosity is a common remedy, and several methods reduce its coefficient during training. We examine adaptive moment estimation (Adam) followed by limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) optimisation. The schedule followed during Adam, the viscosity used during L-BFGS, and the checkpoint supplied between them can change together. We separate these factors for the one-dimensional Buckley–Leverett equation using a fixed multilayer perceptron with hyperbolic-tangent activations and 15 paired seeds per configuration. The central experiments impose a zero-diffusive-flux outlet condition. Lowering the L-BFGS viscosity from 3.5×10−3 to 1.67×10−4 increased full-grid relative L2 error by 0.0413±0.0069. The schedule–viscosity interaction was −0.0067±0.0075 and was not detected after multiplicity adjustment. A decay ending at 3.5×10−3 improved on the tested low-endpoint decay before high-viscosity polishing, while its difference from constant high viscosity remained inconclusive. The higher of the two tested constant viscosities improved aggregate and pre-breakthrough accuracy but increased post-breakthrough error. Stage-matched curricula showed no detected accuracy advantage; their greater executed cost arose from intermediate diagnostic L-BFGS phases whose states were discarded. These results identify the terminal viscosity, checkpoint rule, outlet treatment, and optimiser exposure as controls needed to interpret schedule comparisons in this pipeline.