The effect of physical constraints on the loss function landscapes of deep learning models

Deep learning models have demonstrated remarkable capabilities at producing fast predictions of complex flow fields. However, incorporating known physics is essential to ensure physically consistent solutions generalize to out-of-sample data. This research investigates the impact of different approaches to impose flow incompressibility and no-penetration boundary conditions on deep learning flow field predictions. This study finds that hard constraints lead to a notably more complex loss landscape, making it more difficult to fit a low-error model. This is compared to the loss-function landscape resulting from a soft constraints approach, where the data loss-function is augmented with additional field and boundary terms, such as in physics-informed neural networks. Finally, the importance of these constraint strategies is studied during extrapolation and prediction of physical quantities, such as lift and drag in an airfoil. This work's findings shed light on the challenges and trade-offs involved in incorporating physics into deep learning models, offering valuable insights for future research in physics-informed machine learning.