Investigating the Effects of Labeled Data on Parameterized Physics-Informed Neural Networks for Surrogate Modeling: Design Optimization for Drag Reduction over a Forward-Facing Step

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Physics-informed neural networks (PINNs) are gaining traction as surrogate models for fluid dynamics problems, combining machine learning with physics-based constraints. This study investigates the impact of labeled data on the performance of parameterized physics-informed neural networks (PINNs) for surrogate modeling and design optimization. Different training approaches, including physics-only, data-only, and several combinations of both, are evaluated using fully connected (FCNN) and Fourier neural network (FNN) architectures. The test case focuses on reducing drag over a forward-facing step through optimal placement and sizing of an upstream obstacle. Results demonstrate that the inclusion of labeled data significantly enhances the accuracy and convergence rates of FCNNs, particularly in predicting flow separation and recirculation regions, and improves the stability of design optimization outcomes. Conversely, FNNs exhibit inconsistent responses to parameter changes when trained with labeled data, suggesting limitations in their applicability for certain design optimization tasks. The findings reveal that FCNNs trained with a balanced integration of data and physics constraints outperform both data-only and physics-only models, highlighting the importance of optimizing the training approach based on the specific requirements of fluid mechanics applications.