Quantitative schlieren with physics-informed neural networks

This article introduces physics-informed quantitative schlieren (PIQS), a data-assimilation framework that transforms standard qualitative schlieren images into quantitative flow fields. By leveraging physics-informed neural networks (PINNs), the method treats the unknown calibration factor between image intensity and density gradient as a learnable parameter that is inferred during the training process. The solution is anchored using Rankine–Hugoniot jump relations across identified shock waves to solve the inherent scale ambiguity. The framework is validated against a numerical ground truth for inviscid supersonic flow over a rhombus airfoil, where it recovers the true calibration factor and density field with high accuracy ( R² > 0.99 R 2 > 0.99 ). Application to a turbulent Mach 2 shock-wave/boundary-layer interaction (SBLI) demonstrates that PIQS yields reconstructions consistent with independent quantitative background-oriented schlieren (BOS) measurements. Finally, the method is applied to previously published schlieren imagery of a supersonic projectile, successfully recovering quantitative density profiles that match analytical Taylor–Maccoll solutions. These results demonstrate that PIQS offers a robust pathway for extracting quantitative data from both contemporary and historical schlieren visualizations without requiring dedicated calibration hardware.