PDE-Constrained Optimization of Subgrid‑Scale Models Applied to a Viscous Shu–Osher Analog

We demonstrate an application of the deep learning PDE model (DPM) framework for subgrid-scale (SGS) modeling in shock-dominated compressible flows. This work extends the classical Shu-Osher benchmark to include viscous effects. A viscous shock profile with imposed perturbations is computed via the one-dimensional Navier-Stokes equations to emulate shock-boundary layer interactions. We compare a conventional offline SGS model, trained a priori on filtered fine-mesh data, with an embedded SGS model that integrates closure terms directly into the governing equations using adjoint-based optimization. Our results show that the embedded SGS model provides superior representation of shock-induced SGS dynamics and improved numerical stability on coarse grids, relative to the offline approach and analogous simulations without closures. These results motivate future efforts in developing embedded SGS models for shock-boundary layer interactions.