Data-driven shape inference in three-dimensional steady state supersonic flows using ODIL and JAX-Fluids

We present a novel data- and first-principles-driven method for inferring the shape of a solid obstacle and its flow field in three-dimensional steady-state supersonic flows. The method combines the Optimizing a Discrete Loss (ODIL) technique with the automatically differentiable JAX-Fluids CFD solver to jointly reconstruct flow fields and obstacle shapes. ODIL minimizes the discrete residual of the governing PDE via gradient descent-based algorithms and inherits the consistency and stability of the chosen numerical discretization. Discrete residuals and their gradients are computed using JAX-Fluids, which features nonlinear shock-capturing schemes and level-set-based immersed solid boundaries. We validate our method on synthetic data for challenging inverse problems, including shape inference of solid obstacles in 3D steady-state supersonic flows. In particular, we study flow around a cylinder, sphere, and ellipse. Two shape representations are investigated: (1) parametric, where the shape is described by a small set of parameters (e.g., radius of the cylinder or sphere) optimized jointly with the flow field, and (2) free-form, where the level-set function is optimized pointwise over the mesh without predefined shapes. For the parametric case, we provide a detailed comparison with Physics-Informed Neural Networks. We demonstrate that the combination of nonlinear shock-capturing discretization and level-set-based interface representation enables accurate inference of obstacle shapes and flow fields via the ODIL method. This approach opens new avenues for solving complex inverse problems in supersonic aerodynamics.