Data-driven shape inference in three-dimensional steady-state supersonic flows: Optimizing a discrete loss with JAX-Fluids

We present a 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 proposed method combines the optimizing a discrete loss (ODIL) technique with the automatically differentiable JAX-Fluids computational fluid dynamics (CFD) solver to study the joint reconstruction of flow fields and obstacle shapes. ODIL minimizes the discrete residual of the governing partial differential equation (PDE) by gradient descent-based algorithms. The ODIL framework inherits the characteristics of the chosen numerical discretization of the underlying PDE, including its consistency and stability. Discrete residuals and their automatic differentiation gradients are computed by the JAX-Fluids solver, which provides nonlinear shock-capturing schemes and level-set-based immersed solid boundaries. We use synthetic data to validate this approach on challenging inverse problems, including the shape inference of a solid obstacle in three-dimensional steady-state supersonic flow. Specifically, we study flows around a cylinder, a sphere, and an ellipse. We investigate two distinct approaches for the obstacle shape representation: (1) , where the obstacle is described by a small set of parameters (e.g., the radius of the cylinder and the sphere) that are optimized together with the flow field, and (2) , where the level-set function is directly optimized at each point of the computational mesh, without relying on predefined shapes. For the former, a thorough comparison with physics-informed neural networks is provided. We show that the nonlinear shock-capturing discretization in combination with the level-set-based interface representation allows for accurate inference of the obstacle shape and its flow field for the ODIL method. The proposed approach opens new avenues for solving complex inverse problems in supersonic aerodynamics.