Data assimilation for compressible flows by optimizing a discrete loss (ODIL) with automatic differentiation

We solve data assimilation and other inverse problems for compressible fluid flows including flow reconstruction from sparse measurements, inference of body shape from supersonic flow past the body, and inference of material parameters for two-phase flows. Our method is a combination of the ODIL (Optimizing a DIscrete Loss) framework to formulate the inverse problem through optimization and the fully-differentiable JAX-Fluids CFD package to obtain gradients of the residuals of the governing equations. The loss function includes residuals of the Euler equations, terms to impose known flow measurements, and regularization terms. The ODIL framework employs multiresolution techniques in the loss function and representation of unknown discrete fields to speed up the convergence of standard gradient-based optimizers. We identify combinations of boundary conditions and measurements necessary to infer the flow field, and study the effects of regularization terms on the convergence speed and reconstruction accuracy. Our results suggest that our framework can incorporate noisy and incomplete data into flow simulations and therefore complement experimental measurements. In addition, we provide a comparative study with the popular Physics-Informed Neural Networks (PINNs) method.