Optimizing the Discrete Loss for the Solution for Inverse problems in fluid mechanics: multiresolution and automatic differentiation

We present a potent method, Optimizing the DIscrete Loss (ODIL) for the solution of inverse problems in fluid mechanics .In ODIL inverse problems are formulated in terms of a deterministic loss function, based on a discete version of the governing equations, that can accommodate data and regularization terms. ODIL is based on similar ideas as the popular Physics Informed Neural Networks (PINNS) but does not deploy neural networks. A multigrid decomposition accelerates the convergence of gradient-based methods for optimization problems with parameters on a grid. The multiresolution ODIL (mODIL) improves the avoidance of local minima while automatic differentiation used for calculating the gradients of the loss function facilitates implementation of the framework. We demonstrate the capabilities of ODIL and mODIL on a variety of inverse and flow reconstruction problems: solution reconstruction for the Burgers equation, inferring conductivity from temperature measurements, and inferring the body shape from wake velocity measurements in three dimensions. A comparative study demonstrates that mODIL is 1000x to 100'000X faster than PINNs in a number of benchmark problems ranging from simple PDEs to lid-driven cavity problems. We discuss the advantages and defficiencies of the method. Our results suggest that mODIL is a very potent, fast and consistent method for solving inverse problems in fluid mechanics.