**Title: **Model Inversion via Machine Learning: Algorithms and Applications to Fluids and Materials sponsored by AFOSR Computational Mathematics Program

**Abstract: **This goal of this project is to develop a computational framework for model inversion based on multi-fidelity information fusion and stochastic optimization. This framework serves as a powerful tool in probabilistic design and parameter inference, especially in the presence of uncertainties, for applications in fluid mechanics and materials science. We incorporate new advances from machine learning using Gaussian processes (GP) and auto-regressive stochastic schemes but also new concepts from the theory of deep networks. The two new main contributions of this project are the drastic reduction of computational complexity by orders of magnitude and the ability to search for optimum functional forms and not simply parameters using the deep networks. To test the proposed framework we consider two different types of problems from fluid mechanics and soft matter. In particular, we consider turbulent flow past a cylinder and we attempt to learn the turbulence closure, which can be used in large-eddy simulations. We also consider polymers and employ mesoscopic simulations using the Dissipative Particle Method (DPD) to predict desired macroscopic material properties by learning the micro-parameters of DPD. This project is relevant to the Air Force and DoD in general as multi-fidelity simulation with total uncertainty quantification can be a big breakthrough for realistic simulations of industrial-complexity problems in fluid mechanics and materials science. In this new paradigm, no simulation is left behind as even very under-resolved simulations or very simplified mathematical models, or even empirical correlations can be employed to construct an accurate stochastic response surface. Although our main focus here is on parameter and boundary conditions estimation for fluids and materials, the implications of the proposed methodology are far reaching, and practically applicable to a wide class of inverse problems.