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Inverse problems in fluid dynamics are ubiquitous in science and engineering,with applications ranging from electronic cooling system design to oceanmodeling. We propose a general and robust approach for solving inverse problemsin the steady-state Navier-Stokes equations by combining deep neural networksand numerical partial differential equation (PDE) schemes. Our approachexpresses numerical simulation as a computational graph with differentiableoperators. We then solve inverse problems by constrained optimization, usinggradients calculated from the computational graph with reverse-mode automaticdifferentiation. This technique enables us to model unknown physical propertiesusing deep neural networks and embed them into the PDE model. We demonstratethe effectiveness of our method by computing spatially-varying viscosity andconductivity fields with deep neural networks (DNNs) and training the DNNsusing partial observations of velocity fields. We show that the DNNs arecapable of modeling complex spatially-varying physical fields with sparse andnoisy data. Our implementation leverages the open access ADCME, a library forsolving inverse modeling problems in scientific computing using automaticdifferentiation.

Solving Inverse Problems in Steady-State Navier-Stokes Equations using Deep Neural Networks