Inverse problems in fluid dynamics are ubiquitous in science and engineering, with applications ranging from electronic cooling system design to ocean modeling. We propose a general and robust approach for solving inverse problems for the steady state Navier-Stokes equations by combining deep neural networks and numerical PDE schemes. Our approach expresses numerical simulation as a computational graph with differentiable operators. We then solve inverse problems by constrained optimization, using gradients calculated from the computational graph with reverse-mode automatic differentiation. This technique enables us to model unknown physical properties using deep neural networks and embed them into the PDE model. Specifically, we express the Newton's iteration to solve the highly nonlinear Navier-Stokes equations as a computational graph. We demonstrate the effectiveness of our method by computing spatially-varying viscosity and conductivity fields with deep neural networks (DNNs) and training  DNNs using partial observations of velocity fields. We show that  DNNs are capable of modeling complex spatially-varying physical field with sparse and noisy data. We implement our method using ADCME, a library for solving inverse modeling problems in scientific computing using automatic differentiation.
Inverse problems in fluid dynamics are ubiquitous in science and engineering, with applications ranging from electronic cooling system design to ocean modeling. We propose a general and robust approach for solving inverse problems in the steady-state Navier-Stokes equations by combining deep neural networks and numerical partial differential equation (PDE) schemes. Our approach expresses numerical simulation as a computational graph with differentiable operators. We then solve inverse problems by constrained optimization, using gradients calculated from the computational graph with reverse-mode automatic differentiation. This technique enables us to model unknown physical properties using deep neural networks and embed them into the PDE model. We demonstrate the effectiveness of our method by computing spatially-varying viscosity and conductivity fields with deep neural networks (DNNs) and training the DNNs using partial observations of velocity fields. We show that the DNNs are capable of modeling complex spatially-varying physical fields with sparse and noisy data. Our implementation leverages the open access ADCME, a library for solving inverse modeling problems in scientific computing using automatic differentiation.
R
for the steady state
in the steady-state
R
PDE
partial differential equation (PDE)
D
Specifically, we express the Newton's iteration to solve the highly nonlinear Navier-Stokes equations as a computational graph.
A
the
A
the
R
field
fields
R
We implement our method using
Our implementation leverages the open access