Optimization of complex fluids flows using end to end differentiable immersed boundary algorithm

Inverse design of fluid flows of complex fluids is important in several applications such as controlling the fluid rheological response, optimizng the design of microfluidic devices, etc. Solving a PDE-constrained optimization problem using direct optimization approaches where the jacobian is calcualted from a forward simulation is easy to implement and flexible but is prohibitively computationally expensive. Automatic differentiation which is the key driver to train deep neural networks alllows for efficient computation of the Jacobian making it possible to use direct optimization approaches to tackle inverse flow problems. In this work, we demonstrate how auomatic differentation can be used to solve high-dimensional optimization problems related to complex flows. We developed a differentiable solver that implements the immersed boundary method. By differentiating through the solver, we optimize the flow of a couple of canonical problems such as flow in porous media, swimming of active matter, and mixing in low Reynolds numbers.