End-to-End Mesh Optimization of a Hybrid Deep Learning Black-Box PDE
  Solver
Shaocong Ma James Diffenderfer Bhavya Kailkhura Yi Zhou
Deep learning has been widely applied to solve partial differential equations
(PDEs) in computational fluid dynamics. Recent research proposed a PDE
correction framework that leverages deep learning to correct the solution
obtained by a PDE solver on a coarse mesh. However, end-to-end training of such
a PDE correction model over both solver-dependent parameters such as mesh
parameters and neural network parameters requires the PDE solver to support
automatic differentiation through the iterative numerical process. Such a
feature is not readily available in many existing solvers. In this study, we
explore the feasibility of end-to-end training of a hybrid model with a
black-box PDE solver and a deep learning model for fluid flow prediction.
Specifically, we investigate a hybrid model that integrates a black-box PDE
solver into a differentiable deep graph neural network. To train this model, we
use a zeroth-order gradient estimator to differentiate the PDE solver via
forward propagation. Although experiments show that the proposed approach based
on zeroth-order gradient estimation underperforms the baseline that computes
exact derivatives using automatic differentiation, our proposed method
outperforms the baseline trained with a frozen input mesh to the solver.
Moreover, with a simple warm-start on the neural network parameters, we show
that models trained by these zeroth-order algorithms achieve an accelerated
convergence and improved generalization performance.