End-to-End Mesh Optimization of a Hybrid Deep Learning Black-Box PDE Solver
<|reference_start|>End-to-End Mesh Optimization of a Hybrid Deep Learning Black-Box PDE Solver: 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.<|reference_end|>
@article{ma2024end-to-end,
    title={End-to-End Mesh Optimization of a Hybrid Deep Learning Black-Box PDE
  Solver},
    author={Shaocong Ma, James Diffenderfer, Bhavya Kailkhura, Yi Zhou},
    journal={arXiv preprint arXiv:2404.11766},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.11766},
    primaryClass={cs.LG cs.NA math.NA math.OC}
}