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.

翻译：深度学习已被广泛应用于计算流体力学中求解偏微分方程（PDE）。近年研究提出了PDE修正框架，利用深度学习对粗网格上PDE求解器所得解进行修正。然而，对这类PDE修正模型进行端到端训练（同时优化求解器相关参数（如网格参数）和神经网络参数）要求PDE求解器能够支持对迭代数值过程进行自动微分。现有许多求解器并不具备此特性。本研究探索了将黑箱PDE求解器与深度学习模型混合的端到端训练在流体流动预测中的可行性。具体而言，我们研究了一种将黑箱PDE求解器集成到可微深度图神经网络中的混合模型。为训练该模型，我们采用零阶梯度估计器通过前向传播对PDE求解器进行微分。实验表明，尽管基于零阶梯度估计的所提方法性能劣于使用自动微分精确计算梯度的基线方法，但其性能优于使用冻结输入网格训练的基线方法。此外，通过对神经网络参数进行简单的暖启动，我们证明经这些零阶算法训练的模型可实现加速收敛和更优的泛化性能。