We present a novel physics-informed deep learning framework for solving steady-state incompressible flow on multiple sets of irregular geometries by incorporating two main elements: using a point-cloud based neural network to capture geometric features of computational domains, and using the mean squared residuals of the governing partial differential equations, boundary conditions, and sparse observations as the loss function of the network to capture the physics. While the solution of the continuity and Navier-Stokes equations is a function of the geometry of the computational domain, current versions of physics-informed neural networks have no mechanism to express this functionally in their outputs, and thus are restricted to obtain the solutions only for one computational domain with each training procedure. Using the proposed framework, three new facilities become available. First, the governing equations are solvable on a set of computational domains containing irregular geometries with high variations with respect to each other but requiring training only once. Second, after training the introduced framework on the set, it is now able to predict the solutions on domains with unseen geometries from seen and unseen categories as well. The former and the latter both lead to savings in computational costs. Finally, all the advantages of the point-cloud based neural network for irregular geometries, already used for supervised learning, are transferred to the proposed physics-informed framework. The effectiveness of our framework is shown through the method of manufactured solutions and thermally-driven flow for forward and inverse problems.

我们提出了一种新颖的基于物理的深度学习框架，通过结合两个主要元素来解决多组不规则几何上的稳态不可压缩流：使用基于点云的神经网络来捕获计算域的几何特征，以及使用均方控制偏微分方程、边界条件和稀疏观测的残差作为网络的损失函数来捕获物理。虽然连续性和 Navier-Stokes 方程的解是计算域几何的函数，但当前版本的物理信息神经网络没有机制在其输出中以函数方式表达这一点，因此仅限于获得解对于每个训练过程的一个计算域。使用建议的框架，三个新设施可用。首先，控制方程可以在一组计算域上求解，这些域包含不规则的几何形状，彼此之间有很大的变化，但只需要训练一次。其次，在对集合上引入的框架进行训练之后，它现在能够从可见和不可见类别中预测具有不可见几何形状的域上的解决方案。前者和后者都可以节省计算成本。最后，已经用于监督学习的基于点云的不规则几何神经网络的所有优势都转移到了所提出的物理信息框架中。我们的框架的有效性通过制造解决方案的方法和正向和逆向问题的热驱动流动来证明。控制方程可以在一组计算域上求解，这些域包含不规则的几何形状，彼此之间有很大的变化，但只需要训练一次。其次，在对集合上引入的框架进行训练之后，它现在能够从可见和不可见类别中预测具有不可见几何形状的域上的解决方案。前者和后者都可以节省计算成本。最后，已经用于监督学习的基于点云的不规则几何神经网络的所有优势都转移到了所提出的物理信息框架中。我们的框架的有效性通过制造解决方案的方法和正向和逆向问题的热驱动流动来证明。控制方程可以在一组计算域上求解，这些域包含不规则的几何形状，彼此之间有很大的变化，但只需要训练一次。其次，在对集合上引入的框架进行训练之后，它现在能够从可见和不可见类别中预测具有不可见几何形状的域上的解决方案。前者和后者都可以节省计算成本。最后，已经用于监督学习的基于点云的不规则几何神经网络的所有优势都转移到了所提出的物理信息框架中。我们的框架的有效性通过制造解决方案的方法和正向和逆向问题的热驱动流动来证明。