Mosaic Flows：一个用于求解未知领域偏微分方程的可转移深度学习框架
Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains
摘要：
物理信息神经网络（PINN）越来越多地被用来取代/增强求解偏微分方程（PDE）的传统数值方法。虽然国家的最先进的PINN有许多有吸引力的功能，他们近似的PDE系统的具体实现，因此是特定的问题。也就是说，每次边界条件（BC）和域形状/大小改变时，模型都需要重新训练。这种限制禁止PINN的应用，以现实或大规模的工程问题，特别是因为与他们的培训成本和努力是相当大的。我们引入了一个通过深度神经网络解决边值问题（BVP）的可转移框架，该框架可以训练一次，并永远用于各种看不见的域和BC。我们首先介绍了基因组流网络（GFNet），这是一个神经网络，它可以在一个叫做基因组的正方形区域上，通过任意的边界条件来推断边值问题的解。然后，我们提出了马赛克流（MF）预测器，这是一种新的迭代算法，它可以在具有不可见大小/形状和BC的大型域上组装GFNet对BVP的推断，同时保持解决方案的空间规则性。我们证明了我们的框架可以估计拉普拉斯和Navier-Stokes方程的解，这些解分别是训练域的1200倍和12倍。由于我们的框架消除了重新训练未知域和BC模型的需要，因此与最先进的技术相比，它表现出高达3个数量级的加速。
英文摘要：
Physics-informed neural networks (PINNs) are increasingly employed to replace/augment traditional numerical methods in solving partial differential equations (PDEs). While state-of-the-art PINNs have many attractive features, they approximate a specific realization of a PDE system and hence are problem-specific. That is, the model needs to be re-trained each time the boundary conditions (BCs) and domain shape/size change. This limitation prohibits the application of PINNs to realistic or large-scale engineering problems especially since the costs and efforts associated with their training are considerable. We introduce a transferable framework for solving boundary value problems (BVPs) via deep neural networks which can be trained once and used forever for various unseen domains and BCs. We first introduce genomic flow network(GFNet), a neural network that can infer the solution of a BVP across arbitrary BCson a small square domain called genome. Then, we proposed mosaic flow(MF) predictor, a novel iterative algorithm that assembles the GFNet's inferences for BVPs on large domains with unseen sizes/shapes and BCs while preserving the spatial regularity of the solution. We demonstrate that our framework can estimate the solution of Laplace and Navier-Stokes equations in domains of unseen shapes and BCs that are, respectively, 1200 and 12 times larger than the training domains. Since our framework eliminates the need to re-train models for unseen domains and BCs, it demonstrates up to 3 orders-of-magnitude speedups compared to the state-of-the-art.
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