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.
翻译:物理知情神经网络(PINNs)越来越多地被用来取代/加强解决部分差异方程式(PDEs)的传统数字方法。虽然最先进的PINNs具有许多吸引人的功能,但它们接近PDE系统的具体实现,因此是特定问题。也就是说,每次边界条件(BCs)和域形/大小变化,模型都需要经过再培训。这一限制禁止将PINNs应用于现实的或大规模工程问题,特别是因为与其培训相关的成本和努力相当。我们引入了一个通过深层神经网络解决边界值问题的可转让框架(BVPS),这些网络可以一次培训,并永远用于各种隐蔽域和不列颠区域。我们首先引入基因流动网络(GFNet),这是一个神经网络,可以将BVP的解决方案推导出在任意的BCSonformormation(BVPs)上,然后我们建议采用新的迭代数模型,将GFNet的推算结果汇集到与GFNFNFPs(BP-LA-LA-BS-S-S-S-S-C-LS-S-LOLOLS-S-S-S-S-S-S-S-S-LS-S-S-S-Slental-S-S-S-S-S-S-S-SLS-S-S-S-S-S-S-SDS-S-S-S-S-S-S-S-S-S-S-S-S-SLisal-SLS-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S