Mosaic flows: A transferable deep learning framework for solving PDEs on unseen domains

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 BCs on a small square domain called genome. Then, we propose 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. (c) 2021 Elsevier B.V. All rights reserved.

Automated summary

Traditional numerical methods replacing problem-specific PINNs in solving PDEs pose limitations. A framework is proposed using deep neural networks trained once for various unknown domains and BCs. The new GFNet and MF predictor can estimate solutions in unknown domains, demonstrating up to 3 orders-of-magnitude speedups compared to state-of-the-art.