Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains

TL;DR

This paper presents a transferable deep learning framework that can solve PDEs on unseen domains and boundary conditions without retraining, significantly reducing computational costs and enabling large-scale engineering applications.

Contribution

The introduction of GFNet and mosaic flow predictor enables solving PDEs on arbitrary domains and BCs with a single trained model, a novel approach in physics-informed neural networks.

Findings

Successfully applied to Laplace and Navier-Stokes equations.

Achieved up to 3 orders-of-magnitude speedup over existing methods.

Handled domains 1200 and 12 times larger than training domains.

Abstract

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.