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Recently, the advent of deep learning has spurred interest in the developmentof physics-informed neural networks (PINN) for efficiently solving partialdifferential equations (PDEs), particularly in a parametric setting. Among alldifferent classes of deep neural networks, the convolutional neural network(CNN) has attracted increasing attention in the scientific machine learningcommunity, since the parameter-sharing feature in CNN enables efficientlearning for problems with large-scale spatiotemporal fields. However, one ofthe biggest challenges is that CNN only can handle regular geometries withimage-like format (i.e., rectangular domains with uniform grids). In thispaper, we propose a novel physics-constrained CNN learning architecture, aimingto learn solutions of parametric PDEs on irregular domains without any labeleddata. In order to leverage powerful classic CNN backbones, elliptic coordinatemapping is introduced to enable coordinate transforms between the irregularphysical domain and regular reference domain. The proposed method has beenassessed by solving a number of PDEs on irregular domains, including heatequations and steady Navier-Stokes equations with parameterized boundaryconditions and varying geometries. Moreover, the proposed method has also beencompared against the state-of-the-art PINN with fully-connected neural network(FC-NN) formulation. The numerical results demonstrate the effectiveness of theproposed approach and exhibit notable superiority over the FC-NN based PINN interms of efficiency and accuracy.

PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain