Recently, deep learning (DL) emerges as a promising approach for solving forward and inverse problems governed by partial differential equations (PDEs). The underlying mechanism is to train a DL model (i.e., neural networks) for approximating the solutions by minimizing the violation of physical laws defined by PDE residuals. Although great potential has been demonstrated in canonical problems, grant challenges are still present for existing physics-informed deep learning (PIDL) frameworks when it comes to complex systems: (1) poor scalability with respect to the problem size, (2) multi-loss incompatibility introduced by considering boundary enforcement, data assimilation, and physics constraints simultaneously, (3) difficulties in dealing with irregular geometries using classic convolutional neural network (CNN). 

In this work, we introduce an innovative and scalable PIDL framework based on graph convolutional network, aiming to solve the PDE-governed forward and inverse problems in a unified manner. In particular, a highly-scalable discrete learning architecture is constructed, which is compatible with most mainstream discretization methods, i.e., finite difference (FD), finite volume (FV), and finite element (FE) schemes. Moreover, a unified loss function is designed to impose boundary conditions and assimilate labeled data in a hard manner, which avoids introducing penalty coefficients used in most existing PIDL works and can significantly improve the convergence/robustness for both forward and inverse modeling. Furthermore, the construction of graph convolutional networks enable us to deal with complex computational domains with three-dimensional (3-D) irregular geometries. 

To demonstrate the effectiveness and merit of the proposed framework, a series of comprehensive test cases are designed, involving a number of nonlinear PDE systems that commonly used to describe physical phenomena in heat transfer, solid, and fluid mechanics. Full-order numerical simulations based on FV/FE methods are conducted as benchmark solutions to validate the learning accuracy and efficiency. Moreover, synthetic labeled data are also generated from FV/FE-simulated results with added artificial noise. The numerical results show that our proposed method can not only learn the forward solutions in 2-D and 3-D irregular domains but also be able to infer the unknowns, e.g., material properties, initial/boundary conditions, and operating parameters. Moreover, the proposed PIDL framework has been demonstrated to outperform existing approaches, e.g., classic physics-informed neural networks (PINN) with fully-connected formulation, with respect to the metrics of accuracy and efficiency. In general, our proposed method shows great promise in modeling complex, nonlinear physical phenomena with 3-D irregular geometries, where underlying governing PDEs and modeling conditions are known or partially known, but sparse (possibly noisy) observation data are available. 


Physics-Informed Graph Convolutional Neural Networks: A Unified Framework for Solving PDE-Governed Forward and Inverse Problems

Division / Sponsors: 
Aerospace, Bioengineering, Applied Mechanics, Design Engineering Division, Advanced Energy Systems, Manufacturing Engineering Division, Noise Control and Acoustics, Nondestructive Evaluation, Diagnosis, and Prognosis (NDPD), Nuclear Engineering Division, Process Industries Division, Fluids Engineering, Heat Transfer, Materials, Microelectromechanical Systems, Safety Engineering and Risk Analysis, Computers & Information in Engineering


Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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