PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain (2004.13145v2)

Abstract: Recently, the advent of deep learning has spurred interest in the development of physics-informed neural networks (PINN) for efficiently solving partial differential equations (PDEs), particularly in a parametric setting. Among all different classes of deep neural networks, the convolutional neural network (CNN) has attracted increasing attention in the scientific machine learning community, since the parameter-sharing feature in CNN enables efficient learning for problems with large-scale spatiotemporal fields. However, one of the biggest challenges is that CNN only can handle regular geometries with image-like format (i.e., rectangular domains with uniform grids). In this paper, we propose a novel physics-constrained CNN learning architecture, aiming to learn solutions of parametric PDEs on irregular domains without any labeled data. In order to leverage powerful classic CNN backbones, elliptic coordinate mapping is introduced to enable coordinate transforms between the irregular physical domain and regular reference domain. The proposed method has been assessed by solving a number of PDEs on irregular domains, including heat equations and steady Navier-Stokes equations with parameterized boundary conditions and varying geometries. Moreover, the proposed method has also been compared against the state-of-the-art PINN with fully-connected neural network (FC-NN) formulation. The numerical results demonstrate the effectiveness of the proposed approach and exhibit notable superiority over the FC-NN based PINN in terms of efficiency and accuracy.

An Overview of PhyGeoNet: A Physics-Informed CNN for PDEs on Irregular Domains

The paper "PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain" by Han Gao, Luning Sun, and Jian-Xun Wang presents a novel approach in scientific machine learning for solving parameterized steady-state partial differential equations (PDEs) on irregular domains without labeled data. This method leverages the robust capabilities of convolutional neural networks (CNNs) to address the limitations of existing physics-informed neural networks (PINNs), particularly those based on fully-connected neural networks (FC-NNs).

Key Contributions

1. Geometry-Adaptive CNN Architecture: The core innovation lies in adapting the CNN architecture to handle irregular geometries inherent in scientific computations, which traditional CNNs find challenging due to their image-based nature requiring rectangular domains with uniform grids. By employing elliptic coordinate mapping, the proposed method allows transformations between irregular physical domains and regular reference domains, effectively bridging this gap.
2. Physics-Constrained Learning: The proposed PhyGeoNet integrates the elliptic coordinate transformations into the network, enabling physics-constrained learning on the reference domain using a physics-based loss function derived from the reformulated PDEs. This transformation facilitates the application of CNNs to scientific problems characterized by irregular geometries and non-uniform meshes.
3. Hard BC Enforcement: A significant feature of PhyGeoNet is its capability to strictly enforce boundary conditions through hard padding techniques, both for Dirichlet and Neumann types. This ensures that the BCs are exactly satisfied, which is crucial for obtaining accurate solutions in the absence of labeled data.
4. Efficiency and Scalability: PhyGeoNet demonstrates substantial improvements in computational efficiency and scalability compared to existing PINN models using FC-NN formulations. The paper reports significant reductions in training time alongside improved accuracy, highlighting the utility of CNNs for large-scale PDE problems.

Numerical Results and Implications

The paper provides extensive numerical validation across different PDE systems, including heat equations and Navier-Stokes equations, under varying geometric configurations and parameter settings. These tests verify the effectiveness of PhyGeoNet in dealing with complex geometries, showing superior performance over traditional methods.

The implications of this research are far-reaching, particularly for applications requiring rapid, accurate solutions of PDEs across variable geometric domains. The architecture's inherent scalability and efficiency hold promise for real-time applications in optimization, uncertainty quantification, and fluid dynamics, among others.

Future Directions

While the paper demonstrates significant progress, several avenues for future work are identified, such as extending PhyGeoNet to handle time-dependent PDEs and more complex, multi-physics problems. Additionally, this approach opens possibilities for integrating multi-fidelity data to further refine predictions and enhance model robustness.

In conclusion, PhyGeoNet represents a meaningful step forward in applying CNNs to physics-informed scenarios, overcoming challenges associated with irregular domain geometries inherent in scientific computations. Its potential to serve as a surrogate model in real-time applications positions it as a valuable tool in machine learning and computational physics, paving the way for advancements in scientific computing.