Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems (2107.12146v1)

Abstract: Despite the great promise of the physics-informed neural networks (PINNs) in solving forward and inverse problems, several technical challenges are present as roadblocks for more complex and realistic applications. First, most existing PINNs are based on point-wise formulation with fully-connected networks to learn continuous functions, which suffer from poor scalability and hard boundary enforcement. Second, the infinite search space over-complicates the non-convex optimization for network training. Third, although the convolutional neural network (CNN)-based discrete learning can significantly improve training efficiency, CNNs struggle to handle irregular geometries with unstructured meshes. To properly address these challenges, we present a novel discrete PINN framework based on graph convolutional network (GCN) and variational structure of PDE to solve forward and inverse partial differential equations (PDEs) in a unified manner. The use of a piecewise polynomial basis can reduce the dimension of search space and facilitate training and convergence. Without the need of tuning penalty parameters in classic PINNs, the proposed method can strictly impose boundary conditions and assimilate sparse data in both forward and inverse settings. The flexibility of GCNs is leveraged for irregular geometries with unstructured meshes. The effectiveness and merit of the proposed method are demonstrated over a variety of forward and inverse computational mechanics problems governed by both linear and nonlinear PDEs.

• The paper introduces a unified framework that integrates graph convolutional networks with Galerkin methods to solve both forward and inverse PDE problems on unstructured meshes.
• It employs a Galerkin formulation to reduce dimensionality and eliminate dense collocation, enhancing performance on irregular geometries.
• Numerical tests on Poisson, elasticity, and Navier-Stokes equations validate the framework's efficiency and accuracy against traditional solvers.

Insightful Overview of "Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems"

The paper introduces a novel computational paradigm utilizing Physics-Informed Graph Neural Galerkin Networks (PI-GNNs) for addressing forward and inverse problems governed by Partial Differential Equations (PDEs). This methodological innovation is rooted in combining the strengths of graph convolutional networks (GCNs) with Galerkin methods traditionally used in finite element analysis, offering a robust framework to handle complex PDEs on unstructured meshes and irregular geometries efficiently.

Key Contributions

The paper presents several advancements in the field of physics-informed neural networks (PINNs):

1. Graph Neural Network Architecture for Unstructured Data: The authors propose using GCNs to facilitate operations on non-Euclidean data (i.e., graphs), thereby extending traditional PINNs to efficiently manage unstructured meshes and complex domains. This avoids the need for data rasterization or coordinate transformations, which are typically required in convolutional neural networks (CNNs) handling irregular meshes.
2. Galerkin Formulation for PDE Residuals: By leveraging the Galerkin method, which uses a variational (weak) formulation for computing PDE residuals, this work eliminates the need for dense collocation points. The method reduces the dimensions of the solution search space by adopting piecewise polynomial basis functions, improving the feasibility and convergence of the physics-informed training process.
3. Unified Forward and Inverse Problem Approach: The proposed framework simultaneously solves for unknown field variables and parameters in PDEs by assimilating boundary and observational data directly in the network’s formulation. This circumvents the need for penalty-weighted objectives commonly used in traditional PINNs, thereby simplifying hyperparameter tuning and potentially enhancing solution stability and accuracy.

Numerical Demonstrations

The effectiveness of the PI-GNN framework is validated through multiple test cases highlighting the framework’s capacity to solve complex linear and nonlinear PDEs. Notable experiments include:

• Poisson's equation on varied geometries (simple and complex), where observed errors are minimal compared to analytical solutions or finite element method (FEM) benchmarks.
• Linear elasticity equations in both forward problem configurations and inverse problem settings, wherein unknown material properties (Lamé parameters) are accurately inferred alongside displacement fields.
• Incompressible Navier-Stokes equations in classic configurations like lid-driven cavity flow, which underscore the capabilities of PI-GNNs to manage the nonlinearity and high dimensionality of such systems with competitive accuracy against traditional solvers.

Implications and Future Directions

The developed PI-GNN framework underscores substantial implications for computational mechanics and physics-based learning:

• Practical Efficiency: By reducing the computational complexity associated with problem-specific setups and integration of boundary conditions, the framework shows promise for real-world applications necessitating rapid and reliable PDE solutions under constraints.
• Versatility and Scalability: The use of GCNs provides a scalable solution for complex problem geometries beyond typical Cartesian grids, advancing the potential applications of PINNs in scientific computing domains.
• Potential Theoretical Extensions: Future work can build on this framework to explore time-dependent PDEs, domain-adaptive polynomial bases, and integration with other machine learning structures, potentially bringing further advancements in efficiency and generalization across diverse computational tasks in engineering and physics.

In conclusion, the paper provides a comprehensive and technically sound approach to solving PDE-governed problems using graph-based neural network architectures, offering significant contributions to the domain of physics-informed machine learning with potential far-reaching impacts on both industrial applications and foundational computational theories.