Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions (2205.11912v2)

Abstract: Graph neural network (GNN) is a promising approach to learning and predicting physical phenomena described in boundary value problems, such as partial differential equations (PDEs) with boundary conditions. However, existing models inadequately treat boundary conditions essential for the reliable prediction of such problems. In addition, because of the locally connected nature of GNNs, it is difficult to accurately predict the state after a long time, where interaction between vertices tends to be global. We present our approach termed physics-embedded neural networks that considers boundary conditions and predicts the state after a long time using an implicit method. It is built based on an E(n)-equivariant GNN, resulting in high generalization performance on various shapes. We demonstrate that our model learns flow phenomena in complex shapes and outperforms a well-optimized classical solver and a state-of-the-art machine learning model in speed-accuracy trade-off. Therefore, our model can be a useful standard for realizing reliable, fast, and accurate GNN-based PDE solvers. The code is available at https://github.com/yellowshippo/penn-neurips2022.

• The paper presents a novel framework that integrates physical constraints directly into graph neural networks to rigorously satisfy both Dirichlet and Neumann boundary conditions.
• It leverages an E(n)-equivariant architecture and neural nonlinear solver with global pooling to enhance long-term prediction capabilities in complex PDE scenarios.
• Experimental results show significantly reduced mean squared error in gradient predictions and superior performance in incompressible fluid flow tasks compared to traditional solvers.

Analyzing Physics-Embedded Neural Networks for PDE Solving

The presented paper explores the integration of Graph Neural Networks (GNNs) with physical laws to solve boundary value problems governed by partial differential equations (PDEs). This innovative approach, termed Physics-Embedded Neural Networks (PENNs), offers a novel treatment for PDEs with mixed boundary conditions, specifically by embedding the physics directly into the neural architecture rather than solely within the loss function. This method addresses shortcomings of typical machine learning models in PDE applications, such as inadequate boundary condition treatments and limitations in long-term predictions due to local connectivity.

Key Contributions and Methodology

At the core of PENNs is an $\mathrm{E}(n)$-equivariant GNN framework, which accommodates complex symmetries and local structures inherent in the physical problems described by PDEs. The authors introduce several methodologically significant components:

• Mixed Boundary Condition FulfiLLMent: The model employs a boundary encoder and pseudoinverse decoder to rigorously satisfy Dirichlet and Neumann boundary conditions. This is achieved by embedding these conditions into the model's architecture, ensuring they are inherently respected throughout the computation.
• Neural Nonlinear Solver: By leveraging a neural approach to solving nonlinear PDEs, the model incorporates global pooling operations, enhancing its capacity to predict state evolutions over extended timeframes where global interactions become significant. This is realized through iterative optimization, akin to the Barzilai-Borwein method.
• Compatibility with Physical Symmetries: The model's design relies heavily on leveraging the inherent equivariance properties of physical systems, translating these into computational efficiencies and improved generalization capabilities across various geometries and boundary conditions.

Experimental Validation

The paper showcases PENNs' efficacy through rigorous experiments, including gradient prediction tasks and solving fluid dynamics problems characterized by incompressible flow. Notably, the results demonstrate PENNs' ability to outperform conventional solvers, both in terms of speed and accuracy, particularly when tested across diverse configurations and transformations—a testament to their generalization potential.

1. Gradient Dataset: PENNs significantly reduced mean squared error (MSE) in gradient predictions compared to traditional IsoGCN models, particularly in regions near boundaries, reflecting its robustness in adhering to boundary conditions.
2. Incompressible Fluid Flow: The experiments with fluid dynamics emphasized PENNs' superior performance in speed-accuracy trade-off. Remarkably, the models remained consistent under various domain transformations, a crucial improvement over competing methods like MP-PDE solvers.

Implications and Future Directions

The implications of this work extend beyond enhanced PDE solving; they signal a shift towards incorporating domain knowledge within neural architectures themselves. This has several prospective benefits:

• Enhanced Reliability: By embedding hard-constraint conditions within the model, PENNs ensure physically realistic predictions, making them highly applicable in real-world applications that demand adherence to strict physical laws, such as aerodynamics simulations or climate modeling.
• Scalability and Flexibility: Given their success in generalizing across meshes and boundary conditions, there's potential for PENNs to be adapted for even more complex systems involving multi-physics simulations or real-time adaptive mesh refinements.

Future work could delve into integrating physics-informed training loss functions with PENNs to bridge the advantages of both PINNs and PENNs, particularly for scenarios where governing equations might be partially known. Moreover, optimization in computational efficiency and further explorations into hybrid methods where data-driven and physics-based models coalesce could significantly amplify capabilities in tackling PDEs in broader and more diverse domains.

In conclusion, Physics-Embedded Neural Networks present an advanced methodological leap in solving PDEs with embedded physical constraints, exemplifying adaptability and precision in complex systems, and laying groundwork for future research in AI-driven simulation technologies.