Enhancing Physics-Informed Neural Networks for Solving Partial Differential Equations Using Symmetry Group-Based Domain Decomposition

Core Concepts

A symmetry group-based domain decomposition strategy is proposed to enhance physics-informed neural networks for solving forward and inverse problems of partial differential equations possessing a Lie symmetry group.

Abstract

The paper presents a symmetry group-based domain decomposition (sdPINN) strategy to enhance physics-informed neural networks (PINNs) for solving forward and inverse problems of partial differential equations (PDEs) that possess a Lie symmetry group. For the forward problem: The symmetry group is used to generate dividing lines that separate the whole domain into non-overlapping sub-domains. In each sub-domain, a PINN or a symmetry-enhanced PINN (sPINN) is used to learn the solution, and the overall solution is obtained by stitching the sub-domain solutions. The interfaces between sub-domains have exact labeled data, allowing for independent training in each sub-domain. For the inverse problem: The symmetry group is used to generate labeled data in the interior domain from the initial and boundary conditions. The inverse problem is solved by training the neural networks only in a sub-domain, not the whole domain. The proposed sdPINN method can predict high-accuracy solutions of PDEs that the vanilla PINN and the extended PINN (XPINN) fail to achieve in the whole domain. Numerical results for the Korteweg-de Vries equation and the nonlinear viscous fluid equation demonstrate significant improvements in accuracy compared to the PINN and XPINN methods.

Stats

The L2 relative error of the predicted solution by the PINN method for the Korteweg-de Vries equation with b = 20 is around 0.1. The L2 relative error of the predicted solution by the XPINN method for the nonlinear viscous fluid equation is around 0.36. The L2 relative error of the predicted solution by the sdPINN method for the Korteweg-de Vries equation with b = 20 is around 0.0067. The L2 relative error of the predicted solution by the sdPINN-isc method for the nonlinear viscous fluid equation is around 0.00031.

Quotes

"The key point of the introduced method is the symmetry group which endows each sub-domain the exact initial and boundary conditions and thus makes the training in each sub-domain completely independent." "The distinct merit of the proposed method is the interfaces of two adjacent sub-domains which have the exact labeled data, and thus in each sub-domain one can perform completely different neural networks including the choice of different network architecture, number of training points and activation functions, optimization algorithm, etc."

Key Insights Distilled From

Symmetry group based domain decomposition to enhance physics-informed neural networks for solving partial differential equations

by Ye Liu,Jie-Y... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.18538.pdf

Deeper Inquiries

How can the proposed sdPINN method be extended to solve PDEs that do not possess a clear Lie symmetry group

The proposed sdPINN method can be extended to solve PDEs that do not possess a clear Lie symmetry group by exploring alternative symmetry properties or transformations that may exist in the equations. While Lie symmetry groups are a powerful tool for reducing the complexity of PDEs and providing insights into their solutions, not all PDEs exhibit clear Lie symmetries. In such cases, other types of symmetries or invariant properties can be leveraged to decompose the domain and enhance the training of neural networks. One approach could be to investigate other types of symmetries, such as scale invariance, rotational invariance, or reflection symmetry, that may be present in the PDEs. By identifying these alternative symmetries, the domain can still be decomposed into sub-domains based on the invariant properties, allowing for independent training in each sub-domain. This approach would require a thorough analysis of the specific PDEs under consideration to uncover any hidden symmetries that could be utilized in the domain decomposition strategy. Additionally, techniques from group theory and symmetry analysis can be applied to identify and exploit any underlying symmetries that may not be immediately apparent. By extending the concept of symmetry group-based domain decomposition to encompass a broader range of symmetry properties, the sdPINN method can be adapted to effectively solve a wider variety of PDEs, even those without a clear Lie symmetry group.

What are the potential limitations or drawbacks of the symmetry group-based domain decomposition approach, and how can they be addressed

The symmetry group-based domain decomposition approach, while effective in enhancing the training of physics-informed neural networks for solving PDEs, may have some potential limitations or drawbacks that need to be addressed: Dependency on Symmetry Groups: One limitation is the reliance on the existence of clear symmetry groups in the PDEs. Not all PDEs exhibit obvious symmetry properties, which can restrict the applicability of the domain decomposition method. To address this limitation, alternative symmetry properties or transformations need to be explored, as mentioned in the previous response. Complexity of Symmetry Analysis: Identifying symmetry groups and invariant properties in PDEs can be a complex and computationally intensive task, especially for high-dimensional or nonlinear equations. This complexity can hinder the practical implementation of the symmetry group-based approach. One way to address this is to develop automated algorithms or tools for symmetry analysis to streamline the process. Interface Handling: Ensuring continuity and smooth transitions at the interfaces between sub-domains is crucial for the success of domain decomposition methods. Inaccuracies or discontinuities at the interfaces can lead to errors in the overall solution. Techniques such as incorporating interface terms in the loss function or refining the interface handling mechanisms can help address this issue. Scalability: The scalability of the domain decomposition approach to large-scale or complex PDEs needs to be considered. As the number of sub-domains increases, the computational cost and memory requirements may become prohibitive. Implementing efficient parallelization strategies and optimizing the training process can help improve scalability. By addressing these limitations through advanced symmetry analysis techniques, improved interface handling methods, and enhanced scalability measures, the symmetry group-based domain decomposition approach can be further optimized for solving a wide range of PDEs.

How can the insights from this work on leveraging symmetry properties be applied to other machine learning tasks beyond solving PDEs

The insights gained from leveraging symmetry properties in solving PDEs using the sdPINN method can be applied to other machine learning tasks beyond PDEs in various ways: Feature Engineering: Symmetry properties can be utilized in feature engineering for machine learning tasks. By identifying and incorporating relevant symmetries in the data, the model can capture important patterns and relationships, leading to improved performance and generalization. Data Augmentation: Symmetry-based data augmentation techniques can be employed to generate additional training data by exploiting symmetries in the dataset. This can help enhance the robustness and accuracy of machine learning models, especially in scenarios with limited data availability. Regularization: Symmetry constraints can be integrated into regularization techniques to enforce desired properties in the model. By incorporating symmetry regularization, the model can learn more stable and interpretable representations, reducing overfitting and improving generalization. Transfer Learning: Insights from symmetry analysis can inform transfer learning strategies by identifying common symmetries across different tasks or domains. Leveraging shared symmetries can facilitate knowledge transfer and adaptation of pre-trained models to new tasks more effectively. Model Interpretability: Symmetry-based approaches can enhance the interpretability of machine learning models by providing insights into the underlying structure of the data. Understanding and leveraging symmetries can lead to more transparent and explainable models. By applying the principles of symmetry analysis and domain decomposition from solving PDEs to other machine learning tasks, researchers can develop more robust, efficient, and interpretable models across a wide range of applications.