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Exploring prototype-guided strategy for domain decomposition in physics-informed neural network
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AI
DOI：10.1007/s11071-025-10871-4.png)
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Abstract
En 中文
We propose an adaptive domain decomposition framework called Prototype-guided Physics-Informed Neural Network (Pro-PINN) for solving partial differential equations (PDEs). Popular domain decomposition methods (DDMs), such as XPINN and APINN, rely on hand-designed components that encode prior knowledge of underlying PDEs. In contrast, we achieve prior-knowledge-free domain decomposition based on prototype similarity by incorporating the concept of prototype learning into PINN. Pro-PINN is an encoder-decoder architecture, and its training process consists of two stages: Domain prototypes generation and Prototype-based learning. Specifically, in stage I, Pro-PINN employs a shared encoder which captures spatial differences in the entire domain to generate representative domain prototypes. In stage II, the domain prototypes guide the decoder (sub-nets) to perform prototype-based learning to obtain the approximate solution. The process of prototype-based learning is gradually integrated into stage II through the incorporation of Domain Prototype Alignment Method (DPAM), which we have discovered to greatly enhance training stability. Moreover, to enable sub-nets to focus on domain-specific knowledge and fully utilize the data from other sub-nets, we propose Orthogonal Constraint of Domain Prototypes (OCDP) to increase the discrepancy among domain prototypes and Entropy-based Weight Balancing (EWB) to maintain the interconnection among sub-nets. Comprehensive numerical experiments on six challenging PDEs in various dimensions demonstrate the advantages of our Pro-PINN in terms of approximation accuracy and generalization ability.
Keywords:
Physics-informed neural networks
Domain decomposition
Prototype learning
Forward modeling
PDEs