Beyond hard constraint: unified knowledge-embedding physics informed neural networks for multi-domain system

PINNs, enabling the assimilation of physical laws and sparse observational data into deep models, have been a powerful method for rapid prediction of complex physical phenomena. However, due to high dimensionality, nonlinearity and multi-scale behaviors in physical systems, the training of PINNs may become ill-posed. One of the primary reasons is the multi-objective loss competition arising from heterogeneity of Partial differential equations (PDEs) and boundary conditions (BCs). For multi-domain coupled systems, coupling relations among domains further complicates the problem. Current methods struggle to achieve satisfactory computation speed and accuracy. Inspired by hard-constraint PINNs, we propose a unified knowledge-embedding PINNs (UKE-PINNs) that defines a generalized graph-structured PINNs framework. UKE-PINNs integrates boundary conditions and implicit coupling relations, defined as essential knowledge, into PINNs ansatz, thereby alleviating the loss competition. Additionally, we introduce a residual learning method to learn the difference from rough knowledge, enabling the embedding of various forms of boundary dynamics and domain distribution patterns. Our method rigorously identifies and eliminates the inherent initialization errors in traditional hard constraint approaches, establishing a novel mechanism that accommodates rough knowledge of varying granularities. We validate UKE-PINNs on benchmark systems with both linear and nonlinear constraints. Tests include flow systems (4-node, 25-node) with linear constraints and standard nonlinear problems (Burgers, Allen-Cahn equations). Compared to pure PINNs and conventional hard-constrained methods, UKE-PINNs demonstrate significant improvements in accuracy and computational efficiency. The method achieves fine-grained accuracy while enabling substantial computational acceleration. Moreover, experiments also confirm that finer-grained rough knowledge embedding significantly enhances UKE-PINNs' performance.