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

Area of Science:

• Computational physics
• Deep learning for scientific computing

Background:

• Physics-Informed Neural Networks (PINNs) integrate physical laws and data for complex phenomena prediction.
• PINN training is often ill-posed due to high dimensionality, nonlinearity, and multi-scale behaviors.
• Loss competition from heterogeneous PDEs and boundary conditions complicates multi-domain systems.

Purpose of the Study:

• To develop a novel PINN framework, UKE-PINNs, to address training challenges in complex physical systems.
• To alleviate multi-objective loss competition by integrating essential knowledge into the PINN ansatz.
• To enhance computational speed and prediction accuracy compared to existing methods.

Main Methods:

• Proposed a generalized graph-structured PINN framework: Unified Knowledge-Embedding PINNs (UKE-PINNs).
• Integrated boundary conditions and implicit coupling relations as essential knowledge into the PINN ansatz.
• Introduced a residual learning method for embedding various boundary dynamics and domain patterns, accommodating rough knowledge.

Main Results:

• UKE-PINNs demonstrated significant improvements in accuracy and computational efficiency on benchmark systems.
• Validated on flow systems (4-node, 25-node) and nonlinear problems (Burgers, Allen-Cahn equations).
• Achieved fine-grained accuracy and substantial computational acceleration compared to pure PINNs and conventional hard-constrained methods.

Conclusions:

• UKE-PINNs effectively alleviate loss competition and improve training stability for complex physical systems.
• The proposed method offers a novel mechanism for accommodating rough knowledge of varying granularities.
• Finer-grained knowledge embedding significantly enhances UKE-PINNs performance, offering a promising direction for scientific machine learning.