Physics Constraint‐Guided Neural Network for Solving Partial Differential Equations

ABSTRACT

Partial differential equations (PDEs) are key tools for modeling continuous physical processes, but traditional solvers are costly for high‐dimensional problems with complex boundaries. Existing neural network solvers usually add physical constraints only as loss terms, which limits physical constraint embedding learning and can cause local violations. To address this issue, this paper proposes a Physics Constraint‐Guided Network (PCGN) for deep PDE solving. Its main novelty is to introduce physical information at three levels: feature representation, optimization, and output correction. First, governing equations, boundary conditions, and initial conditions are encoded into propagatable constraint features, and neighborhood propagation improves local consistency. Second, adaptive residual balancing adjusts different constraint terms, reducing instability from uneven residual scales. Third, a differentiable constraint projection layer corrects predictions toward feasible solutions. Experiments on Burgers' equation and Darcy flow show that PCGN achieves lower absolute and relative errors than existing deep learning solvers, while improving training stability and physical consistency.