Boundary-information-enhanced physics-informed neural network for solving partial differential equations

In the rapidly evolving field of scientific machine learning, physics-informed neural networks (PINNs) have emerged as a powerful paradigm for numerically solving partial differential equations (PDEs). Within this framework, boundary conditions serve as pivotal physical prior knowledge; the approximate enforcement of these physical constraints in regions adjacent to the domain boundaries frequently constitutes a critical bottleneck that limits solution accuracy. To address this challenge, we propose a boundary-information-enhanced physics-informed neural network (
-PINN) by embedding boundary-derivative restraints (bd-restraints) into the loss function to systematically strengthen the impact of boundary physics. These derivative relationships typically lack explicit representation in raw boundary data, thus the bd-restraints are derived from governing physical laws or differential-geometric properties. We present a detailed analysis on how to derive bd-restraints for real-world problems. We validate the effectiveness of the
-PINN through various benchmark problems that span diverse physical regimes including Poisson equation, elasticity, heat conduction problem, and the KdV equation. Extensive numerical comparisons demonstrate that the
-PINN achieves a one to two-order of magnitude reduction in boundary error and consistently improves accuracy across the entire computational domain compared to conventional PINNs. Our findings highlight the strengths of
-PINN: exceptional computational efficiency and superior performance in high-fidelity simulations of complex problems under conditions of limited physical knowledge, thereby providing a reliable and efficient solution strategy for real-world engineering challenges.