ABSTRACT 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.
Li et al. (2026) studied this question.