Physics-informed neural networks (PINNs) have gained significant attention as a surrogate modeling strategy for partial differential equations (PDEs), particularly in regimes where labeled data are scarce and physical constraints can be leveraged to regularize the learning process. In practice, however, PINNs are frequently trained using experimental or numerical data that are not fully consistent with the governing equations due to measurement noise, discretization errors, or modeling assumptions. The implications of such data-to-PDE inconsistencies on the accuracy and convergence of PINNs remain insufficiently understood. In this work, we systematically analyze how data inconsistency fundamentally limits the attainable accuracy of PINNs. We introduce the concept of a consistency barrier, defined as an intrinsic lower bound on the error that arises from mismatches between the fidelity of the data and the exact enforcement of the PDE residual. To isolate and quantify this effect, we consider the 1D viscous Burgers equation with a manufactured analytical solution, which enables full control over data fidelity and residual errors. PINNs are trained using datasets of progressively increasing numerical accuracy, as well as perfectly consistent analytical data. Results show that while the inclusion of the PDE residual allows PINNs to partially mitigate low-fidelity data and recover the dominant physical structure, the training process ultimately saturates at an error level dictated by the data inconsistency. When high-fidelity numerical data are employed, PINN solutions become indistinguishable from those trained on analytical data, indicating that the consistency barrier is effectively removed. These findings clarify the interplay between data quality and physics enforcement in PINNs providing practical guidance for the construction and interpretation of physics-informed surrogate models.

翻译：物理信息神经网络（PINNs）作为一种偏微分方程（PDE）的代理建模策略，在标签数据稀缺且可利用物理约束正则化学习过程的领域中受到广泛关注。然而在实际应用中，PINNs 常使用与支配方程不完全一致的实验或数值数据进行训练，这种不一致源于测量噪声、离散化误差或建模假设。此类数据与PDE之间的不一致性对PINNs精度与收敛性的影响尚未得到充分理解。本文系统分析了数据不一致性如何从根本上限制PINNs可达到的精度。我们提出了“一致性壁垒”的概念，其定义为数据保真度与PDE残差精确执行之间的失配所导致的固有误差下界。为分离并量化该效应，我们以具有解析解的1D粘性Burgers方程为研究对象，从而实现对数据保真度与残差误差的完全控制。通过使用数值精度逐步提升的数据集以及完全一致的解析数据对PINNs进行训练，结果表明：虽然引入PDE残差可使PINNs部分抵消低保真度数据并恢复主导物理结构，但训练过程最终会达到由数据不一致性决定的误差饱和水平。当采用高保真度数值数据时，PINN解与基于解析数据训练所得解无法区分，表明一致性壁垒被有效消除。这些发现阐明了PINNs中数据质量与物理约束执行之间的相互作用，为构建和解释物理信息代理模型提供了实践指导。