Parametric partial differential equations (PDEs) serve as fundamental mathematical tools for modeling complex physical phenomena, yet repeated high-fidelity numerical simulations across parameter spaces remain computationally prohibitive. In this work, we propose a physical law-corrected prior Gaussian process (LC-prior GP) for efficient surrogate modeling of parametric PDEs. The proposed method employs proper orthogonal decomposition (POD) to represent high-dimensional discrete solutions in a low-dimensional modal coefficient space, significantly reducing the computational cost of kernel optimization compared with standard GP approaches in full-order spaces. The governing physical laws are further incorporated to construct a law-corrected prior to overcome the limitation of existing physics-informed GP methods that rely on linear operator invariance, which enables applications to nonlinear and multi-coupled PDE systems without kernel redesign. Furthermore, the radial basis function-finite difference (RBF-FD) method is adopted for generating training data, allowing flexible handling of irregular spatial domains. The resulting differentiation matrices are independent of solution fields, enabling efficient optimization in the physical correction stage without repeated assembly. The proposed framework is validated through extensive numerical experiments, including nonlinear multi-parameter systems and scenarios involving multi-coupled physical variables defined on different two-dimensional irregular domains to highlight the accuracy and efficiency compared with baseline approaches.

翻译：参数化偏微分方程（PDEs）是模拟复杂物理现象的基础数学工具，然而在参数空间内重复进行高保真数值模拟仍面临巨大的计算成本。本文提出一种物理规律修正先验的高斯过程（LC-prior GP），用于参数化PDEs的高效代理建模。该方法采用本征正交分解（POD）将高维离散解表示在低维模态系数空间中，相较于全阶空间中的标准GP方法，显著降低了核优化的计算开销。进一步融入控制物理规律构建规律修正先验，克服了现有物理信息驱动GP方法依赖线性算子不变性的局限性，从而无需重新设计核函数即可应用于非线性及多耦合PDE系统。此外，采用径向基函数-有限差分（RBF-FD）方法生成训练数据，可灵活处理不规则空间区域。所生成的差分矩阵独立于解场，使得物理修正阶段的优化无需重复组装即可高效进行。通过广泛的数值实验验证了所提框架的有效性，包括非线性多参数系统及定义在二维不规则区域上的多耦合物理变量场景，与基线方法相比突出了其精度与效率优势。