Linear partial differential equations (PDEs) are an important, widely applied class of mechanistic models, describing physical processes such as heat transfer, electromagnetism, and wave propagation. In practice, specialized numerical methods based on discretization are used to solve PDEs. They generally use an estimate of the unknown model parameters and, if available, physical measurements for initialization. Such solvers are often embedded into larger scientific models with a downstream application and thus error quantification plays a key role. However, by ignoring parameter and measurement uncertainty, classical PDE solvers may fail to produce consistent estimates of their inherent approximation error. In this work, we approach this problem in a principled fashion by interpreting solving linear PDEs as physics-informed Gaussian process (GP) regression. Our framework is based on a key generalization of the Gaussian process inference theorem to observations made via an arbitrary bounded linear operator. Crucially, this probabilistic viewpoint allows to (1) quantify the inherent discretization error; (2) propagate uncertainty about the model parameters to the solution; and (3) condition on noisy measurements. Demonstrating the strength of this formulation, we prove that it strictly generalizes methods of weighted residuals, a central class of PDE solvers including collocation, finite volume, pseudospectral, and (generalized) Galerkin methods such as finite element and spectral methods. This class can thus be directly equipped with a structured error estimate. In summary, our results enable the seamless integration of mechanistic models as modular building blocks into probabilistic models by blurring the boundaries between numerical analysis and Bayesian inference.

翻译：线性偏微分方程（PDE）是一类重要且广泛应用机理模型，用于描述热传导、电磁学和波传播等物理过程。实践中，基于离散化的专门数值方法被用于求解偏微分方程。这些方法通常使用未知模型参数的估计值以及（若可用）物理测量值进行初始化。此类求解器常作为模块嵌入具有下游应用的大型科学模型中，因此误差量化起着关键作用。然而，经典偏微分方程求解器忽略参数和测量不确定性，可能无法一致地估计其固有近似误差。本文通过将线性偏微分方程求解解释为物理信息高斯过程（GP）回归，以原理性方式解决此问题。我们的框架基于高斯过程推断定理的关键推广，使其适用于通过任意有界线性算子进行观测的情形。至关重要的是，这种概率视角允许：(1) 量化固有离散化误差；(2) 将模型参数的不确定性传播至解；(3) 对含噪测量进行条件化处理。为展示该形式的优势，我们证明它严格推广了加权残量法——这类核心偏微分方程求解方法包括配置法、有限体积法、伪谱法以及（广义）伽辽金法（如有限元法和谱方法）。因此，此类方法可直接配备结构化误差估计。总之，我们的结果通过模糊数值分析与贝叶斯推断之间的界限，实现了将机理模型作为模块化构建单元无缝集成到概率模型中。