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（PDEs）是一类重要且广泛应用的理论模型，描述了传热、电磁学和波传播等物理过程。实践中，基于离散化的专用数值方法被用于求解偏微分方程。这些方法通常利用未知模型参数的估计值，并在可用时使用物理测量值进行初始化。此类求解器常被嵌入到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，由于忽略参数和测量不确定性，经典偏微分方程求解器可能无法对其固有近似误差产生一致的估计。在本工作中，我们通过将线性偏微分方程的求解解释为物理信息高斯过程回归，以一种严谨的方式处理该问题。我们的框架基于将高斯过程推断定理推广到通过任意有界线性算子进行的观测这一关键推广。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到解中；以及（3）基于含噪测量进行条件化。为证明该公式的优越性，我们严格证明它推广了加权残量法——这一偏微分方程求解器的核心类别，包括配点法、有限体积法、伪谱法以及（广义）伽辽金方法（如有限元法和谱方法）。因此，该类方法可直接配备结构化误差估计。综上所述，我们的结果通过模糊数值分析与贝叶斯推断之间的界限，实现了将理论模型作为模块化构建块无缝集成到概率模型中。