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 such that 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 a widely-applied theorem for conditioning GPs on direct measurements 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：线性偏微分方程是一类重要且广泛应用的现象学模型，描述了传热、电磁学和波传播等物理过程。实际应用中，基于离散化的专门数值方法被用于求解偏微分方程。这些方法通常利用未知模型参数的估计值，并在有物理测量数据时将其用于初值设定。此类求解器常被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由于忽略参数和测量不确定性，可能无法对其固有近似误差给出一致的估计。在本工作中，我们通过将求解线性偏微分方程解释为物理信息高斯过程回归，以系统化的方式处理这一问题。我们的框架基于一个关键推广：将广泛应用的直接测量条件下高斯过程条件定理，扩展至通过任意有界线性算子进行观测的情形。关键在于，这一概率视角使我们能够：（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）以含噪测量数据为条件。为展示该公式的效力，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器包括配置法、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。因此这类方法可直接配备结构化误差估计。总之，我们的研究结果通过模糊数值分析与贝叶斯推断的界限，实现了将现象学模型作为模块化构建模块无缝集成到概率模型中。