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中观模型，描述热传导、电磁学和波传播等物理过程。实践中，基于离散化的专用数值方法被用于求解PDEs。这些方法通常利用未知模型参数的估计值，并可能结合物理测量数据进行初始化。此类求解器常嵌入具有下游应用的大型科学模型中，因此误差量化至关重要。然而，经典PDE求解器因忽略参数和测量不确定性，可能无法对其固有近似误差生成一致的估计。本文通过将线性PDE求解解释为物理信息高斯过程（GP）回归，以原则性方法处理该问题。我们的框架基于高斯过程推理定理的关键推广，使其适用于通过任意有界线性算子获取的观测数据。关键的是，这一概率视角允许（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）基于含噪声测量进行条件化。为展示该公式的强度，我们证明其严格泛化了加权残差法——这一PDE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）伽辽金法（如有限元法和谱方法）。此类方法因此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理的边界，实现了将中观模型作为模块化构建块无缝集成到概率模型中。