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:线性偏微分方程是一类重要且广泛应用于物理过程中的机理模型,描述了热传递、电磁学和波传播等现象。在实践中,通常采用基于离散化的专业数值方法来解决偏微分方程。它们通常使用未知模型参数的估计值和(如果可用)物理测量值进行初始化。这样的求解器通常嵌入到具有下游应用的较大科学模型中,因此误差量化起着关键作用。然而,忽略参数和测量不确定性,传统的偏微分方程求解器可能无法产生其固有近似误差的一致估计。在这项工作中,我们以一种原则性的方式来解决这个问题,将线性偏微分方程的求解解释为物理知识指导的高斯过程回归(GP回归)。我们的框架基于高斯过程推断定理对通过任意有界线性算子进行的观测的关键推广。关键在于,这种概率观点允许(1)量化固有离散化误差;(2)将关于模型参数的不确定性传播到解决方案中;(3)基于噪声测量进行条件性建模。通过演示这种形式的优势,我们证明了它严格泛化了加权残差的方法,这是包括插值、有限体积、赝谱和(广义)伽辽金方法(例如有限元和谱方法)在内的PDE求解器的核心类。因此,该类求解器可以直接配备有结构化的误差估计。总之,我们的结果通过模糊数值分析和贝叶斯推断之间的界限,使机理模型成为概率模型中的模块化构建单元,从而实现了机理模型的无缝集成。