Partial differential equations (PDEs) are widely used for the description of physical and engineering phenomena. Some key parameters involved in PDEs, which represent certain physical properties with important scientific interpretations, are difficult or even impossible to measure directly. Estimating these parameters from noisy and sparse experimental data of related physical quantities is an important task. Many methods for PDE parameter inference involve a large number of evaluations for numerical solutions to PDE through algorithms such as the finite element method, which can be time-consuming, especially for nonlinear PDEs. In this paper, we propose a novel method for the inference of unknown parameters in PDEs, called the PDE-Informed Gaussian Process (PIGP) based parameter inference method. Through modeling the PDE solution as a Gaussian process (GP), we derive the manifold constraints induced by the (linear) PDE structure such that, under the constraints, the GP satisfies the PDE. For nonlinear PDEs, we propose an augmentation method that transforms the nonlinear PDE into an equivalent PDE system linear in all derivatives, which our PIGP-based method can handle. The proposed method can be applied to a broad spectrum of nonlinear PDEs. The PIGP-based method can be applied to multi-dimensional PDE systems and PDE systems with unobserved components. Like conventional Bayesian approaches, the method can provide uncertainty quantification for both the unknown parameters and the PDE solution. The PIGP-based method also completely bypasses the numerical solver for PDEs. The proposed method is demonstrated through several application examples from different areas.

翻译：偏微分方程广泛应用于描述物理和工程现象。其中涉及的关键参数代表具有重要科学解释的特定物理性质，但通常难以甚至无法直接测量。从含噪声且稀疏的实验数据中估计相关物理量的参数是一项重要任务。许多偏微分方程参数推断方法需要通过有限元法等算法对偏微分方程数值解进行大量评估，这对于非线性偏微分方程尤其耗时。本文提出一种新颖的偏微分方程未知参数推断方法，称为基于偏微分方程信息的高斯过程（PIGP）参数推断方法。通过将偏微分方程解建模为高斯过程，我们推导出（线性）偏微分方程结构诱导的流形约束，使得在该约束下高斯过程满足偏微分方程。针对非线性偏微分方程，我们提出一种增广方法，将非线性偏微分方程转化为所有导数呈线性的等效偏微分方程组，从而适配基于PIGP的方法。所提方法可应用于广泛的非线性偏微分方程。基于PIGP的方法适用于多维偏微分方程组及含未观测分量的偏微分方程组。与传统贝叶斯方法类似，该方法能为未知参数和偏微分方程解提供不确定性量化。同时，基于PIGP的方法完全避开了偏微分方程的数值求解器。通过多个不同领域的应用实例验证了该方法的有效性。