The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula. This paper extends earlier work on linear PDEs to a general class of initial value problems specified by nonlinear PDEs, motivated by problems for which evaluations of the right-hand-side, initial conditions, or boundary conditions of the PDE have a high computational cost. The proposed method can be viewed as exact Bayesian inference under an approximate likelihood, which is based on discretisation of the nonlinear differential operator. Proof-of-concept experimental results demonstrate that meaningful probabilistic uncertainty quantification for the unknown solution of the PDE can be performed, while controlling the number of times the right-hand-side, initial and boundary conditions are evaluated. A suitable prior model for the solution of PDEs is identified using novel theoretical analysis of the sample path properties of Matérn processes, which may be of independent interest.

可以将微分方程的数值解公式化为可以应用形式统计方法的推理问题。然而，从推理的角度来看，非线性偏微分方程 (PDE) 带来了巨大的挑战，最明显的是缺乏显式条件公式。本文将线性偏微分方程的早期工作扩展到非线性偏微分方程指定的一类初值问题，其动机是对偏微分方程的右侧、初始条件或边界条件的评估具有高计算成本的问题。所提出的方法可以看作是近似似然下的精确贝叶斯推理，它基于非线性微分算子的离散化。概念验证实验结果表明，可以对 PDE 的未知解进行有意义的概率不确定性量化，同时控制评估右侧、初始和边界条件的次数。使用对 Matérn 过程的样本路径属性进行新的理论分析来确定 PDE 解的合适先验模型，这可能是独立的兴趣。