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Bayesian numerical methods for nonlinear partial differential equations

机译：非线性偏微分方程的贝叶斯数值方法

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摘要

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 Matern processes, which may be of independent interest.
机译：微分方程的数值解配制成可以应用正式统计方法的推理问题。然而，非线性偏微分方程（PDE）从推理的透视中提出了大量的挑战，最重要的是没有明确的调理公式。本文在线性PDE延伸到非线性PDE指定的一般初始值问题的一般工作，这是由PDE的右手侧，初始条件或边界条件的评估的问题产生高计算成本的问题。在近似可能性下，所提出的方法可以被视为精确的贝叶斯推断，这是基于非线性差分运算符的离散性。概念证明实验结果表明，可以执行针对PDE的未知解决方案的有意义的概率不确定性定量，同时评估右手侧，初始和边界条件的次数。利用Mattn工艺的样品路径性质的新颖理论分析来鉴定用于PDE的溶液的合适的先前模型，其可以具有独立的兴趣。

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