论文标题
增强高斯随机字段：理论与计算
Augmented Gaussian Random Field: Theory and Computation
论文作者
论文摘要
我们提出了新型的增强高斯随机场（AGRF），这是一个通用框架，结合了可观察到的任何顺序的数据。建立了严格的理论。我们证明，在某些条件下，可观察到的任何顺序及其衍生物都由一个高斯随机场约束，即上述AGRF。作为推论，该陈述``高斯过程的衍生物仍然是高斯过程''，因为衍生物由AGRF的一部分表示。此外，构建了与通用AGRF框架相对应的计算方法。都考虑了无声和嘈杂的场景。后分布的公式以不错的封闭形式推导。我们的计算方法的一个重要优点是，通用AGRF框架提供了一种自然的方式来合并任意订单衍生品和处理丢失的数据。我们使用四个数值示例来证明计算方法的有效性。数值示例是复合函数，阻尼的谐波振荡器，korteweg-de vries方程和汉堡方程。
We propose the novel augmented Gaussian random field (AGRF), which is a universal framework incorporating the data of observable and derivatives of any order. Rigorous theory is established. We prove that under certain conditions, the observable and its derivatives of any order are governed by a single Gaussian random field, which is the aforementioned AGRF. As a corollary, the statement ``the derivative of a Gaussian process remains a Gaussian process'' is validated, since the derivative is represented by a part of the AGRF. Moreover, a computational method corresponding to the universal AGRF framework is constructed. Both noiseless and noisy scenarios are considered. Formulas of the posterior distributions are deduced in a nice closed form. A significant advantage of our computational method is that the universal AGRF framework provides a natural way to incorporate arbitrary order derivatives and deal with missing data. We use four numerical examples to demonstrate the effectiveness of the computational method. The numerical examples are composite function, damped harmonic oscillator, Korteweg-De Vries equation, and Burgers' equation.
