Augmented Gaussian Random Field: Theory and Computation

TL;DR

This paper introduces the augmented Gaussian random field (AGRF), a comprehensive framework that models observable data and derivatives of any order, supported by rigorous theory and a practical computational method for both noiseless and noisy data scenarios.

Contribution

The paper develops the AGRF framework, proving its theoretical foundation and providing a novel computational approach that handles derivatives and missing data effectively.

Findings

AGRF unifies observable data and derivatives under a single Gaussian field.

The computational method accurately handles noisy and noiseless data.

Numerical examples demonstrate the method's effectiveness across different equations.

Abstract

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.