  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.
