1D Linear operator with two parameters

In the previous examples we focus on systems with only one parameter. To see if the framework can be applied to a more general case, we use a 1D linear operator on two parameters, and verify the prediction.

Here we set up the linear operator as follows:

:raw-latex:`\begin{align*} \mathcal{L}_x^\phi u(x) &= f(x) \\ \mathcal{L}_x^\phi &:= \phi_1 \cdot + \phi_2\frac{d}{dx}\cdot \end{align*}`

A suitable solution can be:

:raw-latex:`\begin{align*} u ( x ) & = \sin ( x ) \\ f ( x ) & = \phi _ { 1 } \sin ( x ) + \phi _ { 2 } \cos ( x ) \\ x & \in [ 0,1 ] \end{align*}`

In this example, we assume \(\phi_1 = 2\), \(\phi_2 = 5\), and estimate \(\phi_1\) and \(\phi_2\).

Simulate data

##Initiating f(x) and u(x) with 20 data points
x = np.random.rand(20)
phi1 = 2.0
phi2 = 5.0
y_u = np.sin(x)
y_f = phi1*np.sin(x) + phi2*np.cos(x)

Evaluate kernels

Corresponding kernels are defined as following:

\(k _ { u u } \left( x _ { i } , x _ { j } ; \theta \right) = \theta \exp \left( - \frac { 1 } { 2 l } \left( x _ { i } - x _ { j } \right) ^ { 2 } \right)\)

\(\left. \begin{array} { l } { k _ { f f } \left( x _ { i } , x _ { j } ; \theta , \phi _ { 1 } , \phi _ { 2 } \right) } \\ { = \mathcal { L } _ { x _ { i } } ^ { \phi } \mathcal { L } _ { x _ { j } } ^ { \phi } k _ { u u } \left( x _ { i } , x _ { j } ; \theta \right) } \\ { = \mathcal { L } _ { x _ { i } } ^ { \phi } \left( \phi _ { 1 } k _ { u u } + \phi _ { 2 } \frac { \partial } { \partial x _ { j } } k _ { u u } \right) } \\ { = \phi _ { 1 } ^ { 2 } k _ { u u } + \phi _ { 1 } \phi _ { 2 } \frac { \partial } { \partial x _ { j } } k _ { u u } + \phi _ { 1 } \phi _ { 2 } \frac { \partial } { \partial x _ { i } } k _ { u u } + \phi _ { 2 } ^ { 2 } \frac { \partial } { \partial x _ { i } } \frac { \partial } { \partial x _ { i } } k _ { u u } } \end{array} \right.\)

\(k_{fu}(x_i,x_j;\theta,\phi_1, \phi_2) \\ = \mathcal{L}_{x_i}^\phi k_{uu}(x_i, x_j; \theta) \\ = \phi_1 k_{uu} + \phi_2 \frac{\partial}{\partial x_i}k_{uu}\)

\(\left. \begin{array} { l } { k _ { u f } \left( x _ { i } , x _ { j } ; \theta , \phi _ { 1 } , \phi _ { 2 } \right) } \\ { = \mathcal { L } _ { x _ { j } } ^ { \phi } k _ { u u } \left( x _ { i } , x _ { j } ; \theta \right) } \end{array} \right.\)

Optimize hyperparameters

phi ## Estimated phi1 and phi2 using noiseless data points

[1.9999955379492513, 5.000001322351089]

Parameter	Value
\(\phi_1\)	1.9999
\(\phi_2\)	5.0000

We see that the error rate is less than 0.01% for the hyperparameter estimation. This example shows, that one can use our framework for multiple parameter estimation.