Creating a training dataset¶

In this section, we explain how to generate a training dataset in MATLAB to learn the Green’s function associated to an ODE or a system of ODEs.

This step requires the MATLAB package called Chebfun (see https://www.chebfun.org/download/ for installation instructions).

Definition of the differential operator¶

The definition of the differential operator can be done by creating a MATLAB script in a folder examples/. In the following example, we add a script called helmholtz.m in the folder with the following content:

function output_example = helmholtz()
% Helmholtz equation

% Define the domain.
dom = [0,1];

% Parameter of the equation
K = 15;

% Differential operator
N = chebop(@(x,u) diff(u,2)+K^2*u, dom);

% Boundary conditions
N.bc = @(x,u) [u(dom(1)); u(dom(2))];

% Output
output_example = {N};
end

Here, we define a Helmholtz operator \(\mathcal{L}u = \frac{d^2u}{dx^2}+K^2 u\) with Helmholtz frequency \(K=15\), on the interval \([0,1]\), with homogeneous Dirichlet boundary conditions.

One could, for instance, define the boundary conditions to be \(u(0)=-1\), \(u(1)=2\) as follows:
% Boundary conditions N.bc = @(x,u) [u(dom(1))+1; u(dom(2))-2];

It is also possible to impose \(u(0)=0\), \(u'(0)=1\) with the following line:

% Boundary conditions
N.bc = @(x,u) [u(dom(1)); feval(diff(u),dom(1))-1];

If the exact Green’s function is known, one can in addition provide its expression (as a string in numpy format) to compare with the learned Green’s function:

% Exact Green's function
G = sprintf('(%d*np.sin(%d))**(-1)*np.sin(%d*x)*np.sin(%d*(y-1))*(x<=y)+(%d*np.sin(%d))**(-1)*np.sin(%d*y)*np.sin(%d*(x-1))*(x>y)',K,K,K,K,K,K,K,K);

% Output
output_example = {N, "ExactGreen", G};

System of differential operators¶

The syntax is similar to define a system of differential operators. In this example, we define the following system:

\[\begin{split}\mathcal{L}(u,v) = \left(\begin{array}{c} \frac{d^2u}{dx^2}-v\\ -\frac{d^2v}{dx^2}+xu \end{array}\right),\end{split}\]

on the domain \([-1,1]\) with boundary conditions:

\[u(-1)=1,\,u(1)=-1,\,v(-1)=v(1)=-2.\]

function output_example = ODE_system()
% System of ODEs

% Define the domain.
dom = [-1, 1];

% Differential operator
N = chebop(@(x,u,v) [diff(u,2)-v; -diff(v,2)+x.*u], dom);

% Boundary conditions
N.bc = @(x,u, v) [u(-1)-1; u(1)+1; v(-1)+2; v(1)+2];

% Output
output_example = {N};
end

Generating the dataset¶

Make sure that the MATLAB codes generate_gl_example.m and generate_gl_datasets.m are present in the parent directory of examples/ and run the following command in a MATLAB terminal:

generate_gl_example("helmholtz");

Alternatively, all the datasets corresponding to the examples in the folder examples/ can be generated as follows:

generate_gl_datasets();

The datasets are saved as .mat files at the location examples/datasets/.

By default, the code generates \(100\) sampled functions \(f\) from a squared-exponential Gaussian process, and solve the equation \(\mathcal{L}u=f\) for the different forcing terms to obtain the training solutions \(u\). Then, the forcing terms are evaluated at \(200\) uniform points in the domain \(\Omega\), while the solutions are evaluated at \(100\) points.
It is possible to edit the MATLAB script generate_gl_example.m to add noise to the output functions, or change the different parameters such as number of forcing terms, spatial points, …