diff --git a/synthetic_data/data_generation/data_generation_2.m b/synthetic_data/data_generation/data_generation_2.m
new file mode 100644
index 00000000..a1a5a369
--- /dev/null
+++ b/synthetic_data/data_generation/data_generation_2.m
@@ -0,0 +1,86 @@
+% This script generate a synthetic dataset with three different
+% possibilities. In any cases we consider the three following functions 
+%                   f_1 = @(a) alpha * sin(a); 
+%                   f_2 = @(a) beta * cos(a);
+%                   f_3 = @(a) gamma * sin(a) + delta * cos(a);
+% where alpha, beta, gamma and delta are scalars.
+% The three possibilities are :
+%   1) Generate train/test datasets by solving an ODE system of the form
+%                       dX/dt = A*X + g(t)
+%   where A is a N*N symmetric negative definite matrix and g is such that 
+%                       g(t) = b*t
+%   where b is a random N*1 vector.
+%   The labels are generated by taking for each output one of the above
+%   functions at random, i.e for each input x_i and output p_i,
+%                       p_i = f_j(x_i), for j in {1,2,3}
+%                                               
+%   2) Generate train/test datasets in the same way as 1), except that X is
+%   now symmetric positive definite. Apply the same mapping for the labels.
+%
+%   3) Generate train/test datasets in the same way as 2). This time the
+%   labels are generated by evaluating the functions on a linear
+%   combination of the input dataset.
+%
+% Also, before generating the labels, there is the possibility to add
+% gaussian noise the dataset X.
+
+clc
+clear all
+close all
+
+N = 3; % Number of input/output parameters
+Nh = 70; % Resolution of the grid for the ODE system
+N_samples = 2000; % Number of samples that we want to create
+t0 = 0; tf = 5; % Initial time and final time
+
+path = '../'; % Path to the files
+
+
+%% Definition of different functions
+
+% Definition of the time linear part of the ODE
+b = rand(N, 1);
+g = @(t) b * t;
+
+% System of ODEs
+fun = @(t, X) A*X + g(t);
+
+
+%% Reading the matrix A of the ODE system
+
+% Choose the matrix to be spd or snd 
+spd = 1; % Boolean for the type of the matrix
+i = 0; % For reading files
+A = read_matrix_ODE(spd, N, i);
+
+% Choose to apply a linear combination or not
+lin_combin = 1;
+
+% Get the random linear combination matrix
+B = random_linear_combination(N, lin_combin);
+
+%% Files creation
+
+% Decide whether we add noise or not
+add_noise = 0;
+
+% Define the index for the control variable
+control_index = 2;%randi(N,1);
+
+% Create directory to write dataset 
+name = 'files';
+% make directory 'files_N'
+make_directories(path, name, N)
+
+% Generate train set 
+mode = 'train';
+[X_train, t] = file_creation_2(A, B, g, t0, tf, N, Nh, N_samples, control_index, add_noise, path, mode);
+
+% Generate test set
+mode = 'test';
+[X_test, t] = file_creation_2(A, B, g, t0, tf, N, Nh, N_samples, control_index, add_noise, path, mode);
+
+
+
+
+
diff --git a/synthetic_data/data_generation/data_generation_2.m~ b/synthetic_data/data_generation/data_generation_2.m~
new file mode 100644
index 00000000..15236610
--- /dev/null
+++ b/synthetic_data/data_generation/data_generation_2.m~
@@ -0,0 +1,83 @@
+% This script generate a synthetic dataset with three different
+% possibilities. In any cases we consider the three following functions 
+%                   f_1 = @(a) alpha * sin(a); 
+%                   f_2 = @(a) beta * cos(a);
+%                   f_3 = @(a) gamma * sin(a) + delta * cos(a);
+% where alpha, beta, gamma and delta are scalars.
+% The three possibilities are :
+%   1) Generate train/test datasets by solving an ODE system of the form
+%                       dX/dt = A*X + g(t)
+%   where A is a N*N symmetric negative definite matrix and g is such that 
+%                       g(t) = b*t
+%   where b is a random N*1 vector.
+%   The labels are generated by taking for each output one of the above
+%   functions at random, i.e for each input x_i and output p_i,
+%                       p_i = f_j(x_i), for j in {1,2,3}
+%                                               
+%   2) Generate train/test datasets in the same way as 1), except that X is
+%   now symmetric positive definite. Apply the same mapping for the labels.
+%
+%   3) Generate train/test datasets in the same way as 2). This time the
+%   labels are generated by evaluating the functions on a linear
+%   combination of the input dataset.
+%
+% Also, before generating the labels, there is the possibility to add
+% gaussian noise the dataset X.
+
+clc
+clear all
+close all
+
+N = 3; % Number of input/output parameters
+Nh = 70; % Resolution of the grid for the ODE system
+N_samples = 2000; % Number of samples that we want to create
+t0 = 0; tf = 5; % Initial time and final time
+
+path = '../'; % Path to the files
+
+
+%% Definition of different functions
+
+% Definition of the time linear part of the ODE
+b = rand(N, 1);
+g = @(t) b * t;
+
+% System of ODEs
+fun = @(t, X) A*X + g(t);
+
+
+%% Reading the matrix A of the ODE system
+
+% Choose the matrix to be spd or snd 
+spd = 0; % Boolean for the type of the matrix
+i = 0; % For reading files
+A = read_matrix_ODE(spd, N, i);
+
+% Choose to apply a linear combination or not
+lin_combin = 1;
+
+% Get the random linear combination matrix
+B = random_linear_combination(N, lin_combin);
+
+%% Files creation
+
+% Decide whether we add noise or not
+add_noise = 0;
+
+% Create directory to write dataset 
+name = 'files';
+% make directory 'files_N'
+make_directories(path, name, N)
+
+% Generate train set 
+mode = 'train';
+[X_train, y_train, t] = file_creation_2(A, B, g, t0, tf, N, Nh, N_samples, control_index, add_noise, path, mode);
+
+% Generate test set
+mode = 'test';
+[X_test, y_test, t] = file_creation_2(A, B, g, t0, tf, N, Nh, N_samples, control_index, add_noise, path, mode);
+
+
+
+
+
diff --git a/synthetic_data/data_generation/file_creation_2.m b/synthetic_data/data_generation/file_creation_2.m
new file mode 100644
index 00000000..bc084144
--- /dev/null
+++ b/synthetic_data/data_generation/file_creation_2.m
@@ -0,0 +1,86 @@
+function [X, t] = file_creation_2(A, B, g, t0, tf, N, Nh, N_samples, control_index, add_noise, path, mode)
+% Creates train/test dataset
+% Inputs :
+%           A : Matrix of the ODE system
+%           B : Matrix of linear combination
+%           g : Linear part in the ODE system
+%           choices : choices for functions
+%           t0 : Initial time            
+%           tf : Final time
+%           N : Number of input/output parameters
+%           Nh : Resolution of the grid
+%           N_samples : Number of samples desired
+%           add_noise : Boolean, for adding noise or not
+%           path : path to the folder where we create the dataset
+%           mode : 'test' or 'train'
+%
+% Outputs :
+%           X : Dataset
+%           y : Labels
+%           t : time interval
+
+
+if add_noise
+    noise = random('Normal', 0, 1e-0, [N,Nh+1]);
+else
+    noise = 0;
+end
+
+% Iterate over the samples
+for i=0:N_samples-1
+    % Generate random initial condition
+    y0 = randi(20,1)*rand(N,1);
+        
+    % Solve the system of ODEs using Backward Euler
+    [t, X] = backward_euler(A, g, y0, t0, tf, Nh);
+    
+    % Add noise
+    X = X + noise;
+    
+    % Apply the linear combination of the inputs
+    C = X' * B;
+    C = C';
+    
+    % Open the target and input files to write the solutions
+    fileID2 = fopen(strcat(path, 'files_', num2str(N), '/', mode, '/target/file_', num2str(i), '.txt'),'w');
+    fileID3 = fopen(strcat(path, 'files_', num2str(N), '/', mode, '/input/file_', num2str(i), '.txt'),'w');
+
+    for j = 1:N
+        
+        if j == control_index
+            % Write in the files...
+            for t_=1:Nh+1
+                % ... the input.
+                fprintf(fileID3, '%f1 ', X(j,t_));
+            end
+        else
+            % Write in the files...
+            for t_=1:Nh+1
+                % ... the target.
+                fprintf(fileID2, '%f2 ', X(j,t_));
+            end
+        end
+        
+        fprintf(fileID2,'\n');
+        fprintf(fileID3,'\n');
+    end
+    fclose(fileID2);
+    fclose(fileID3);
+    
+end
+
+not_control_indexes = [];
+for j = 1:N
+    if j ~= control_index
+        not_control_indexes = [not_control_indexes, j];
+    end
+end
+% Plot a sample of the inputs and the targets
+figure
+plot(t, X(not_control_indexes(1), :), '-r', t, X(not_control_indexes(2), :), 'b-', t, X(control_index, :), 'c--');
+title(mode)
+legend('sample\_output1', 'sample\_ouput2', 'sample\_label1')
+grid; 
+
+
+return
\ No newline at end of file
diff --git a/synthetic_data/data_generation/file_creation_2.m~ b/synthetic_data/data_generation/file_creation_2.m~
new file mode 100644
index 00000000..927d676b
--- /dev/null
+++ b/synthetic_data/data_generation/file_creation_2.m~
@@ -0,0 +1,85 @@
+function [X, y, t] = file_creation_2(A, B, g, t0, tf, N, Nh, N_samples, control_index, add_noise, path, mode)
+% Creates train/test dataset
+% Inputs :
+%           A : Matrix of the ODE system
+%           B : Matrix of linear combination
+%           g : Linear part in the ODE system
+%           choices : choices for functions
+%           t0 : Initial time            
+%           tf : Final time
+%           N : Number of input/output parameters
+%           Nh : Resolution of the grid
+%           N_samples : Number of samples desired
+%           add_noise : Boolean, for adding noise or not
+%           path : path to the folder where we create the dataset
+%           mode : 'test' or 'train'
+%
+% Outputs :
+%           X : Dataset
+%           y : Labels
+%           t : time interval
+
+
+if add_noise
+    noise = random('Normal', 0, 1e-0, [N,Nh+1]);
+else
+    noise = 0;
+end
+
+% Iterate over the samples
+for i=0:N_samples-1
+    % Generate random initial condition
+    y0 = randi(20,1)*rand(N,1);
+        
+    % Solve the system of ODEs using Backward Euler
+    [t, X] = backward_euler(A, g, y0, t0, tf, Nh);
+    
+    % Add noise
+    X = X + noise;
+    
+    % Apply the linear combination of the inputs
+    C = X' * B;
+    C = C';
+    
+    % Open the target and input files to write the solutions
+    fileID2 = fopen(strcat(path, 'files_', num2str(N), '/', mode, '/target/file_', num2str(i), '.txt'),'w');
+    fileID3 = fopen(strcat(path, 'files_', num2str(N), '/', mode, '/input/file_', num2str(i), '.txt'),'w');
+
+    for j = 1:N
+        
+        % if choice is 1 we choose sin, if 2 we choose cos etc.
+        if j == control_index
+            % Write in the files...
+            for t_=1:Nh+1
+                % ... the target.
+                fprintf(fileID2, '%f ', X(j,t_));
+            end
+        else
+            % Write in the files...
+            for t_=1:Nh+1
+                % ... the input.
+                fprintf(fileID3, '%f ', X(j,t_));
+            end
+        end
+        
+        fprintf(fileID2,'\n');
+        fprintf(fileID3,'\n');
+    end
+    fclose(fileID2);
+    fclose(fileID3);
+    
+end
+
+not_control_indexes = [];
+for j = 1:N
+    if j ~= control_index
+        not_control_indexes = [not_control_indexes, j];
+% Plot a sample of the inputs and the targets
+figure
+plot(t, X(not_control_indexes(1), :), '-r', t, X(2, :), 'b-', t, X(control_index, :), 'c--', t);
+title(mode)
+legend('sample\_input1', 'sample\_input2', 'sample\_label1', 'sample\_label2')
+grid; 
+
+
+return
\ No newline at end of file