How to use Neural network for non binary input and output - matlab

Question

I tried to use the modified version of NN back propagation code by Phil Brierley
(www.philbrierley.com). When i try to solve the XOR problem it works perfectly. but when i try to solve a problem of the form output = x1^2 + x2^2 (ouput = sum of squares of input), the results are not accurate. i have scaled the input and ouput between -1 and 1. I get different results every time i run the same program (i understand its due to random wts initialization), but results are very different. i tried changing learning rate but still results converge.
have given the code below
%---------------------------------------------------------
% MATLAB neural network backprop code
% by Phil Brierley
%--------------------------------------------------------
clear; clc; close all;
%user specified values
hidden_neurons = 4;
epochs = 20000;
input = [];
for i =-10:2.5:10
for j = -10:2.5:10
input = [input;i j];
end
end
output = (input(:,1).^2 + input(:,2).^2);
output1 = output;
% Maximum input and output limit and scaling factors
m1 = -10; m2 = 10;
m3 = 0; m4 = 250;
c = -1; d = 1;
%Scale input and output
for i =1:size(input,2)
I = input(:,i);
scaledI = ((d-c)*(I-m1) ./ (m2-m1)) + c;
input(:,i) = scaledI;
end
for i =1:size(output,2)
I = output(:,i);
scaledI = ((d-c)*(I-m3) ./ (m4-m3)) + c;
output(:,i) = scaledI;
end
train_inp = input;
train_out = output;
%read how many patterns and add bias
patterns = size(train_inp,1);
train_inp = [train_inp ones(patterns,1)];
%read how many inputs and initialize learning rate
inputs = size(train_inp,2);
hlr = 0.1;
%set initial random weights
weight_input_hidden = (randn(inputs,hidden_neurons) - 0.5)/10;
weight_hidden_output = (randn(1,hidden_neurons) - 0.5)/10;
%Training
err = zeros(1,epochs);
for iter = 1:epochs
alr = hlr;
blr = alr / 10;
%loop through the patterns, selecting randomly
for j = 1:patterns
%select a random pattern
patnum = round((rand * patterns) + 0.5);
if patnum > patterns
patnum = patterns;
elseif patnum < 1
patnum = 1;
end
%set the current pattern
this_pat = train_inp(patnum,:);
act = train_out(patnum,1);
%calculate the current error for this pattern
hval = (tanh(this_pat*weight_input_hidden))';
pred = hval'*weight_hidden_output';
error = pred - act;
% adjust weight hidden - output
delta_HO = error.*blr .*hval;
weight_hidden_output = weight_hidden_output - delta_HO';
% adjust the weights input - hidden
delta_IH= alr.*error.*weight_hidden_output'.*(1-(hval.^2))*this_pat;
weight_input_hidden = weight_input_hidden - delta_IH';
end
% -- another epoch finished
%compute overall network error at end of each epoch
pred = weight_hidden_output*tanh(train_inp*weight_input_hidden)';
error = pred' - train_out;
err(iter) = ((sum(error.^2))^0.5);
%stop if error is small
if err(iter) < 0.001
fprintf('converged at epoch: %d\n',iter);
break
end
end
%Output after training
pred = weight_hidden_output*tanh(train_inp*weight_input_hidden)';
Y = m3 + (m4-m3)*(pred-c)./(d-c);
% Testing for a new set of input
input_test = [6 -3.1; 0.5 1; -2 3; 3 -2; -4 5; 0.5 4; 6 1.5];
output_test = (input_test(:,1).^2 + input_test(:,2).^2);
input1 = input_test;
%Scale input
for i =1:size(input1,2)
I = input1(:,i);
scaledI = ((d-c)*(I-m1) ./ (m2-m1)) + c;
input1(:,i) = scaledI;
end
%Predict output
train_inp1 = input1;
patterns = size(train_inp1,1);
bias = ones(patterns,1);
train_inp1 = [train_inp1 bias];
pred1 = weight_hidden_output*tanh(train_inp1*weight_input_hidden)';
%Rescale
Y1 = m3 + (m4-m3)*(pred1-c)./(d-c);
analy_numer = [output_test Y1']
plot(err)
This is the sample output i get for problem
state after 20000 epochs
analy_numer =
45.6100 46.3174
1.2500 -2.9457
13.0000 11.9958
13.0000 9.7097
41.0000 44.9447
16.2500 17.1100
38.2500 43.9815
if i run once more i get different results. as can be observed for small values of input i get totally wrong ans (negative ans not possible). for other values accuracy is still poor.
can someone tell what i am doing wrong and how to correct.
thanks
raman