An artificial neural network is a piece of software that adapts and learns from sets of sample data to determine how to react to new inputs. These networks are highly effective and are increasingly being used in various applications, including facial recognition systems, self-driving cars, financial market forecasting, sports betting predictions, and personalized product recommendations on websites.
The XOR problem stands as a fundamental challenge in the field of artificial neural network (ANN) research. It involves training a neural network to accurately predict the results of XOR logic gates based on two binary inputs. Essentially, an XOR function outputs a true value when the inputs differ and a false value when they are identical.
This is the code that we upload to arduino:
//Author: Ralph Heymsfeld
//28/06/2018
#include <math.h>
/******************************************************************
* Network Configuration - customized per network
******************************************************************/
const int PatternCount = 10;
const int InputNodes = 7;
const int HiddenNodes = 8;
const int OutputNodes = 4;
const float LearningRate = 0.3;
const float Momentum = 0.9;
const float InitialWeightMax = 0.5;
const float Success = 0.0004;
const byte Input[PatternCount][InputNodes] = {
{ 1, 1, 1, 1, 1, 1, 0 }, // 0
{ 0, 1, 1, 0, 0, 0, 0 }, // 1
{ 1, 1, 0, 1, 1, 0, 1 }, // 2
{ 1, 1, 1, 1, 0, 0, 1 }, // 3
{ 0, 1, 1, 0, 0, 1, 1 }, // 4
{ 1, 0, 1, 1, 0, 1, 1 }, // 5
{ 0, 0, 1, 1, 1, 1, 1 }, // 6
{ 1, 1, 1, 0, 0, 0, 0 }, // 7
{ 1, 1, 1, 1, 1, 1, 1 }, // 8
{ 1, 1, 1, 0, 0, 1, 1 } // 9
};
const byte Target[PatternCount][OutputNodes] = {
{ 0, 0, 0, 0 },
{ 0, 0, 0, 1 },
{ 0, 0, 1, 0 },
{ 0, 0, 1, 1 },
{ 0, 1, 0, 0 },
{ 0, 1, 0, 1 },
{ 0, 1, 1, 0 },
{ 0, 1, 1, 1 },
{ 1, 0, 0, 0 },
{ 1, 0, 0, 1 }
};
/******************************************************************
* End Network Configuration
******************************************************************/
int i, j, p, q, r;
int ReportEvery1000;
int RandomizedIndex[PatternCount];
long TrainingCycle;
float Rando;
float Error;
float Accum;
float Hidden[HiddenNodes];
float Output[OutputNodes];
float HiddenWeights[InputNodes+1][HiddenNodes];
float OutputWeights[HiddenNodes+1][OutputNodes];
float HiddenDelta[HiddenNodes];
float OutputDelta[OutputNodes];
float ChangeHiddenWeights[InputNodes+1][HiddenNodes];
float ChangeOutputWeights[HiddenNodes+1][OutputNodes];
void setup(){
Serial.begin(9600);
randomSeed(analogRead(3));
ReportEvery1000 = 1;
for( p = 0 ; p < PatternCount ; p++ ) {
RandomizedIndex[p] = p ;
}
}
void loop (){
/******************************************************************
* Initialize HiddenWeights and ChangeHiddenWeights
******************************************************************/
for( i = 0 ; i < HiddenNodes ; i++ ) {
for( j = 0 ; j <= InputNodes ; j++ ) {
ChangeHiddenWeights[j][i] = 0.0 ;
Rando = float(random(100))/100;
HiddenWeights[j][i] = 2.0 * ( Rando - 0.5 ) * InitialWeightMax ;
}
}
/******************************************************************
* Initialize OutputWeights and ChangeOutputWeights
******************************************************************/
for( i = 0 ; i < OutputNodes ; i ++ ) {
for( j = 0 ; j <= HiddenNodes ; j++ ) {
ChangeOutputWeights[j][i] = 0.0 ;
Rando = float(random(100))/100;
OutputWeights[j][i] = 2.0 * ( Rando - 0.5 ) * InitialWeightMax ;
}
}
Serial.println("Initial/Untrained Outputs: ");
toTerminal();
/******************************************************************
* Begin training
******************************************************************/
for( TrainingCycle = 1 ; TrainingCycle < 2147483647 ; TrainingCycle++) {
/******************************************************************
* Randomize order of training patterns
******************************************************************/
for( p = 0 ; p < PatternCount ; p++) {
q = random(PatternCount);
r = RandomizedIndex[p] ;
RandomizedIndex[p] = RandomizedIndex[q] ;
RandomizedIndex[q] = r ;
}
Error = 0.0 ;
/******************************************************************
* Cycle through each training pattern in the randomized order
******************************************************************/
for( q = 0 ; q < PatternCount ; q++ ) {
p = RandomizedIndex[q];
/******************************************************************
* Compute hidden layer activations
******************************************************************/
for( i = 0 ; i < HiddenNodes ; i++ ) {
Accum = HiddenWeights[InputNodes][i] ;
for( j = 0 ; j < InputNodes ; j++ ) {
Accum += Input[p][j] * HiddenWeights[j][i] ;
}
Hidden[i] = 1.0/(1.0 + exp(-Accum)) ;
}
/******************************************************************
* Compute output layer activations and calculate errors
******************************************************************/
for( i = 0 ; i < OutputNodes ; i++ ) {
Accum = OutputWeights[HiddenNodes][i] ;
for( j = 0 ; j < HiddenNodes ; j++ ) {
Accum += Hidden[j] * OutputWeights[j][i] ;
}
Output[i] = 1.0/(1.0 + exp(-Accum)) ;
OutputDelta[i] = (Target[p][i] - Output[i]) * Output[i] * (1.0 - Output[i]) ;
Error += 0.5 * (Target[p][i] - Output[i]) * (Target[p][i] - Output[i]) ;
}
/******************************************************************
* Backpropagate errors to hidden layer
******************************************************************/
for( i = 0 ; i < HiddenNodes ; i++ ) {
Accum = 0.0 ;
for( j = 0 ; j < OutputNodes ; j++ ) {
Accum += OutputWeights[i][j] * OutputDelta[j] ;
}
HiddenDelta[i] = Accum * Hidden[i] * (1.0 - Hidden[i]) ;
}
/******************************************************************
* Update Inner-->Hidden Weights
******************************************************************/
for( i = 0 ; i < HiddenNodes ; i++ ) {
ChangeHiddenWeights[InputNodes][i] = LearningRate * HiddenDelta[i] + Momentum * ChangeHiddenWeights[InputNodes][i] ;
HiddenWeights[InputNodes][i] += ChangeHiddenWeights[InputNodes][i] ;
for( j = 0 ; j < InputNodes ; j++ ) {
ChangeHiddenWeights[j][i] = LearningRate * Input[p][j] * HiddenDelta[i] + Momentum * ChangeHiddenWeights[j][i];
HiddenWeights[j][i] += ChangeHiddenWeights[j][i] ;
}
}
/******************************************************************
* Update Hidden-->Output Weights
******************************************************************/
for( i = 0 ; i < OutputNodes ; i ++ ) {
ChangeOutputWeights[HiddenNodes][i] = LearningRate * OutputDelta[i] + Momentum * ChangeOutputWeights[HiddenNodes][i] ;
OutputWeights[HiddenNodes][i] += ChangeOutputWeights[HiddenNodes][i] ;
for( j = 0 ; j < HiddenNodes ; j++ ) {
ChangeOutputWeights[j][i] = LearningRate * Hidden[j] * OutputDelta[i] + Momentum * ChangeOutputWeights[j][i] ;
OutputWeights[j][i] += ChangeOutputWeights[j][i] ;
}
}
}
/******************************************************************
* Every 1000 cycles send data to terminal for display
******************************************************************/
ReportEvery1000 = ReportEvery1000 - 1;
if (ReportEvery1000 == 0)
{
Serial.println();
Serial.println();
Serial.print ("TrainingCycle: ");
Serial.print (TrainingCycle);
Serial.print (" Error = ");
Serial.println (Error, 5);
toTerminal();
if (TrainingCycle==1)
{
ReportEvery1000 = 999;
}
else
{
ReportEvery1000 = 1000;
}
}
/******************************************************************
* If error rate is less than pre-determined threshold then end
******************************************************************/
if( Error < Success ) break ;
}
Serial.println ();
Serial.println();
Serial.print ("TrainingCycle: ");
Serial.print (TrainingCycle);
Serial.print (" Error = ");
Serial.println (Error, 5);
toTerminal();
Serial.println ();
Serial.println ();
Serial.println ("Training Set Solved! ");
Serial.println ("--------");
Serial.println ();
Serial.println ();
ReportEvery1000 = 1;
}
void toTerminal()
{
for( p = 0 ; p < PatternCount ; p++ ) {
Serial.println();
Serial.print (" Training Pattern: ");
Serial.println (p);
Serial.print (" Input ");
for( i = 0 ; i < InputNodes ; i++ ) {
Serial.print (Input[p][i], DEC);
Serial.print (" ");
}
Serial.print (" Target ");
for( i = 0 ; i < OutputNodes ; i++ ) {
Serial.print (Target[p][i], DEC);
Serial.print (" ");
}
/******************************************************************
* Compute hidden layer activations
******************************************************************/
for( i = 0 ; i < HiddenNodes ; i++ ) {
Accum = HiddenWeights[InputNodes][i] ;
for( j = 0 ; j < InputNodes ; j++ ) {
Accum += Input[p][j] * HiddenWeights[j][i] ;
}
Hidden[i] = 1.0/(1.0 + exp(-Accum)) ;
}
/******************************************************************
* Compute output layer activations and calculate errors
******************************************************************/
for( i = 0 ; i < OutputNodes ; i++ ) {
Accum = OutputWeights[HiddenNodes][i] ;
for( j = 0 ; j < HiddenNodes ; j++ ) {
Accum += Hidden[j] * OutputWeights[j][i] ;
}
Output[i] = 1.0/(1.0 + exp(-Accum)) ;
}
Serial.print (" Output ");
for( i = 0 ; i < OutputNodes ; i++ ) {
Serial.print (Output[i], 5);
Serial.print (" ");
}
}
}After running the code we can check the progress of learning in the serial montor:
After thousands of cycles that run in a couple of minutes it gets solved:
The network undergoes several rounds of training. During each round, it processes various training examples. For each example, the network computes its output based on the given input, checks this output against the desired outcome, and modifies its parameters using a learning algorithm. This cycle is repeated until the error is reduced to an acceptable level (indicating success) or until it has completed a predetermined maximum number of training rounds.
After that we change the code according to the XOR table:
This change results in much faster training cycles that ultimately lead to a much quicker solution.
Commenti
Posta un commento