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/**
* Example solving the following ODE:
*
* g(t) = (d^2/d^t)y(t) + 4 (d/dt)y(t) + 4y(t) = 0, for t in [0, 4]
* y(0) = 1
* (d/dt)y(0) = 1
*
* -------------------------------------------
* Analytical solution: e^(-2x) * (3x + 1)
* NN representation: sum over [a_i * (1 + e^(-x * w_i + b_i))^(-1)]
* -------------------------------------------
* Optimal NN setting without biases (2 inner neurons)
* Path 1. w = -6.35706416, b = 0.00000000, a = -1.05305639
* Path 2. w = -1.55399893, b = 0.00000000, a = 3.07464411
* -------------------------------------------
* Optimal NN setting with biases (2 inner neurons)
* Path 1. w = 6.75296220, b = -1.63419516, a = 1.71242130
* Path 2. w = 1.86917877, b = 1.09972747, a = -1.70757578
* @author Michal Kravčenko
* @date 17.7.18 -
*/
Michal Kravcenko
committed
#include "Solvers/DESolver.h"
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double eval_f(double x){
return std::pow(E, -2.0 * x) * (3.0 * x + 1.0);
}
double eval_df(double x){
return std::pow(E, -2.0 * x) * (1.0 - 6.0 * x);
}
double eval_ddf(double x){
return 4.0 * std::pow(E, -2.0 * x) * (3.0 * x - 2.0);
}
double eval_approx_f(double x, size_t n_inner_neurons, std::vector<double> ¶meters){
double value= 0.0, wi, ai, bi, ei, ei1;
for(size_t i = 0; i < n_inner_neurons; ++i){
wi = parameters[3 * i];
ai = parameters[3 * i + 1];
bi = parameters[3 * i + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
value += ai / (ei1);
}
return value;
}
double eval_approx_df(double x, size_t n_inner_neurons, std::vector<double> ¶meters){
double value= 0.0, wi, ai, bi, ei, ei1;
for(size_t i = 0; i < n_inner_neurons; ++i){
wi = parameters[3 * i];
ai = parameters[3 * i + 1];
bi = parameters[3 * i + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
value += ai * wi * ei / (ei1 * ei1);
}
return value;
}
double eval_approx_ddf(double x, size_t n_inner_neurons, std::vector<double> ¶meters){
double value= 0.0, wi, ai, bi, ewx, eb;
for(size_t i = 0; i < n_inner_neurons; ++i){
wi = parameters[3 * i];
ai = parameters[3 * i + 1];
bi = parameters[3 * i + 2];
eb = std::pow(E, bi);
ewx = std::pow(E, wi * x);
value += -(ai*wi*wi*eb*ewx*(ewx - eb))/((eb + ewx)*(eb + ewx)*(eb + ewx));
}
return value;
}
//NN partial derivative (wi): (ai * x * e^(bi - wi * x)) * (e^(bi - wi * x) + 1)^(-2)
double eval_approx_dw_f(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, ai, bi, ei, ei1;
wi = parameters[3 * neuron_idx];
ai = parameters[3 * neuron_idx + 1];
bi = parameters[3 * neuron_idx + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
return (ai * x * ei) / (ei1 * ei1);
}
//dNN partial derivative (wi): -(a w x e^(b - w x))/(e^(b - w x) + 1)^2 + (2 a w x e^(2 b - 2 w x))/(e^(b - w x) + 1)^3 + (a e^(b - w x))/(e^(b - w x) + 1)^2
double eval_approx_dw_df(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, ai, bi, ei, ei1;
wi = parameters[3 * neuron_idx];
ai = parameters[3 * neuron_idx + 1];
bi = parameters[3 * neuron_idx + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
return -(ai * wi * x * ei)/(ei1 * ei1) + (2.0*ai*wi*x*ei*ei)/(ei1 * ei1 * ei1) + (ai* ei)/(ei1 * ei1);
}
//ddNN partial derivative (wi): -(a w^2 x e^(b + 2 w x))/(e^b + e^(w x))^3 - (a w^2 x e^(b + w x) (e^(w x) - e^b))/(e^b + e^(w x))^3 + (3 a w^2 x e^(b + 2 w x) (e^(w x) - e^b))/(e^b + e^(w x))^4 - (2 a w e^(b + w x) (e^(w x) - e^b))/(e^b + e^(w x))^3
double eval_approx_dw_ddf(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, ai, bi, eb, ewx;
wi = parameters[3 * neuron_idx];
ai = parameters[3 * neuron_idx + 1];
bi = parameters[3 * neuron_idx + 2];
eb = std::pow(E, bi);
ewx = std::pow(E, wi * x);
return -(ai*wi*wi* x * eb*ewx*ewx)/((eb + ewx)*(eb + ewx)*(eb + ewx)) - (ai*wi*wi*x*eb*ewx*(ewx - eb))/((eb + ewx)*(eb + ewx)*(eb + ewx)) + (3*ai*wi*wi*x*eb*ewx*ewx*(ewx - eb))/((eb + ewx)*(eb + ewx)*(eb + ewx)*(eb + ewx)) - (2*ai*wi*eb*ewx*(ewx - eb))/((eb + ewx)*(eb + ewx)*(eb + ewx));
}
//NN partial derivative (ai): (1 + e^(-x * wi + bi))^(-1)
double eval_approx_da_f(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, bi, ei, ei1;
wi = parameters[3 * neuron_idx];
bi = parameters[3 * neuron_idx + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
return 1.0 / ei1;
}
//dNN partial derivative (ai): (w e^(b - w x))/(e^(b - w x) + 1)^2
double eval_approx_da_df(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, bi, ei, ei1;
wi = parameters[3 * neuron_idx];
bi = parameters[3 * neuron_idx + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
return (wi*ei)/(ei1 * ei1);
}
//ddNN partial derivative (ai): -(w^2 e^(b + w x) (e^(w x) - e^b))/(e^b + e^(w x))^3
double eval_approx_da_ddf(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, bi, eip, ewx, eb;
wi = parameters[3 * neuron_idx];
bi = parameters[3 * neuron_idx + 2];
eip = std::pow(E, bi + wi * x);
eb = std::pow(E, bi);
ewx = std::pow(E, wi * x);
return -(wi*wi*eip*(ewx - eb))/((eb + ewx)*(eb + ewx)*(eb + ewx));
}
//NN partial derivative (bi): -(ai * e^(bi - wi * x)) * (e^(bi - wi * x) + 1)^(-2)
double eval_approx_db_f(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, bi, ei, ai, ei1;
wi = parameters[3 * neuron_idx];
ai = parameters[3 * neuron_idx + 1];
bi = parameters[3 * neuron_idx + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
return -(ai * ei)/(ei1 * ei1);
}
//dNN partial derivative (bi): (a w e^(b + w x) (e^(w x) - e^b))/(e^b + e^(w x))^3
double eval_approx_db_df(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, bi, ai, ewx, eb;
wi = parameters[3 * neuron_idx];
ai = parameters[3 * neuron_idx + 1];
bi = parameters[3 * neuron_idx + 2];
eb = std::pow(E, bi);
ewx = std::pow(E, wi*x);
return (ai* wi* eb*ewx* (ewx - eb))/((eb + ewx)*(eb + ewx)*(eb + ewx));
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//ddNN partial derivative (bi): -(a w^2 e^(b + w x) (-4 e^(b + w x) + e^(2 b) + e^(2 w x)))/(e^b + e^(w x))^4
double eval_approx_db_ddf(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, bi, ai, ewx, eb;
wi = parameters[3 * neuron_idx];
ai = parameters[3 * neuron_idx + 1];
bi = parameters[3 * neuron_idx + 2];
eb = std::pow(E, bi);
ewx = std::pow(E, wi*x);
return -(ai* wi*wi* eb*ewx* (-4.0* eb*ewx + eb*eb + ewx*ewx))/((eb +ewx)*(eb +ewx)*(eb +ewx)*(eb +ewx));
}
double eval_error_function(std::vector<double> ¶meters, size_t n_inner_neurons, std::vector<double> test_points){
double output = 0.0, approx, frac = 1.0 / (test_points.size());
for(auto x: test_points){
/* governing equation */
approx = 4.0 * eval_approx_f(x, n_inner_neurons, parameters) + 4.0 * eval_approx_df(x, n_inner_neurons, parameters) + eval_approx_ddf(x, n_inner_neurons, parameters);
output += (0.0 - approx) * (0.0 - approx) * frac;
}
/* BC */
approx = eval_approx_f(0.0, n_inner_neurons, parameters);
output += (1.0 - approx) * (1.0 - approx);
approx = eval_approx_df(0.0, n_inner_neurons, parameters);
output += (1.0 - approx) * (1.0 - approx);
return output;
}
void test_analytical_gradient_y(std::vector<double> &guess, double accuracy, size_t n_inner_neurons, size_t train_size, bool opti_w, bool opti_b, double d1_s, double d1_e,
size_t test_size, double ts, double te) {
/* SETUP OF THE TRAINING DATA */
std::vector<double> inp, out;
double frac, alpha, x;
double grad_norm = accuracy * 10.0, mem, ai, bi, wi, error, derror, approx, xj, gamma, total_error, sk, sy, sx, beta;
double grad_norm_prev = grad_norm;
size_t i, j, iter_idx = 0;
x = (std::cos(PI - alpha * i) + 1.0) * frac + d1_s;
data_points[i] = x;
}
std::vector<double> *gradient_current = new std::vector<double>(3 * n_inner_neurons);
std::vector<double> *gradient_prev = new std::vector<double>(3 * n_inner_neurons);
std::vector<double> *params_current = new std::vector<double>(guess);
std::vector<double> *params_prev = new std::vector<double>(guess);
std::vector<double> *conjugate_direction_current = new std::vector<double>(3 * n_inner_neurons);
std::vector<double> *conjugate_direction_prev = new std::vector<double>(3 * n_inner_neurons);
std::vector<double> *ptr_mem;
std::fill(gradient_current->begin(), gradient_current->end(), 0.0);
std::fill(gradient_prev->begin(), gradient_prev->end(), 0.0);
std::fill(conjugate_direction_current->begin(), conjugate_direction_current->end(), 0.0);
std::fill(conjugate_direction_prev->begin(), conjugate_direction_prev->end(), 0.0);
for (i = 0; i < n_inner_neurons; ++i) {
wi = (*params_current)[3 * i];
ai = (*params_current)[3 * i + 1];
bi = (*params_current)[3 * i + 2];
printf("Path %3d. w = %15.8f, b = %15.8f, a = %15.8f\n", (int)(i + 1), wi, bi, ai);
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gamma = 1.0;
while( grad_norm > accuracy) {
iter_idx++;
/* current gradient */
std::fill(gradient_current->begin(), gradient_current->end(), 0.0);
/* error boundary condition: y(0) = 1 => e1 = (1 - y(0))^2 */
xj = 0.0;
mem = (1.0 - eval_approx_f(xj, n_inner_neurons, *params_current));
derror = 2.0 * mem;
total_error = mem * mem;
for(i = 0; i < n_inner_neurons; ++i){
if(opti_w){
(*gradient_current)[3 * i] -= derror * eval_approx_dw_f(xj, i, *params_current);
(*gradient_current)[3 * i + 1] -= derror * eval_approx_da_f(xj, i, *params_current);
}
if(opti_b){
(*gradient_current)[3 * i + 2] -= derror * eval_approx_db_f(xj, i, *params_current);
}
}
// for(auto e: *gradient_current){
// printf("[%10.8f]", e);
// }
// printf("\n");
/* error boundary condition: y'(0) = 1 => e2 = (1 - y'(0))^2 */
mem = (1.0 - eval_approx_df(xj, n_inner_neurons, *params_current));
derror = 2.0 * mem;
total_error += mem * mem;
for(i = 0; i < n_inner_neurons; ++i){
if(opti_w){
(*gradient_current)[3 * i] -= derror * eval_approx_dw_df(xj, i, *params_current);
(*gradient_current)[3 * i + 1] -= derror * eval_approx_da_df(xj, i, *params_current);
}
if(opti_b){
(*gradient_current)[3 * i + 2] -= derror * eval_approx_db_df(xj, i, *params_current);
}
}
// for(auto e: *gradient_current){
// printf("[%10.8f]", e);
// }
// printf("\n");
for(j = 0; j < data_points.size(); ++j){
xj = data_points[i];
/* error of the governing equation: y''(x) + 4y'(x) + 4y(x) = 0 => e3 = 1/n * (0 - y''(x) - 4y'(x) - 4y(x))^2 */
approx= eval_approx_ddf(xj, n_inner_neurons, *params_current) + 4.0 * eval_approx_df(xj, n_inner_neurons, *params_current) + 4.0 * eval_approx_f(xj, n_inner_neurons, *params_current);
mem = 0.0 - approx;
error = 2.0 * mem / train_size;
for(i = 0; i < n_inner_neurons; ++i){
if(opti_w){
(*gradient_current)[3 * i] -= error * (eval_approx_dw_ddf(xj, i, *params_current) + 4.0 * eval_approx_dw_df(xj, i, *params_current) + 4.0 * eval_approx_dw_f(xj, i, *params_current));
(*gradient_current)[3 * i + 1] -= error * (eval_approx_da_ddf(xj, i, *params_current) + 4.0 * eval_approx_da_df(xj, i, *params_current) + 4.0 * eval_approx_da_f(xj, i, *params_current));
}
if(opti_b){
(*gradient_current)[3 * i + 2] -= error * (eval_approx_db_ddf(xj, i, *params_current) + 4.0 * eval_approx_db_df(xj, i, *params_current) + 4.0 * eval_approx_db_f(xj, i, *params_current));
}
}
total_error += mem * mem / train_size;
}
// /* error of the unknown function: y(x) = e^(-2x)*(3x+1) => e3 = 1/n * (e^(-2x)*(3x+1) - y(x))^2 */
// for(j = 0; j < data_points.size(); ++j) {
// xj = data_points[i];
// approx= eval_approx_f(xj, n_inner_neurons, *params_current);
// mem = (eval_f(xj) - approx);
// error = 2.0 * mem / train_size;
// for(i = 0; i < n_inner_neurons; ++i){
// if(opti_w){
// (*gradient_current)[3 * i] -= error * (eval_approx_dw_f(xj, i, *params_current));
// (*gradient_current)[3 * i + 1] -= error * (eval_approx_da_f(xj, i, *params_current));
// }
// if(opti_b){
// (*gradient_current)[3 * i + 2] -= error * (eval_approx_db_f(xj, i, *params_current));
// }
// }
// total_error += mem * mem / train_size;
// }
// /* error of the unknown function: y'(x) = e^(-2x)*(1-6x) => e3 = 1/n * (e^(-2x)*(1-6x) - y'(x))^2 */
// for(j = 0; j < data_points.size(); ++j) {
// xj = data_points[i];
// approx= eval_approx_df(xj, n_inner_neurons, *params_current);
// mem = (eval_df(xj) - approx);
// error = 2.0 * mem / train_size;
// for(i = 0; i < n_inner_neurons; ++i){
// if(opti_w){
// (*gradient_current)[3 * i] -= error * (eval_approx_dw_df(xj, i, *params_current));
// (*gradient_current)[3 * i + 1] -= error * (eval_approx_da_df(xj, i, *params_current));
// }
// if(opti_b){
// (*gradient_current)[3 * i + 2] -= error * (eval_approx_db_df(xj, i, *params_current));
// }
// }
// total_error += mem * mem / train_size;
// }
// /* error of the unknown function: y''(x) = 4e^(-2x)*(3x-2) => e3 = 4/n * (e^(-2x)*(3x-2) - y''(x))^2 */
// for(j = 0; j < data_points.size(); ++j) {
// xj = data_points[i];
// approx= eval_approx_ddf(xj, n_inner_neurons, *params_current);
// mem = (eval_ddf(xj) - approx);
// error = 2.0 * mem / train_size;
// for(i = 0; i < n_inner_neurons; ++i){
// if(opti_w){
// (*gradient_current)[3 * i] -= error * (eval_approx_dw_ddf(xj, i, *params_current));
// (*gradient_current)[3 * i + 1] -= error * (eval_approx_da_ddf(xj, i, *params_current));
// }
// if(opti_b){
// (*gradient_current)[3 * i + 2] -= error * (eval_approx_db_ddf(xj, i, *params_current));
// }
// }
//// printf("x: %f -> error: %f\n", xj, error);
// total_error += mem * mem / train_size;
// }
// for(auto e: *gradient_current){
// printf("[%10.8f]", e);
// }
// printf("\n");
/* conjugate direction coefficient (Polak-Ribiere): <-grad_curr, -grad_curr + grad_prev> / <-grad_prev, -grad_prev >*/
/* Update of the parameters */
if(iter_idx < 10 || iter_idx % 1 == 0){
for(i = 0; i < conjugate_direction_current->size(); ++i){
(*conjugate_direction_current)[i] = - (*gradient_current)[i];
}
}
else{
/* conjugate gradient */
sk = sy = 0.0;
for(i = 0; i < conjugate_direction_current->size(); ++i){
sk += (*gradient_current)[i] * ((*gradient_current)[i] - (*gradient_prev)[i]);
sy += (*gradient_prev)[i] * (*gradient_prev)[i];
}
beta = std::max(0.0, sk / sy);
/* update of the conjugate direction */
for(i = 0; i < conjugate_direction_current->size(); ++i){
(*conjugate_direction_current)[i] = beta * (*conjugate_direction_prev)[i] - (*gradient_current)[i];
}
}
/* step length calculation */
if(iter_idx < 10){
/* fixed step length */
gamma = 0.000001;
}
else{
// /* Barzilai-Borwein */
// sk = sy = 0.0;
//
// for(i = 0; i < gradient_current->size(); ++i){
// sx = (*params_current)[i] - (*params_prev)[i];
// sg = (*conjugate_direction_current)[i] - (*conjugate_direction_prev)[i];
//
// sk += sx * sg;
// sy += sg * sg;
// }
//
// gamma = -sk / sy;
// /* Line search */
//
// gamma = line_search(10.0, conjugate_direction, *params_current, *gradient_current, n_inner_neurons, ds_00);
}
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// for(auto e: conjugate_direction){
// printf("[%10.8f]", e);
// }
// printf("\n");
//
// for(auto e: *params_current){
// printf("[%10.8f]", e);
// }
// printf("\n");
/* norm of the gradient calculation */
grad_norm_prev = grad_norm;
/* adaptive step-length */
sk = 0.0;
for(i = 0; i < gradient_current->size(); ++i){
sx = (*gradient_current)[i] - (*gradient_prev)[i];
sk += sx * sx;
}
sk = std::sqrt(sk);
if(sk <= 1e-3 || grad_norm < grad_norm_prev){
/* movement on a line */
/* new slope is less steep, speed up */
gamma *= 1.0005;
}
else if(grad_norm > grad_norm_prev){
/* new slope is more steep, slow down*/
gamma /= 1.0005;
}
else{
gamma /= 1.005;
}
grad_norm = 0.0;
for(auto v: *gradient_current){
grad_norm += v * v;
}
grad_norm = std::sqrt(grad_norm);
for(i = 0; i < gradient_current->size(); ++i){
// (*params_prev)[i] = (*params_current)[i] - gamma * (*gradient_current)[i];
(*params_prev)[i] = (*params_current)[i] + gamma * (*conjugate_direction_current)[i];
}
// printf("\n");
/* switcheroo */
ptr_mem = gradient_prev;
gradient_prev = gradient_current;
gradient_current = ptr_mem;
ptr_mem = params_prev;
params_prev = params_current;
params_current = ptr_mem;
ptr_mem = conjugate_direction_prev;
conjugate_direction_prev = conjugate_direction_current;
conjugate_direction_current = ptr_mem;
// for (i = 0; i < n_inner_neurons; ++i) {
// wi = (*params_current)[3 * i];
// ai = (*params_current)[3 * i + 1];
// bi = (*params_current)[3 * i + 2];
//
// printf("Path %3d. w = %15.8f, b = %15.8f, a = %15.8f\n", i + 1, wi, bi, ai);
// }
if(iter_idx % 100 == 0){
printf("Iteration %12d. Step size: %15.8f, Gradient norm: %15.8f. Total error: %10.8f (%10.8f)\n", (int)iter_idx, gamma, grad_norm, total_error, eval_error_function(*params_prev, n_inner_neurons, data_points));
// for (i = 0; i < n_inner_neurons; ++i) {
// wi = (*params_current)[3 * i];
// ai = (*params_current)[3 * i + 1];
// bi = (*params_current)[3 * i + 2];
//
// printf("Path %3d. w = %15.8f, b = %15.8f, a = %15.8f\n", i + 1, wi, bi, ai);
// }
// std::cout << "-----------------------------" << std::endl;
std::cout.flush();
}
printf("Iteration %12d. Step size: %15.8f, Gradient norm: %15.8f. Total error: %10.8f\r\n",(int) iter_idx, gamma, grad_norm, eval_error_function(*params_current, n_inner_neurons, data_points));
std::cout.flush();
for (i = 0; i < n_inner_neurons; ++i) {
wi = (*params_current)[3 * i];
ai = (*params_current)[3 * i + 1];
bi = (*params_current)[3 * i + 2];
printf("Path %3d. w = %15.8f, b = %15.8f, a = %15.8f\n", (int)(i + 1), wi, bi, ai);
}
printf("\n--------------------------------------------------\ntest output for gnuplot\n--------------------------------------------------\n");
if(total_error < 1e-3 || true){
/* ISOTROPIC TEST SET */
frac = (te - ts) / (test_size - 1);
for(j = 0; j < test_size; ++j){
xj = frac * j + ts;
std::cout << j + 1 << " " << xj << " " << eval_f(xj) << " " << eval_approx_f(xj, n_inner_neurons, *params_current) << " " << eval_df(xj) << " " << eval_approx_df(xj, n_inner_neurons, *params_current) << " " << eval_ddf(xj) << " " << eval_approx_ddf(xj, n_inner_neurons, *params_current) << std::endl;
}
}
/* error analysis */
double referential_error = 0.0;
mem = eval_approx_df(0.0, n_inner_neurons, *params_current);
referential_error += (mem - 1.0) * (mem - 1.0);
mem = eval_approx_f(0.0, n_inner_neurons, *params_current);
referential_error += (mem - 1.0) * (mem - 1.0);
frac = 1.0 / train_size;
for(j = 0; j < data_points.size(); ++j){
xj = data_points[j];
mem = 4.0 * eval_approx_f(xj, n_inner_neurons, *params_current) + 4.0 * eval_approx_df(xj, n_inner_neurons, *params_current) + eval_approx_ddf(xj, n_inner_neurons, *params_current);
referential_error += mem * mem * frac;
printf("Total error (as used in the NN example): %10.8f\n", referential_error);
delete gradient_current;
delete gradient_prev;
delete params_current;
delete params_prev;
delete conjugate_direction_current;
delete conjugate_direction_prev;
void test_odr(size_t n_inner_neurons){
size_t n_inputs = 1;
size_t n_equations = 3;
DESolver solver_01( n_equations, n_inputs, n_inner_neurons );
Michal Kravcenko
committed
/* SETUP OF THE EQUATIONS */
MultiIndex alpha_0( n_inputs );
MultiIndex alpha_1( n_inputs );
MultiIndex alpha_2( n_inputs );
alpha_2.set_partial_derivative(0, 2);
alpha_1.set_partial_derivative(0, 1);
/* the governing differential equation */
solver_01.add_to_differential_equation( 0, alpha_2, 1.0 );
solver_01.add_to_differential_equation( 0, alpha_1, 4.0 );
solver_01.add_to_differential_equation( 0, alpha_0, 4.0 );
/* dirichlet boundary condition */
solver_01.add_to_differential_equation( 1, alpha_0, 1.0 );
/* neumann boundary condition */
solver_01.add_to_differential_equation( 2, alpha_1, 1.0 );
/* SETUP OF THE TRAINING DATA */
std::vector<double> inp, out;
Michal Kravcenko
committed
/* TRAIN DATA FOR THE GOVERNING DE */
std::vector<std::pair<std::vector<double>, std::vector<double>>> data_vec_g;
/* ISOTROPIC TRAIN SET */
frac = (d1_e - d1_s) / (train_size - 1);
for(unsigned int i = 0; i < train_size; ++i){
inp = {frac * i};
out = {0.0};
data_vec_g.emplace_back(std::make_pair(inp, out));
}
/* CHEBYSCHEV TRAIN SET */
// alpha = PI / (train_size - 1);
// frac = 0.5 * (d1_e - d1_s);
// for(unsigned int i = 0; i < train_size; ++i){
// inp = {(std::cos(alpha * i) + 1.0) * frac + d1_s};
// out = {0.0};
// data_vec_g.emplace_back(std::make_pair(inp, out));
// }
Michal Kravcenko
committed
DataSet ds_00(&data_vec_g);
/* TRAIN DATA FOR DIRICHLET BC */
std::vector<std::pair<std::vector<double>, std::vector<double>>> data_vec_y;
inp = {0.0};
out = {1.0};
data_vec_y.emplace_back(std::make_pair(inp, out));
Michal Kravcenko
committed
DataSet ds_01(&data_vec_y);
/* TRAIN DATA FOR NEUMANN BC */
std::vector<std::pair<std::vector<double>, std::vector<double>>> data_vec_dy;
inp = {0.0};
out = {1.0};
data_vec_dy.emplace_back(std::make_pair(inp, out));
Michal Kravcenko
committed
DataSet ds_02(&data_vec_dy);
Michal Kravcenko
committed
/* Placing the conditions into the solver */
solver_01.set_error_function( 0, ErrorFunctionType::ErrorFuncMSE, &ds_00 );
solver_01.set_error_function( 1, ErrorFunctionType::ErrorFuncMSE, &ds_01 );
solver_01.set_error_function( 2, ErrorFunctionType::ErrorFuncMSE, &ds_02 );
Michal Kravcenko
committed
/* PARTICLE SWARM TRAINING METHOD SETUP */
//must encapsulate each of the partial error functions
double *domain_bounds = new double[ 6 * n_inner_neurons ];
for(unsigned int i = 0; i < 3 * n_inner_neurons; ++i){
domain_bounds[2 * i] = -800.0;
domain_bounds[2 * i + 1] = 800.0;
}
Michal Kravcenko
committed
double gamma = 0.5, epsilon = 0.02, delta = 0.9;
Michal Kravcenko
committed
solver_01.solve_via_particle_swarm( domain_bounds, c1, c2, w, n_particles, max_iters, gamma, epsilon, delta );
Michal Kravcenko
committed
NeuralNetwork *solution = solver_01.get_solution();
std::vector<double> parameters(3 * n_inner_neurons);//w1, a1, b1, w2, a2, b2, ... , wm, am, bm
std::vector<double> *weight_params = solution->get_parameter_ptr_weights();
std::vector<double> *biases_params = solution->get_parameter_ptr_biases();
for(size_t i = 0; i < n_inner_neurons; ++i){
parameters[3 * i] = weight_params->at(i);
parameters[3 * i + 1] = weight_params->at(i + n_inner_neurons);
parameters[3 * i + 2] = biases_params->at(i);
}
for(unsigned int i = 0; i < n_test_points; ++i){
x = i * ((d1_e - d1_s) / (n_test_points - 1)) + d1_s;
input[0] = x;
std::cout << i + 1 << " " << x << " " << std::pow(E, -2*x) * (3*x + 1)<< " " << output[0] << " " << std::pow(E, -2*x) * (1 - 6*x)<< " " << eval_approx_df(x, n_inner_neurons, parameters) << " " << 4 * std::pow(E, -2*x) * (3*x - 2)<< " " << eval_approx_ddf(x, n_inner_neurons, parameters) << std::endl;
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// bool optimize_weights = true;
// bool optimize_biases = true;
// unsigned int train_size = 150;
// double accuracy = 1e-5;
// double ds = 0.0;
// double de = 4.0;
//
// unsigned int test_size = 300;
// double ts = ds;
// double te = de;
//
//// std::vector<double> init_guess = {0.35088209, -0.23738505, 0.14160885, 3.72785473, -6.45758308, 1.73769138};
// std::vector<double> init_guess(3 * n_inner_neurons);
//
// std::random_device seeder;
// std::mt19937 gen(seeder());
// std::uniform_real_distribution<double> dist(-1.0, 1.0);
// for(unsigned int i = 0; i < 3 * n_inner_neurons; ++i){
// init_guess[i] = dist(gen);
// init_guess[i] = dist(gen);
// }
// if(!optimize_biases){
// for(unsigned int i = 0; i < n_inner_neurons; ++i){
// init_guess[3 * i + 2] = 0.0;
// }
// }
// if(!optimize_weights){
// for(unsigned int i = 0; i < n_inner_neurons; ++i){
// init_guess[3 * i] = 0.0;
// init_guess[3 * i + 1] = 0.0;
// }
// }
//
// test_analytical_gradient_y(init_guess, accuracy, n_inner_neurons, train_size, optimize_weights, optimize_biases, ds, de, test_size, ts, te);