def VisuError(Error, testcases, Savetofile=False):
name = testcases.name
Vals = testcases.Vals
symbols = ('^','o', 's','D','p','H',)
lines = ('-', '--',':', '-.','.',)
newfig(width=1.3)
for i in range(len(Vals)):
namestr = ', ' + '$N_s$=' + str(Vals[i])
if name != 'SampleNum' and i ==0:
plt.semilogy(M_Vec,Error[i,:,4], 'k-.',label='Projection' )
plt.semilogy(M_Vec,Error[i,:,0], 'r-.',label='POD-G ' )
elif name == 'SampleNum':
plt.semilogy(M_Vec,Error[i,:,4], 'k-.'+symbols[i] ,label='Projection'+namestr )
plt.semilogy(M_Vec,Error[i,:,0], 'r-.'+symbols[i] ,label='POD-G' +namestr )
if not( name == 'NResi' ):
plt.semilogy(M_Vec,Error[i,:,1], 'y:'+symbols[i] ,label='PDNN' +namestr )
plt.semilogy(M_Vec,Error[i,:,2], 'g--'+symbols[i] ,label='PINN'+namestr )
plt.semilogy(M_Vec,Error[i,:,3], 'g-'+symbols[i] ,label='PRNN'+namestr )
plt.xlabel('$m$')
plt.ylabel('Error')
#plt.title(name)
plt.legend(loc="lower left", ncol=1, handlelength=3)
plt.show()
if Savetofile:
savefig("fig/ErrorComparsion_"+name)
def VisuError(Error, testcases, Savetofile=False):
name = testcases.name
Vals = testcases.Vals
symbols = ('^','o', 's','D','p','H',)
lines = ('-', '--',':', '-.','.',)
#plt.close('all')i
if name == 'SampleNum':
tmp = '$N_s$'
width = 1.3
elif name =='NetSize':
tmp = '$n_H$'
width = 1
elif name == 'NResi':
tmp = '$N_{Resi}$'
width = 1
newfig(width=width)
for i in range(len(Vals)):
namestr = ', ' + tmp+'=' + str(Vals[i])
if name != 'SampleNum' and i ==0:
plt.semilogy(M_Vec,Error[i,:,4], 'k-.',label='Projection' )
plt.semilogy(M_Vec,Error[i,:,0], 'r-.',label='POD-G ' )
elif name == 'SampleNum':
plt.semilogy(M_Vec,Error[i,:,4], 'k-.'+symbols[i] ,label='Projection'+namestr )
plt.semilogy(M_Vec,Error[i,:,0], 'r-.'+symbols[i] ,label='POD-G' +namestr )
if not( name == 'NResi' ):
plt.semilogy(M_Vec,Error[i,:,1], 'y:'+symbols[i] ,label='PDNN' +namestr )
plt.semilogy(M_Vec,Error[i,:,2], 'g--'+symbols[i] ,label='PINN'+namestr )
plt.semilogy(M_Vec,Error[i,:,3], 'g-'+symbols[i] ,label='PRNN'+namestr )
if name == 'NResi':
plt.ylim(bottom=9E-5)
elif name == 'NetSize':
plt.ylim(bottom=7E-5)
plt.xlabel('$m$')
plt.ylabel(r'Error $\varepsilon$')
#plt.title(name)
plt.legend(loc="lower left", ncol=1, handlelength=3)
if name == 'NResi':
plt.legend(loc="best", ncol=1, handlelength=3)
plt.show()
if Savetofile:
savefig("fig/ErrorComparsion_"+name)
np.random.choice(np.arange(1, N - 1), size=N_eqns - 2, replace=False),
np.array([N - 1])
])
model = HiddenPathways(t_train[idx_data], S_train[idx_data, :],
t_train[idx_eqns], layers)
model.train(num_epochs=20000, batch_size=N_eqns, learning_rate=1e-3)
model.train(num_epochs=40000, batch_size=N_eqns, learning_rate=1e-4)
model.train(num_epochs=20000, batch_size=N_eqns, learning_rate=1e-5)
S_pred, S_pred_std = model.predict(t_star[:, None])
####### Plotting ##################
fig, ax = newfig(3.0, 0.3)
gs0 = gridspec.GridSpec(1, 2)
gs0.update(top=0.95,
bottom=0.1,
left=0.1,
right=0.95,
hspace=0.5,
wspace=0.3)
ax = plt.subplot(gs0[0:1, 0:1])
ax.plot(t_star, S_star[:, 4], 'C1', linewidth=2, label='input data')
ax.scatter(t_star[idx_data],
S_star[idx_data, 4],
marker='o',
s=50,
label='sampled input')
ax.set_xlabel('$t\ (min)$', fontsize=18)
M = 1
scheme = 'AM'
model = Multistep_NN(dt, X_train, layers, M, scheme)
N_Iter = 50000
model.train(N_Iter)
def learned_f(x, t):
f = model.predict_f(x[None, :])
return f.flatten()
learned_X_star = odeint(learned_f, x0, t_star)
####### Plotting ##################
fig, ax = newfig(1.0, 0.8)
ax.axis('off')
gs0 = gridspec.GridSpec(1, 2)
gs0.update(top=0.95, bottom=0.1, left=0.0, right=0.90, wspace=0.15)
ax = plt.subplot(gs0[:, 0:1], projection='3d')
ax.w_xaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))
ax.w_yaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))
ax.w_zaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))
colorline3d(ax,
X_star[:, 0],
X_star[:, 1],
X_star[:, 2],
cmap=plt.cm.ocean)
ax.grid(False)
def plot_pressure(p_pred, p_star, X_star, epoch, learning_rate):
# Predict for plotting
lb = X_star.min(0)
ub = X_star.max(0)
nn = 200
x = np.linspace(lb[0], ub[0], nn)
y = np.linspace(lb[1], ub[1], nn)
X, Y = np.meshgrid(x, y)
PP_star = griddata(X_star,
np.ndarray.flatten(p_pred.numpy()), (X, Y),
method='cubic')
P_exact = griddata(X_star,
np.ndarray.flatten(p_star), (X, Y),
method='cubic')
x_star = X_star[:, 0:1]
y_star = X_star[:, 1:2]
fig, ax = newfig(1.015, 0.8)
ax.axis('off')
######## Predicted p(t,x,y) ###########
gs2 = gridspec.GridSpec(1, 2)
gs2.update(top=1, bottom=1 - 1 / 2, left=0.1, right=0.9, wspace=0.5)
ax = plt.subplot(gs2[:, 0])
h = ax.imshow(
PP_star,
interpolation='nearest',
cmap='rainbow',
extent=[x_star.min(),
x_star.max(),
y_star.min(),
y_star.max()],
origin='lower',
aspect='auto')
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(h, cax=cax)
ax.set_xlabel('$x$')
ax.set_ylabel('$y$')
ax.set_aspect('equal', 'box')
ax.set_title('Predicted pressure', fontsize=10)
######## Exact p(t,x,y) ###########
ax = plt.subplot(gs2[:, 1])
h = ax.imshow(
P_exact,
interpolation='nearest',
cmap='rainbow',
extent=[x_star.min(),
x_star.max(),
y_star.min(),
y_star.max()],
origin='lower',
aspect='auto')
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(h, cax=cax)
ax.set_xlabel('$x$')
ax.set_ylabel('$y$')
ax.set_aspect('equal', 'box')
ax.set_title('Exact pressure', fontsize=10)
savefig('./figures/NavierStokes_prediction ' + str(learning_rate) + ' ' +
str(epoch))
plt.close()
pass
def plot_inf_disc_results(x_star,
idx_t_0,
idx_t_1,
x_0,
u_0,
ub,
lb,
u_1_pred,
Exact_u,
x,
t,
save_path=None,
save_hp=None):
fig, ax = newfig(1.0, 1.2)
ax.axis('off')
####### Row 0: h(t,x) ##################
gs0 = gridspec.GridSpec(1, 2)
gs0.update(top=1 - 0.06,
bottom=1 - 1 / 2 + 0.1,
left=0.15,
right=0.85,
wspace=0)
ax = plt.subplot(gs0[:, :])
h = ax.imshow(Exact_u.T,
interpolation='nearest',
cmap='rainbow',
extent=[t.min(),
t.max(),
x_star.min(),
x_star.max()],
origin='lower',
aspect='auto')
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(h, cax=cax)
line = np.linspace(x.min(), x.max(), 2)[:, None]
ax.plot(t[idx_t_0] * np.ones((2, 1)), line, 'w-', linewidth=1)
ax.plot(t[idx_t_1] * np.ones((2, 1)), line, 'w-', linewidth=1)
ax.set_xlabel('$t$')
ax.set_ylabel('$x$')
leg = ax.legend(frameon=False, loc='best')
ax.set_title('$u(t,x)$', fontsize=10)
####### Row 1: h(t,x) slices ##################
gs1 = gridspec.GridSpec(1, 2)
gs1.update(top=1 - 1 / 2 - 0.05,
bottom=0.15,
left=0.15,
right=0.85,
wspace=0.5)
ax = plt.subplot(gs1[0, 0])
ax.plot(x, Exact_u[idx_t_0, :], 'b-', linewidth=2)
ax.plot(x_0, u_0, 'rx', linewidth=2, label='Data')
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.set_title('$t = %.2f$' % (t[idx_t_0]), fontsize=10)
ax.set_xlim([lb - 0.1, ub + 0.1])
ax.legend(loc='upper center',
bbox_to_anchor=(0.8, -0.3),
ncol=2,
frameon=False)
ax = plt.subplot(gs1[0, 1])
ax.plot(x, Exact_u[idx_t_1, :], 'b-', linewidth=2, label='Exact')
ax.plot(x_star, u_1_pred, 'r--', linewidth=2, label='Prediction')
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.set_title('$t = %.2f$' % (t[idx_t_1]), fontsize=10)
ax.set_xlim([lb - 0.1, ub + 0.1])
ax.legend(loc='upper center',
bbox_to_anchor=(0.1, -0.3),
ncol=2,
frameon=False)
if save_path != None and save_hp != None:
saveResultDir(save_path, save_hp)
else:
plt.show()
return self.loss_Eqs(x, self.u_net(x), source, weight)
def loss_Eqs(self, x, lamda, source, weight=1):
# def loss_PINN(self,x,source):
#lamda = self.u_net(x);
fx = torch.matmul(lamda[:, None, None, :], self.A[None, :, :, :])
fx = torch.matmul(fx, lamda[:, None, :, None])
fx = fx.view(lamda.shape)
fx = fx + torch.matmul(lamda, self.B.T) - source
return self.lossfun(weight * fx, torch.zeros_like(fx))
if __name__ == '__main__':
NumSolsdir = 'NumSols'
Nsample = 80
matfile = NumSolsdir + '/' + 'Burges1D_SampleNum=' + str(Nsample) + '.mat'
M = 2
roeqs = CustomedEqs(matfile, M)
# Net = CustomedNet(roeqs=roeqs,layers=[2,20,20,20,M])
# print(Net.labeledLoss)
from plotting import newfig, savefig
import matplotlib.pyplot as plt
newfig(width=0.8)
plt.semilogy(np.arange(roeqs.sigma.shape[0]) + 1, roeqs.sigma, '-ko')
plt.xlabel('$m$')
plt.ylabel('Singular value')
plt.show()
savefig('fig/SingularValues_%d' % (Nsample))
lines = (
'-',
'--',
':',
'-.',
'.',
)
colors = (
'r',
'g',
'b',
'g',
'y',
)
plt.close('all')
newfig(width=1)
for k in range(2):
for i in range(alpha.shape[0]):
alphai = alpha[i:i + 1, :]
lamdai = Net(torch.tensor(alphai).float().to(DEVICE))
phi_proj = np.matmul(lamdai, roeqs.Modes.T)
phi_Exact = roeqs.phix(roeqs.xgrid.T, alphai[:, 0:1], alphai[:, 1:2])
#plot
name = '$\\boldsymbol{\\mu}=(%0.1f,%0.1f)$' % (alphai[0, 0], alphai[0,
1])
if k == 0:
plt.plot(roeqs.xgrid,
phi_Exact.T,
colors[i] + lines[0],
label='PS',
data_vort = scipy.io.loadmat('../Data/cylinder_nektar_t0_vorticity.mat')
x_vort = data_vort['x']
y_vort = data_vort['y']
w_vort = data_vort['w']
modes = np.asscalar(data_vort['modes'])
nel = np.asscalar(data_vort['nel'])
xx_vort = np.reshape(x_vort, (modes + 1, modes + 1, nel), order='F')
yy_vort = np.reshape(y_vort, (modes + 1, modes + 1, nel), order='F')
ww_vort = np.reshape(w_vort, (modes + 1, modes + 1, nel), order='F')
box_lb = np.array([1.0, -2.0])
box_ub = np.array([8.0, 2.0])
fig, ax = newfig(1.0, 1.2)
ax.axis('off')
####### Row 0: Vorticity ##################
gs0 = gridspec.GridSpec(1, 2)
gs0.update(top=1 - 0.06,
bottom=1 - 2 / 4 + 0.12,
left=0.0,
right=1.0,
wspace=0)
ax = plt.subplot(gs0[:, :])
for i in range(0, nel):
h = ax.pcolormesh(xx_vort[:, :, i],
yy_vort[:, :, i],
ww_vort[:, :, i],
def plot_inf_cont_results(X_star, u_pred, v_pred, h_pred, Exact_h, X, T, x, t, ub, lb, x0, tb, save_path=None, save_hp=None):
# Interpolating the results on the whole (x,t) domain.
# griddata(points, values, points at which to interpolate, method)
U_pred = griddata(X_star, u_pred.flatten(), (X, T), method='cubic')
V_pred = griddata(X_star, v_pred.flatten(), (X, T), method='cubic')
H_pred = griddata(X_star, h_pred.flatten(), (X, T), method='cubic')
# FU_pred = griddata(X_star, f_u_pred.flatten(), (X, T), method='cubic')
# FV_pred = griddata(X_star, f_v_pred.flatten(), (X, T), method='cubic')
X0 = np.concatenate((x0, 0*x0), 1) # (x0, 0)
X_lb = np.concatenate((0*tb + lb[0], tb), 1) # (lb[0], tb)
X_ub = np.concatenate((0*tb + ub[0], tb), 1) # (ub[0], tb)
X_u_train = np.vstack([X0, X_lb, X_ub])
fig, ax = newfig(1.0, 0.9)
ax.axis('off')
####### Row 0: h(t,x) ##################
gs0 = gridspec.GridSpec(1, 2)
gs0.update(top=1-0.06, bottom=1-1/3, left=0.15, right=0.85, wspace=0)
ax = plt.subplot(gs0[:, :])
h = ax.imshow(H_pred.T, interpolation='nearest', cmap='YlGnBu',
extent=[lb[1], ub[1], lb[0], ub[0]],
origin='lower', aspect='auto')
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(h, cax=cax)
ax.plot(X_u_train[:,1], X_u_train[:,0], 'kx', label = 'Data (%d points)' % (X_u_train.shape[0]), markersize = 4, clip_on = False)
line = np.linspace(x.min(), x.max(), 2)[:,None]
ax.plot(t[75]*np.ones((2,1)), line, 'k--', linewidth = 1)
ax.plot(t[100]*np.ones((2,1)), line, 'k--', linewidth = 1)
ax.plot(t[125]*np.ones((2,1)), line, 'k--', linewidth = 1)
ax.set_xlabel('$t$')
ax.set_ylabel('$x$')
leg = ax.legend(frameon=False, loc = 'best')
# plt.setp(leg.get_texts(), color='w')
ax.set_title('$|h(t,x)|$', fontsize = 10)
####### Row 1: h(t,x) slices ##################
gs1 = gridspec.GridSpec(1, 3)
gs1.update(top=1-1/3, bottom=0, left=0.1, right=0.9, wspace=0.5)
ax = plt.subplot(gs1[0, 0])
ax.plot(x,Exact_h[:,75], 'b-', linewidth = 2, label = 'Exact')
ax.plot(x,H_pred[75,:], 'r--', linewidth = 2, label = 'Prediction')
ax.set_xlabel('$x$')
ax.set_ylabel('$|h(t,x)|$')
ax.set_title('$t = %.2f$' % (t[75]), fontsize = 10)
ax.axis('square')
ax.set_xlim([-5.1,5.1])
ax.set_ylim([-0.1,5.1])
ax = plt.subplot(gs1[0, 1])
ax.plot(x,Exact_h[:,100], 'b-', linewidth = 2, label = 'Exact')
ax.plot(x,H_pred[100,:], 'r--', linewidth = 2, label = 'Prediction')
ax.set_xlabel('$x$')
ax.set_ylabel('$|h(t,x)|$')
ax.axis('square')
ax.set_xlim([-5.1,5.1])
ax.set_ylim([-0.1,5.1])
ax.set_title('$t = %.2f$' % (t[100]), fontsize = 10)
ax.legend(loc='upper center', bbox_to_anchor=(0.5, -0.8), ncol=5, frameon=False)
ax = plt.subplot(gs1[0, 2])
ax.plot(x,Exact_h[:,125], 'b-', linewidth = 2, label = 'Exact')
ax.plot(x,H_pred[125,:], 'r--', linewidth = 2, label = 'Prediction')
ax.set_xlabel('$x$')
ax.set_ylabel('$|h(t,x)|$')
ax.axis('square')
ax.set_xlim([-5.1,5.1])
ax.set_ylim([-0.1,5.1])
ax.set_title('$t = %.2f$' % (t[125]), fontsize = 10)
if save_path != None and save_hp != None:
saveResultDir(save_path, save_hp)
else:
plt.show()
def plot_inf_cont_results(X_star,
U_pred,
Sigma_pred,
X_u_train,
u_train,
Exact_u,
X,
T,
x,
t,
save_path=None,
save_hp=None):
# Interpolating the results on the whole (x,t) domain.
# griddata(points, values, points at which to interpolate, method)
# U_pred = griddata(X_star, u_pred, (X, T), method='cubic')
# Creating the figures
fig, ax = newfig(1.0, 1.1)
ax.axis('off')
X_u = X_u_train[:, 0:1]
T_u = X_u_train[:, 1:2]
Y_u = u_train
Exact = Exact_u
####### Row 0: u(t,x) ##################
gs0 = gridspec.GridSpec(1, 2)
gs0.update(top=1 - 0.06, bottom=1 - 1 / 3, left=0.15, right=0.85, wspace=0)
ax = plt.subplot(gs0[:, :])
h = ax.imshow(U_pred.T,
interpolation='nearest',
cmap='rainbow',
extent=[t.min(), t.max(), x.min(),
x.max()],
origin='lower',
aspect='auto')
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(h, cax=cax)
ax.plot(T_u,
X_u,
'kx',
label='Data (%d points)' % (Y_u.shape[0]),
markersize=4,
clip_on=False)
line = np.linspace(x.min(), x.max(), 2)[:, None]
ax.plot(t[25] * np.ones((2, 1)), line, 'w-', linewidth=1)
ax.plot(t[50] * np.ones((2, 1)), line, 'w-', linewidth=1)
ax.plot(t[75] * np.ones((2, 1)), line, 'w-', linewidth=1)
ax.set_xlabel('$t$')
ax.set_ylabel('$x$')
ax.legend(frameon=False, loc='best')
ax.set_title('$u(t,x)$', fontsize=10)
####### Row 1: u(t,x) slices ##################
gs1 = gridspec.GridSpec(1, 3)
gs1.update(top=1 - 1 / 3, bottom=0, left=0.1, right=0.9, wspace=0.5)
ax = plt.subplot(gs1[0, 0])
ax.plot(x, Exact[25, :], 'b-', linewidth=2, label='Exact')
ax.plot(x, U_pred[25, :], 'r--', linewidth=2, label='Prediction')
lower = U_pred[25, :] - 2.0 * np.sqrt(Sigma_pred[25, :])
upper = U_pred[25, :] + 2.0 * np.sqrt(Sigma_pred[25, :])
plt.fill_between(x.flatten(),
lower.flatten(),
upper.flatten(),
facecolor='orange',
alpha=0.5,
label="Two std band")
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.set_title('$t = 0.25$', fontsize=10)
ax.axis('square')
ax.set_xlim([-1.1, 1.1])
ax.set_ylim([-1.1, 1.1])
ax = plt.subplot(gs1[0, 1])
ax.plot(x, Exact[50, :], 'b-', linewidth=2, label='Exact')
ax.plot(x, U_pred[50, :], 'r--', linewidth=2, label='Prediction')
lower = U_pred[50, :] - 2.0 * np.sqrt(Sigma_pred[50, :])
upper = U_pred[50, :] + 2.0 * np.sqrt(Sigma_pred[50, :])
plt.fill_between(x.flatten(),
lower.flatten(),
upper.flatten(),
facecolor='orange',
alpha=0.5,
label="Two std band")
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.axis('square')
ax.set_xlim([-1.1, 1.1])
ax.set_ylim([-1.1, 1.1])
ax.set_title('$t = 0.50$', fontsize=10)
ax.legend(loc='upper center',
bbox_to_anchor=(0.5, -0.35),
ncol=5,
frameon=False)
ax = plt.subplot(gs1[0, 2])
ax.plot(x, Exact[75, :], 'b-', linewidth=2, label='Exact')
ax.plot(x, U_pred[75, :], 'r--', linewidth=2, label='Prediction')
lower = U_pred[75, :] - 2.0 * np.sqrt(Sigma_pred[75, :])
upper = U_pred[75, :] + 2.0 * np.sqrt(Sigma_pred[75, :])
plt.fill_between(x.flatten(),
lower.flatten(),
upper.flatten(),
facecolor='orange',
alpha=0.5,
label="Two std band")
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.axis('square')
ax.set_xlim([-1.1, 1.1])
ax.set_ylim([-1.1, 1.1])
ax.set_title('$t = 0.75$', fontsize=10)
# savefig('./Prediction')
# fig, ax = newfig(1.0)
# ax.axis('off')
# ############# Uncertainty ##################
# gs2 = gridspec.GridSpec(1, 2)
# gs2.update(top=1-0.06, bottom=1-1/3, left=0.15, right=0.85, wspace=0)
# ax = plt.subplot(gs2[:, :])
# h = ax.imshow(Sigma_pred.T, interpolation='nearest', cmap='rainbow',
# extent=[t.min(), t.max(), x.min(), x.max()],
# origin='lower', aspect='auto')
# divider = make_axes_locatable(ax)
# cax = divider.append_axes("right", size="5%", pad=0.05)
# fig.colorbar(h, cax=cax)
# ax.set_xlabel('$t$')
# ax.set_ylabel('$x$')
# ax.legend(frameon=False, loc = 'best')
# ax.set_title('Variance of $u(t,x)$', fontsize = 10)
if save_path != None and save_hp != None:
saveResultDir(save_path, save_hp)
else:
plt.show()
def main():
## parameters
device = torch.device(f"cpu")
epochs = 5000
lr = 0.001
num_x = 100
num_y = 100
num_t = 10
num_b_train = 10 # boundary sampling points
num_f_train = 100 # inner sampling points
num_i_train = 100 # initial sampling points
x = np.linspace(0, 1, num=num_x)
y = np.linspace(0, 1, num=num_y)
t = np.linspace(0, 1, num=num_t)
x_grid, y_grid = np.meshgrid(x, y)
# x_test = np.concatenate((t_grid.flatten()[:,None], x_grid.flatten()[:,None], y_grid.flatten()[:,None]), axis=1)
x_2d = np.concatenate(
(x_grid.flatten()[:, None], y_grid.flatten()[:, None]), axis=1)
xt_init = np.concatenate((np.zeros((num_x * num_y, 1)), x_2d), axis=1)
u_init = AC_2D_init(xt_init)[:, None]
x_2d_ext = np.tile(x_2d, [num_t, 1])
t_ext = np.tile(t[:, None], [num_x * num_y, 1])
xt_2d_ext = np.concatenate((t_ext, x_2d_ext), axis=1)
## find a smart way to take boundary point
x_up = np.vstack((x_grid[-1, :], y_grid[-1, :])).T
x_dw = np.vstack((x_grid[0, :], y_grid[0, :])).T
x_l = np.vstack((x_grid[:, 0], y_grid[:, 0])).T
x_r = np.vstack((x_grid[:, -1], y_grid[:, -1])).T
x_bound = np.vstack((x_up, x_dw, x_l, x_r))
x_bound_ext = np.tile(x_bound, [num_t, 1])
t_bound_ext = np.tile(t[:, None], [num_x * 4, 1])
xt_bound_ext = np.concatenate((t_bound_ext, x_bound_ext), axis=1)
u_bound_ext = -1 * np.ones((num_x * 4 * num_t))[:, None]
## sampling
id_f = np.random.choice(num_x * num_y * num_t, num_f_train)
id_b = np.random.choice(num_x * 4, num_b_train) ## Dirichlet
id_i = np.random.choice(num_x * num_y, num_i_train)
x_i = xt_init[id_i, :]
u_i = u_init[id_i, :]
x_f = xt_2d_ext[id_f, :]
x_b = xt_bound_ext[id_b, :]
u_b = u_bound_ext[id_b, :]
## set data as tensor and send to device
x_f_train = torch.tensor(x_f, requires_grad=True,
dtype=torch.float32).to(device)
x_b_train = torch.tensor(x_b, requires_grad=True,
dtype=torch.float32).to(device)
x_test = torch.tensor(xt_2d_ext, requires_grad=True,
dtype=torch.float32).to(device)
x_i_train = torch.tensor(x_i, dtype=torch.float32).to(device)
u_i_train = torch.tensor(u_i, dtype=torch.float32).to(device)
u_b_train = torch.tensor(u_b, dtype=torch.float32).to(device)
## instantiate model
model = Model().to(device)
# Loss and optimizer
optimizer = torch.optim.Adam(model.parameters(), lr=lr)
# training
def train(epoch):
model.train()
def closure():
optimizer.zero_grad()
loss_pde = model.loss_pde(x_f_train)
loss_bc = model.loss_bc(x_b_train, u_b_train)
loss_ic = model.loss_ic(x_i_train, u_i_train)
loss = loss_pde + loss_bc + loss_ic
loss.backward()
return loss
loss = optimizer.step(closure)
loss_value = loss.item() if not isinstance(loss, float) else loss
print(f'epoch {epoch}: loss {loss_value:.6f}')
print('start training...')
tic = time.time()
for epoch in range(1, epochs + 1):
train(epoch)
toc = time.time()
print(f'total training time: {toc-tic}')
## test
u_test = np.zeros((num_t, num_x, num_y))
for i in range(0, 6):
xt = np.concatenate((t[i] * np.ones((num_x * num_y, 1)), x_2d),
axis=1)[:, None]
xt_tensor = torch.tensor(xt, dtype=torch.float32).to(device)
u_test[i, :, :] = to_numpy(model(xt_tensor)).reshape(num_x, num_y)
x_test = to_numpy(x_test)
fig, ax = newfig(2.0, 1.1)
ax.axis('off')
gs0 = gridspec.GridSpec(1, 2)
gs0.update(top=1 - 0.06,
bottom=1 - 1 / 3,
left=0.15,
right=0.85,
wspace=0)
h = ax.imshow(u_test[i, :, :].T,
interpolation='nearest',
cmap='rainbow',
origin='lower',
aspect='auto')
fig.colorbar(h)
ax.plot(x_test[:, 1],
x_test[:, 2],
'kx',
label='Data (%d points)' % (x_test.shape[0]),
markersize=4,
clip_on=False)
line = np.linspace(x_test.min(), x_test.max(), 2)[:, None]
savefig('./u_test_' + str(i))
def plot_ide_cont_results(X_star, u_pred, X_u_train, u_train, Exact_u, X, T, x,
t, lambda_1_value, lambda_1_value_noisy,
lambda_2_value, lambda_2_value_noisy):
fig, ax = newfig(1.0, 1.4)
ax.axis('off')
U_pred = griddata(X_star, u_pred.flatten(), (X, T), method='cubic')
####### Row 0: u(t,x) ##################
gs0 = gridspec.GridSpec(1, 2)
gs0.update(top=1 - 0.06,
bottom=1 - 1.0 / 3.0 + 0.06,
left=0.15,
right=0.85,
wspace=0)
ax = plt.subplot(gs0[:, :])
h = ax.imshow(U_pred.T,
interpolation='nearest',
cmap='rainbow',
extent=[t.min(), t.max(), x.min(),
x.max()],
origin='lower',
aspect='auto')
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(h, cax=cax)
ax.plot(X_u_train[:, 1],
X_u_train[:, 0],
'kx',
label='Data (%d points)' % (u_train.shape[0]),
markersize=2,
clip_on=False)
line = np.linspace(x.min(), x.max(), 2)[:, None]
ax.plot(t[25] * np.ones((2, 1)), line, 'w-', linewidth=1)
ax.plot(t[50] * np.ones((2, 1)), line, 'w-', linewidth=1)
ax.plot(t[75] * np.ones((2, 1)), line, 'w-', linewidth=1)
ax.set_xlabel('$t$')
ax.set_ylabel('$x$')
ax.legend(loc='upper center',
bbox_to_anchor=(1.0, -0.125),
ncol=5,
frameon=False)
ax.set_title('$u(t,x)$', fontsize=10)
####### Row 1: u(t,x) slices ##################
gs1 = gridspec.GridSpec(1, 3)
gs1.update(top=1 - 1.0 / 3.0 - 0.1,
bottom=1.0 - 2.0 / 3.0,
left=0.1,
right=0.9,
wspace=0.5)
ax = plt.subplot(gs1[0, 0])
ax.plot(x, Exact_u[25, :], 'b-', linewidth=2, label='Exact')
ax.plot(x, U_pred[25, :], 'r--', linewidth=2, label='Prediction')
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.set_title('$t = 0.25$', fontsize=10)
ax.axis('square')
ax.set_xlim([-1.1, 1.1])
ax.set_ylim([-1.1, 1.1])
ax = plt.subplot(gs1[0, 1])
ax.plot(x, Exact_u[50, :], 'b-', linewidth=2, label='Exact')
ax.plot(x, U_pred[50, :], 'r--', linewidth=2, label='Prediction')
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.axis('square')
ax.set_xlim([-1.1, 1.1])
ax.set_ylim([-1.1, 1.1])
ax.set_title('$t = 0.50$', fontsize=10)
ax.legend(loc='upper center',
bbox_to_anchor=(0.5, -0.35),
ncol=5,
frameon=False)
ax = plt.subplot(gs1[0, 2])
ax.plot(x, Exact_u[75, :], 'b-', linewidth=2, label='Exact')
ax.plot(x, U_pred[75, :], 'r--', linewidth=2, label='Prediction')
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.axis('square')
ax.set_xlim([-1.1, 1.1])
ax.set_ylim([-1.1, 1.1])
ax.set_title('$t = 0.75$', fontsize=10)
####### Row 3: Identified PDE ##################
gs2 = gridspec.GridSpec(1, 3)
gs2.update(top=1.0 - 2.0 / 3.0, bottom=0, left=0.0, right=1.0, wspace=0.0)
ax = plt.subplot(gs2[:, :])
ax.axis('off')
s1 = r'$\begin{tabular}{ |c|c| } \hline Correct PDE & $u_t + u u_x - 0.0031831 u_{xx} = 0$ \\ \hline Identified PDE (clean data) & '
s2 = r'$u_t + %.5f u u_x - %.7f u_{xx} = 0$ \\ \hline ' % (lambda_1_value,
lambda_2_value)
s3 = r'Identified PDE (1\% noise) & '
s4 = r'$u_t + %.5f u u_x - %.7f u_{xx} = 0$ \\ \hline ' % (
lambda_1_value_noisy, lambda_2_value_noisy)
s5 = r'\end{tabular}$'
s = s1 + s2 + s3 + s4 + s5
ax.text(0.1, 0.1, s)
plt.show()
def plot_ide_disc_results(x_star,
t_star,
idx_t_0,
idx_t_1,
x_0,
u_0,
x_1,
u_1,
ub,
lb,
u_1_pred,
Exact,
lambda_1_value,
lambda_1_value_noisy,
lambda_2_value,
lambda_2_value_noisy,
x,
t,
save_path=None,
save_hp=None):
fig, ax = newfig(1.0, 1.5)
ax.axis('off')
gs0 = gridspec.GridSpec(1, 2)
gs0.update(top=1 - 0.06,
bottom=1 - 1 / 3 + 0.05,
left=0.15,
right=0.85,
wspace=0)
ax = plt.subplot(gs0[:, :])
h = ax.imshow(Exact,
interpolation='nearest',
cmap='rainbow',
extent=[t_star.min(),
t_star.max(), lb[0], ub[0]],
origin='lower',
aspect='auto')
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(h, cax=cax)
line = np.linspace(x_star.min(), x_star.max(), 2)[:, None]
ax.plot(t_star[idx_t_0] * np.ones((2, 1)), line, 'w-', linewidth=1.0)
ax.plot(t_star[idx_t_1] * np.ones((2, 1)), line, 'w-', linewidth=1.0)
ax.set_xlabel('$t$')
ax.set_ylabel('$x$')
ax.set_title('$u(t,x)$', fontsize=10)
gs1 = gridspec.GridSpec(1, 2)
gs1.update(top=1 - 1 / 3 - 0.1,
bottom=1 - 2 / 3,
left=0.15,
right=0.85,
wspace=0.5)
ax = plt.subplot(gs1[0, 0])
ax.plot(x_star,
Exact[:, idx_t_0][:, None],
'b',
linewidth=2,
label='Exact')
ax.plot(x_0, u_0, 'rx', linewidth=2, label='Data')
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.set_title('$t = %.2f$\n%d trainng data' %
(t_star[idx_t_0], u_0.shape[0]),
fontsize=10)
ax = plt.subplot(gs1[0, 1])
ax.plot(x_star,
Exact[:, idx_t_1][:, None],
'b',
linewidth=2,
label='Exact')
ax.plot(x_1, u_1, 'rx', linewidth=2, label='Data')
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.set_title('$t = %.2f$\n%d trainng data' %
(t_star[idx_t_1], u_1.shape[0]),
fontsize=10)
ax.legend(loc='upper center',
bbox_to_anchor=(-0.3, -0.3),
ncol=2,
frameon=False)
gs2 = gridspec.GridSpec(1, 2)
gs2.update(top=1 - 2 / 3 - 0.05,
bottom=0,
left=0.15,
right=0.85,
wspace=0.0)
ax = plt.subplot(gs2[0, 0])
ax.axis('off')
nu = 0.01 / np.pi
s1 = r'$\begin{tabular}{ |c|c| } \hline Correct PDE & $u_t + u u_x + %.6f u_{xx} = 0$ \\ \hline Identified PDE (clean data) & ' % (
nu)
s2 = r'$u_t + %.3f u u_x + %.6f u_{xx} = 0$ \\ \hline ' % (lambda_1_value,
lambda_2_value)
s3 = r'Identified PDE (1\% noise) & '
s4 = r'$u_t + %.3f u u_x + %.6f u_{xx} = 0$ \\ \hline ' % (
lambda_1_value_noisy, lambda_2_value_noisy)
s5 = r'\end{tabular}$'
s = s1 + s2 + s3 + s4 + s5
ax.text(-0.1, 0.2, s)
if save_path != None and save_hp != None:
saveResultDir(save_path, save_hp)
else:
plt.show()
pmin, pmax = Fields_Num[0::6, :].min(), Fields_Num[0::6, :].max()
Tmin, Tmax = Fields_Num[3::6, :].min(), Fields_Num[3::6, :].max()
def contourplot(ax, x, y, phi1, vmin, vmax, phi2, levels, title):
sc1 = ax.contourf(x, y, phi1, vmin=vmin, vmax=vmax, cmap='jet')
sc2 = ax.contour(x, y, phi2, levels=levels, colors=['k'])
s2 = np.sqrt(2) / 2
ax.axis([-s2, s2, -s2, s2])
ax.set_title(title)
ax.axis('off')
return sc1, sc2
plt.close('all')
fig, _ = newfig(width=2, nplots=2 / len(ValidationInd))
gs = gridspec.GridSpec(2, len(ValidationInd))
for i in range(len(ValidationInd)):
alphai = alpha[i:i + 1, :]
Fieldi = Fields[i]
x, y, p, u, v, T, omega, psi = Fieldi[0], Fieldi[1], Fieldi[2], Fieldi[
3], Fieldi[4], Fieldi[5], Fieldi[6], Fieldi[7]
pNum = np.reshape(Fields_Num[0::6, i], x.shape)
uNum = np.reshape(Fields_Num[1::6, i], x.shape)
vNum = np.reshape(Fields_Num[2::6, i], x.shape)
TNum = np.reshape(Fields_Num[3::6, i], x.shape)
omegaNum = np.reshape(Fields_Num[4::6, i], x.shape)
psiNum = np.reshape(Fields_Num[5::6, i], x.shape)
psi_max = psiNum.max()
# Prediction
u_pred, e_pred = model.predict(t, x)
# Error
error_u = np.linalg.norm(u - u_pred, 2) / np.linalg.norm(u, 2)
error_e = np.linalg.norm(e - e_pred, 2) / np.linalg.norm(e, 2)
print('Error p: %e' % (error_u))
print('Error e: %e' % (error_e))
######################################################################
############################# Plotting ###############################
######################################################################
fig, ax = newfig(1.0, 1.2)
ax.axis('off')
gs = gridspec.GridSpec(2, 2)
gs.update(top=0.9, bottom=0.1, left=0.1, right=0.9, wspace=0.5, hspace=0.7)
######## Exact p(t,x) ###########
ax = plt.subplot(gs[0:1, 0])
h = plot_solution(t, x, u, ax)
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(h, cax=cax)
ax.set_xlabel('$t$')
ax.set_ylabel('$x$')
ax.set_title('Exact $P(t,x)$', fontsize=10)
def plotting(t_star, S_star, S_pred, perm, forecast=False):
sns.set()
fig, ax = newfig(2.0, 0.7)
gs0 = gridspec.GridSpec(1, 1)
gs0.update(top=0.9,
bottom=0.15,
left=0.1,
right=0.95,
hspace=0.3,
wspace=0.3)
ax = plt.subplot(gs0[0:1, 0:1])
ax.plot(t_star, S_star[:, 2], 'C1', linewidth=2, label='input data')
ax.scatter(t_star[perm],
S_star[perm, 2],
marker='o',
s=40,
label='sampled input')
ax.set_xlabel('$t\ (min)$', fontsize=18)
ax.set_ylabel('$G\ (mg/dl) $', fontsize=18)
ax.legend(fontsize='large')
####################################
fig, ax = newfig(1.8, 0.75)
gs1 = gridspec.GridSpec(1, 2)
gs1.update(top=0.85,
bottom=0.15,
left=0.1,
right=0.95,
hspace=0.3,
wspace=0.3)
ax = plt.subplot(gs1[0:1, 0:1])
ax.plot(t_star, S_star[:, 0] * Vg2, 'C1', linewidth=2, label='exact')
ax.plot(t_star,
S_pred[:, 0] * Vg2,
'g-.',
linewidth=3,
label='learned')
ax.set_xlabel('$t\ (min)$', fontsize=18)
ax.set_ylabel('$I_p\ (\mu U/ml)$', fontsize=18)
ax.legend(fontsize='large')
ax = plt.subplot(gs1[0:1, 1:2])
ax.plot(t_star, S_star[:, 1] * Vg2, 'C1', linewidth=2)
ax.plot(t_star, S_pred[:, 1] * Vg2, 'g-.', linewidth=3)
ax.set_xlabel('$t\ (min)$', fontsize=18)
ax.set_ylabel('$I_i\ (\mu U/ml)$', fontsize=18)
fig, ax = newfig(1.8, 0.75)
gs2 = gridspec.GridSpec(1, 2)
gs2.update(top=0.85,
bottom=0.15,
left=0.1,
right=0.95,
hspace=0.3,
wspace=0.3)
ax = plt.subplot(gs2[0:1, 0:1])
if not forecast:
ax.scatter(t_star[perm], S_star[perm, 2], marker='o', c='C1', s=30)
else:
ax.plot(t_star, S_star[:, 2], 'C1', linewidth=2)
ax.plot(t_star, S_pred[:, 2], 'g-.', linewidth=3)
ax.set_xlabel('$t\ (min)$', fontsize=18)
ax.set_ylabel('$G\ (mg/dl)$', fontsize=18)
ax = plt.subplot(gs2[0:1, 1:2])
ax.plot(t_star, S_star[:, 6] * Vg2, 'C1', linewidth=2)
ax.plot(t_star, S_pred[:, 6] * Vg2, 'g-.', linewidth=3)
ax.set_xlabel('$t\ (min)$', fontsize=18)
ax.set_ylabel('$I_G\ (mg/min)$', fontsize=18)
print('Error u: %e' % (error_u))
U_pred = griddata(X_star, u_pred.flatten(), (X, T), method='cubic')
FU_pred = griddata(X_star, f_u_pred.flatten(), (X, T), method='cubic')
######################################################################
############################# Plotting ###############################
######################################################################
X0 = np.concatenate((x0, 0 * x0), 1) # (x0, 0)
X_lb = np.concatenate((0 * tb + lb[0], tb), 1) # (lb[0], tb)
X_ub = np.concatenate((0 * tb + ub[0], tb), 1) # (ub[0], tb)
X_u_train = np.vstack([X0, X_lb, X_ub])
fig, ax = newfig(1.3, 1.0)
ax.axis('off')
####### Row 0: h(t,x) ##################
gs0 = gridspec.GridSpec(1, 2)
gs0.update(top=1 - 0.06, bottom=1 - 1 / 3, left=0.15, right=0.85, wspace=0)
ax = plt.subplot(gs0[:, :])
h = ax.imshow(U_pred.T,
interpolation='nearest',
cmap='YlGnBu',
extent=[lb[1], ub[1], lb[0], ub[0]],
origin='lower',
aspect='auto')
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.05)
Exact_idn = np.loadtxt("Results/ReLU/exact_sol.csv", delimiter=',')
U_relu = np.loadtxt("Results/ReLU/idn_relu.csv", delimiter=',')
U_sine = np.loadtxt("Results/Sine/idn_sine.csv", delimiter=',')
U_rat = np.loadtxt("Results/Rational_3_2/idn_rat.csv", delimiter=',')
lb_idn = np.array([0.0, -20.0])
ub_idn = np.array([40.0, 20.0])
# Print mean square errors
E_relu = np.mean((Exact_idn-U_relu)**2)
E_sine = np.mean((Exact_idn-U_sine)**2)
E_rat = np.mean((Exact_idn-U_rat)**2)
print("Error ReLU = %e, Error Sine = %e, Error Rat = %e" % (E_relu, E_sine, E_rat))
# Plot the different solutions
fig, ax = newfig(0.6, 1.2)
ax.axis('off')
gs = gridspec.GridSpec(1, 1)
gs.update(top=0.9, bottom=0.1, left=0.1, right=0.9, wspace=0.7, hspace=0.5)
# ######## Exact solution #######################
ax = plt.subplot(gs[0, 0])
h = ax.imshow(Exact_idn, interpolation='nearest', cmap='jet',
extent=[lb_idn[0], ub_idn[0], lb_idn[1], ub_idn[1]],
origin='lower', aspect='auto')
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(h, cax=cax)
savefig('Results/solution')
u_pred, f_pred = model.idn_predict(t_sol_star, x_sol_star)
u_pred_idn, f_pred_idn = model.sol_predict(t_idn_star, x_idn_star)
error_u = np.linalg.norm(u_sol_star-u_pred,2)/np.linalg.norm(u_sol_star,2)
error_u_idn = np.linalg.norm(u_idn_star-u_pred_idn,2)/np.linalg.norm(u_idn_star,2)
print('Error u: %e' % (error_u))
print('Error u (idn): %e' % (error_u_idn))
U_pred = griddata(X_sol_star, u_pred.flatten(), (T_sol, X_sol), method='cubic')
######################################################################
############################# Plotting ###############################
######################################################################
fig, ax = newfig(1.0, 0.6)
ax.axis('off')
######## Row 2: Pressure #######################
######## Predicted p(t,x,y) ###########
gs = gridspec.GridSpec(1, 2)
gs.update(top=0.8, bottom=0.2, left=0.1, right=0.9, wspace=0.5)
ax = plt.subplot(gs[:, 0])
h = ax.imshow(Exact_sol, interpolation='nearest', cmap='jet',
extent=[lb_sol[0], ub_sol[0], lb_sol[1], ub_sol[1]],
origin='lower', aspect='auto')
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(h, cax=cax)
ax.set_xlabel('$t$')
M = 1
scheme = 'AM'
model = Multistep_NN(dt, X_train, layers, M, scheme)
N_Iter = 50000
model.train(N_Iter)
def learned_f(x, t):
f = model.predict_f(x[None, :])
return f.flatten()
learned_X_star = odeint(learned_f, x0, t_star)
####### Plotting ##################
fig, ax = newfig(1.0, 0.8)
ax.axis('off')
gs0 = gridspec.GridSpec(1, 2)
gs0.update(top=0.95, bottom=0.1, left=0.0, right=0.90, wspace=0.15)
ax = plt.subplot(gs0[:, 0:1], projection='3d')
ax.w_xaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))
ax.w_yaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))
ax.w_zaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))
colorline3d(ax, X_star[:, 0], X_star[:, 1], X_star[:, 2], cmap=plt.cm.jet)
ax.grid(False)
ax.set_xlim([-200, 200])
ax.set_ylim([-200, 200])
ax.set_zlim([-150, 0])
ax.set_xticks([-200, 0, 200])
U2_star = griddata(X_star2, u2_star.flatten(), (X2, T2), method='cubic')
U3_star = griddata(X_star3, u3_star.flatten(), (X3, T3), method='cubic')
U4_star = griddata(X_star4, u4_star.flatten(), (X4, T4), method='cubic')
u1_err = abs(u1_pred - u1_star)
u2_err = abs(u2_pred - u2_star)
u3_err = abs(u3_pred - u3_star)
u4_err = abs(u4_pred - u4_star)
U1_err = griddata(X_star1, u1_err.flatten(), (X1, T1), method='cubic')
U2_err = griddata(X_star2, u2_err.flatten(), (X2, T2), method='cubic')
U3_err = griddata(X_star3, u3_err.flatten(), (X3, T3), method='cubic')
U4_err = griddata(X_star4, u4_err.flatten(), (X4, T4), method='cubic')
############################ Plotting #################>>>>>>>>>>>>>>>>>>
fig, ax = newfig(1.0, 1.1)
# fig = plt.figure()
gridspec.GridSpec(1,1)
ax = plt.subplot2grid((1,1), (0,0))
maxLevel = max(max(u1_star),max(max(u2_star),max(max(u3_star),max(u4_star))))[0]
minLevel = min(min(u1_star),min(min(u2_star), min(min(u3_star), min(u4_star)) ))[0]
levels = np.linspace(minLevel-0.01, maxLevel+0.01, 200)
CS_ext1 = ax.contourf(T, X, U_star, levels=levels, cmap='jet', origin='lower')
cbar = fig.colorbar(CS_ext1)
# , ticks=[-1, -0.5, 0, 0.5, 1])
# cbar.ax.set_yticklabels(['-1', '-0.5', '0', '0.5', '1'])
cbar.ax.tick_params(labelsize=20)
ax.set_xlim(-0.01, 1)
ax.set_ylim(-1.01, 1.02)
#ax_pred.locator_params(nbins=5)
M = 1
scheme = 'AM'
model = Multistep_NN(dt, X_train, layers, M, scheme)
N_Iter = 50000
model.train(N_Iter)
def learned_f(x, t):
f = model.predict_f(x[None, :])
return f.flatten()
learned_X_star = odeint(learned_f, x0, t_star)
####### Plotting ##################
fig, ax = newfig(1.0, 1.55)
ax.axis('off')
gs0 = gridspec.GridSpec(3, 2)
gs0.update(top=0.95,
bottom=0.35,
left=0.1,
right=0.95,
hspace=0.5,
wspace=0.3)
ax = plt.subplot(gs0[0:1, 0:1])
ax.plot(t_star, X_star[:, 0], 'r-')
ax.plot(t_star, learned_X_star[:, 0], 'k--')
ax.set_xlabel('$t$')
ax.set_ylabel('$S_1$')
