def test_est_1d_2():
u = Unknown('u', ldim=1)
phi = Constant('phi')
# ... define a partial differential operator as a lambda function
from sympy.abc import x
L = lambda u: x*dx(u) + phi*u
L_expected = lambda u: x*dx(u) + 2.*u
# ...
# ... symbolic functions for unknown and rhs
from sympy.abc import x
from sympy import sin, cos
u_sym = sin(x)
f_sym = L_expected(u_sym)
# ...
# ... lambdification + evaluation
from numpy import linspace, pi
u_num = lambdify((x), u_sym, "numpy")
f_num = lambdify((x), f_sym, "numpy")
x_u = linspace(0, 2*pi, 10)
x_f = x_u
us = u_num(x_u)
fs = f_num(x_f)
# ...
# compute the likelihood
nlml = NLML(L(u), u, 'RBF')
# set values
nlml.set_u(x_u, us)
nlml.set_f(x_f, fs)
from numpy.random import rand
from numpy import exp, ones
from time import time
# ... using pure python implementation
from symgp.nelder_mead import nelder_mead
x_start = rand(len(nlml.args))
tb = time()
m = nelder_mead(nlml, x_start,
step=0.1, no_improve_thr=10e-6, no_improv_break=10,
max_iter=0, alpha=1., gamma=2., rho=-0.5, sigma=0.5,
verbose=False)
te = time()
elapsed_python = te-tb
args = exp(m[0])
print('> estimated phi = ', nlml.map_args(args)['phi'])
print('> elapsed time python = ', elapsed_python)
def solve(kernel):
print('>>>> using : ', kernel)
# compute the likelihood
nlml = NLML(L(u), u, kernel)
# set values
nlml.set_u(x_u, us)
nlml.set_f(x_f, fs)
x_start = rand(len(nlml.args))
# ... using pure python implementation
from symgp.nelder_mead import nelder_mead
m = nelder_mead(nlml, x_start,
step=0.1, no_improve_thr=10e-6, no_improv_break=10,
max_iter=0, alpha=1., gamma=2., rho=-0.5, sigma=0.5,
verbose=False)
args = exp(m[0])
print('> estimated phi = ', nlml.map_args(args)['phi'])
def test_est_1d_7():
"""Explicit time step for Burgers"""
u = Unknown('u', ldim=1)
nu = Constant('nu')
# ... define a partial differential operator as a lambda function
from sympy.abc import x
from sympy import sin, cos
Dt = 0.0010995574287564279
nu_expected = 0.07
from numpy import genfromtxt
xn = genfromtxt('x.txt')
unew = genfromtxt('unew.txt')
un = genfromtxt('un.txt')
from scipy.interpolate import interp1d
unew = interp1d(xn, unew)
un = interp1d(xn, un)
fn = Function('fn')
L = lambda u: u + Dt*fn(x)*dx(u) + nu*Dt*dx(dx(u))
# ...
# ... lambdification + evaluation
from numpy import linspace, pi
from numpy.random import rand
# x_u = linspace(0, 2*pi, 50)
x_u = rand(50) * 2*pi
x_f = x_u
us = un(x_u)
fs = unew(x_f)
# ...
from numpy.random import rand
from numpy import exp, ones, log
from time import time
from scipy.optimize import minimize
# compute the likelihood
nlml = NLML(L(u), u, 'SE')
# set values
nlml.set_u(x_u, us)
nlml.set_f(x_f, fs)
x_start = rand(len(nlml.args))
# ... using pure python implementation
from symgp.nelder_mead import nelder_mead
x_start = rand(len(nlml.args))
m = nelder_mead(nlml, x_start,
step=1., no_improve_thr=1e-5, no_improv_break=6,
max_iter=0, alpha=1., gamma=1.5, rho=-0.5, sigma=.5,
verbose=False)
args = exp(m[0])
print('> estimated nu = ', nlml.map_args(args)['nu'])
def test_est_2d_2():
u = Unknown('u', ldim=2)
phi = Constant('phi')
# ... define a partial differential operator as a lambda function
L = lambda u: phi * u + dx(u) + dy(dy(u))
L_expected = lambda u: 2. * u + dx(u) + dy(dy(u))
# ...
# ... symbolic functions for unknown and rhs
from sympy.abc import x, y
from sympy import sin, cos
u_sym = x**2 + y
f_sym = L_expected(u_sym)
# ...
# ... lambdification + evaluation
from numpy import linspace, meshgrid, zeros
from numpy.random import rand
u_num = lambdify((x, y), u_sym, "numpy")
f_num = lambdify((x, y), f_sym, "numpy")
x_u = rand(50, 2)
x_f = x_u
us = u_num(x_u[:, 0], x_u[:, 1])
fs = f_num(x_f[:, 0], x_f[:, 1])
# ...
# eps = 1.e-6
eps = 1.e-5
# eps = 1.e-4
niter = 10
# niter = 1
phis = []
scores = []
for i in range(0, niter):
print('> sample ', i)
# compute the likelihood
# nlml = NLML(L(u), u, 'CSE', debug=False)
nlml = NLML(L(u), u, 'SE', debug=False)
# set values
nlml.set_u(x_u, us)
nlml.set_f(x_f, fs)
# ... using pure python implementation
x_start = rand(len(nlml.args))
_nlml = lambda y: nlml(y, s_u=eps, s_f=eps)
m = nelder_mead(_nlml,
x_start,
step=0.1,
no_improve_thr=10e-4,
no_improv_break=4,
max_iter=0,
alpha=.5,
gamma=1.5,
rho=-0.5,
sigma=0.5,
verbose=False)
args = exp(m[0])
score = m[1]
phis.append(nlml.map_args(args)['phi'])
scores.append(score)
# ...
phis = asarray(phis)
scores = asarray(scores)
i_max = argmax(scores)
print(phis[i_max])
print(scores[i_max])
print(scores)
print(phis)
def test_est_2d_3():
u = Unknown('u', ldim=2)
phi = Constant('phi')
# ... define a partial differential operator as a lambda function
L = lambda u: phi * u + dx(u) + dy(dy(u))
L_expected = lambda u: 2. * u + dx(u) + dy(dy(u))
# ...
# ... symbolic functions for unknown and rhs
from sympy.abc import x, y
from sympy import sin, cos
u_sym = x**2 + y
f_sym = L_expected(u_sym)
# ...
# ... lambdification + evaluation
from numpy import linspace, meshgrid, zeros
from numpy.random import rand
u_num = lambdify((x, y), u_sym, "numpy")
f_num = lambdify((x, y), f_sym, "numpy")
x_u = rand(50, 2)
x_f = x_u
us = u_num(x_u[:, 0], x_u[:, 1])
fs = f_num(x_f[:, 0], x_f[:, 1])
# ...
# compute the likelihood
# nlml = NLML(L(u), u, 'SE')
nlml = NLML(L(u), u, 'RBF')
# set values
nlml.set_u(x_u, us)
nlml.set_f(x_f, fs)
from numpy.random import rand, normal
from numpy import exp, ones, log
# ... using pure python implementation
from symgp.nelder_mead import nelder_mead
phi_expected = 2.
phi_var = 0.5
n_samples = 1000
x_starts = rand(len(nlml.args), n_samples)
i_phi = list(nlml.args).index('phi')
x_starts[i_phi, :] = log(normal(phi_expected, phi_var, n_samples))
phis = []
scores = []
phiso = []
scoreso = []
for i in range(0, n_samples):
print('> sample ', i)
x_start = x_starts[:, i]
# ...
def f(params):
params[i_phi] = x_starts[i_phi, i]
return nlml(params)
m = nelder_mead(f,
x_start,
step=0.1,
no_improve_thr=10e-4,
no_improv_break=4,
max_iter=0,
alpha=.5,
gamma=1.5,
rho=-0.5,
sigma=0.5,
verbose=False)
args = exp(m[0])
score = m[1]
phis.append(nlml.map_args(args)['phi'])
scores.append(score)
# ...
# ...
m = nelder_mead(nlml,
x_start,
step=0.1,
no_improve_thr=10e-4,
no_improv_break=4,
max_iter=0,
alpha=.5,
gamma=1.5,
rho=-0.5,
sigma=0.5,
verbose=False)
args = exp(m[0])
score = m[1]
phiso.append(nlml.map_args(args)['phi'])
scoreso.append(score)
# ...
from numpy import savetxt
savetxt('est_2d/phis.txt', asarray(phis))
savetxt('est_2d/scores.txt', asarray(scores))
savetxt('est_2d/phiso.txt', asarray(phiso))
savetxt('est_2d/scoreso.txt', asarray(scoreso))
plt.plot(phis, scores, '.b', label=r'fixed $\phi$', alpha=0.4)
plt.plot(phiso, scoreso, '.r', label=r'free $\phi$', alpha=0.4)
plt.axvline(x=phi_expected, color='green', alpha=0.5)
plt.xlabel(r'$\phi$')
plt.ylabel(r'Likelihood $\mathcal{L}$')
title = '$L u := {}$'.format(latex(L(u)))
title += '\n'
title += r'$\phi_{exact}' + ' := {}$'.format(phi_expected)
plt.title(title)
plt.legend()
plt.show()
def test_est_2d_1():
u = Unknown('u', ldim=2)
phi = Constant('phi')
# ... define a partial differential operator as a lambda function
L = lambda u: phi * u + dx(u) + dy(dy(u))
L_expected = lambda u: 2. * u + dx(u) + dy(dy(u))
# ...
# ... symbolic functions for unknown and rhs
from sympy.abc import x, y
from sympy import sin, cos
u_sym = x**2 + y
f_sym = L_expected(u_sym)
# ...
# ... lambdification + evaluation
from numpy import linspace, meshgrid, zeros
from numpy.random import rand
u_num = lambdify((x, y), u_sym, "numpy")
f_num = lambdify((x, y), f_sym, "numpy")
# t = linspace(0, 1, 5)
# x,y = meshgrid(t, t)
# x_u = zeros((x.size, 2))
# x_u[:,0] = x.reshape(x.size)
# x_u[:,1] = y.reshape(y.size)
x_u = rand(100, 2)
x_f = x_u
us = u_num(x_u[:, 0], x_u[:, 1])
fs = f_num(x_f[:, 0], x_f[:, 1])
# ...
# compute the likelihood
debug = True
nlml = NLML(L(u), u, 'SE', debug=debug)
# set values
nlml.set_u(x_u, us)
nlml.set_f(x_f, fs)
from numpy.random import rand
from numpy import exp, ones, log
# ... using pure python implementation
from symgp.nelder_mead import nelder_mead
x_start = rand(len(nlml.args))
print('> x_start = ', x_start)
m = nelder_mead(nlml,
x_start,
step=0.1,
no_improve_thr=10e-4,
no_improv_break=4,
max_iter=0,
alpha=.5,
gamma=1.5,
rho=-0.5,
sigma=0.5,
verbose=False)
args = exp(m[0])
print('> estimated phi = ', nlml.map_args(args)['phi'])
if debug:
plt.xlabel('number of data')
plt.ylabel(r'eigenvalues in log-scale')
plt.legend()
plt.show()