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dynamics.py
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dynamics.py
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# -*- coding: utf-8 -*-
"""
Created on Wed Sep 23 10:36:47 2020
@author: pccom
"""
import numpy as np
def kuramoto_time(frequencies, max_time, number_of_snapshots, number_of_experiments = 10, K = 2.0, forcing_param = 0.2, relative_tolerance = 1.0e-10, absolute_tolerance = 1.0e-13):
from scipy.integrate import solve_ivp
number_of_oscillators = len(frequencies)
def kuramoto_ode(_, theta):
[theta_i, theta_j] = np.meshgrid(theta, theta)
return frequencies + K / number_of_oscillators * np.sin(theta_j - theta_i).sum(0) + forcing_param * np.sin(theta)
list_snapshots = []
list_derivatives = []
list_times = []
for i in range(number_of_experiments):
theta_init = 2 * np.pi * np.random.rand(number_of_oscillators)
sol = solve_ivp(kuramoto_ode, [0, max_time], theta_init, method='DOP853', t_eval=np.linspace(0, max_time, number_of_snapshots), rtol = relative_tolerance, atol = absolute_tolerance)
snapshots = sol.y
derivatives = np.zeros([number_of_oscillators, number_of_snapshots])
for i in range(number_of_snapshots):
derivatives[:, i] = kuramoto_ode(0, snapshots[:, i])
list_derivatives.append(derivatives)
list_times.append(sol['t'])
list_snapshots.append(snapshots)
return list_snapshots, list_derivatives, list_times
def kuramoto_time_individual(frequencies, initial_position, start_time, end_time, number_of_samples, K = 2.0, forcing_param = 0.2, relative_tolerance = 1.0e-10, absolute_tolerance = 1.0e-13):
from scipy.integrate import solve_ivp
number_of_oscillators = len(frequencies)
def kuramoto_ode(_, theta):
[theta_i, theta_j] = np.meshgrid(theta, theta)
return frequencies + K / number_of_oscillators * np.sin(theta_j - theta_i).sum(0) + forcing_param * np.sin(theta)
sol = solve_ivp(kuramoto_ode, [start_time, end_time], initial_position, method='DOP853', t_eval=np.linspace(start_time, end_time, number_of_samples), rtol = relative_tolerance, atol = absolute_tolerance)
return sol.y, sol['t']
#Convert angles from a
def angle_conversion(angles):
return angles - np.floor(angles/(2*np.pi))*2.0*np.pi
def kuramoto_time_backup(theta_init, frequencies, max_time, number_of_snapshots, K = 2.0, forcing_param = 0.2):
from scipy.integrate import solve_ivp
number_of_oscillators = len(theta_init)
def kuramoto_ode(_, theta):
[theta_i, theta_j] = np.meshgrid(theta, theta)
return frequencies + K / number_of_oscillators * np.sin(theta_j - theta_i).sum(0) + forcing_param * np.sin(theta)
sol = solve_ivp(kuramoto_ode, [0, max_time], theta_init, method='BDF',
t_eval=np.linspace(0, max_time, number_of_snapshots))
snapshots = sol.y
derivatives = np.zeros([number_of_oscillators, number_of_snapshots])
for i in range(number_of_snapshots):
derivatives[:, i] = kuramoto_ode(0, snapshots[:, i])
return snapshots, derivatives, sol['t']
def kuramoto_random(num_oscillators, number_of_snapshots, frequencies, K = 2.0, forcing_param = 0.2):
"""Kuramoto model
Generate data for the Kuramoto model represented by the differential equation
d/dt x_i = w_i + (2/d) * (sin(x_1 - x_i) + ... + sin(x_d - x_i)) + 0.2 * sin(x_i).
See [1]_ and [2]_ for details.
Parameters
----------
theta_init: ndarray
initial distribution of the oscillators
frequencies: ndarray
natural frequencies of the oscillators
time: float
integration time for BDF method
number_of_snapshots: int
number of snapshots
Returns
-------
snapshots: ndarray(number_of_oscillators, number_of_snapshots)
snapshot matrix containing random displacements of the oscillators in [-0.1,0.1]
derivatives: ndarray(number_of_oscillators, number_of_snapshots)
matrix containing the corresponding derivatives
References
----------
.. [1] P. Gelß, S. Klus, J. Eisert, C. Schütte, "Multidimensional Approximation of Nonlinear Dynamical Systems",
arXiv:1809.02448, 2018
.. [2] J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, R. Spigler, "The Kuramoto model: A simple
paradigm for synchronization phenomena", Rev. Mod. Phys. 77, pp. 137-185 , 2005
"""
# snapshots = 2 * np.pi * np.random.rand(num_oscillators, number_of_snapshots) - np.pi
snapshots = 2 * np.pi * np.random.rand(num_oscillators, number_of_snapshots)
derivatives = np.zeros(snapshots.shape)
for i in range(number_of_snapshots):
for j in range(num_oscillators):
derivatives[j,i] = frequencies[j] + K / num_oscillators * np.sin(snapshots[:,i] - snapshots[j,i]).sum() + forcing_param * np.sin(snapshots[j,i])
return snapshots, derivatives
# Build the exact dynamics of the fermi_pasta_ulam if we have polynomials of order up to three.
# In this case the basis functions are sines and cosines.
# The first number_of_oscillators x are cosines, and the rest are sines.
def exact_solution_kuramoto(number_of_oscillators, frequencies, polinomial, K = 2.0, forcing_param = 0.2):
reference_polynomial = polinomial.get_feature_names()
num_basis_functions = len(reference_polynomial)
# Build the exact solution at the boundary
exact_solution = np.zeros((number_of_oscillators, num_basis_functions))
# Build the exact solution.
for i in range(0, number_of_oscillators):
# First order terms.
exact_solution[i, reference_polynomial.index("1")] = frequencies[i]
exact_solution[i, reference_polynomial.index("x" + str(number_of_oscillators + i))] = forcing_param
for j in range(number_of_oscillators):
exact_solution[i, reference_polynomial.index("x" + str(i) + ' x' + str(number_of_oscillators + j))] += K/number_of_oscillators
exact_solution[i, reference_polynomial.index("x" + str(j) + ' x' + str(number_of_oscillators + i))] += -K/number_of_oscillators
return exact_solution
def add_constraints_symmetry_kuramoto(polinomial, number_of_oscillators, normalization_factors):
feature_names = polinomial.get_feature_names()
num_basis_functions = len(feature_names)
pair_symmetries = []
#Constraint on the cos^2 terms:
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
index_first = feature_names.index("x" + str(j) + '^2')
index_first_transformed = index_first + int(num_basis_functions*i)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1, 'constant': 0})
#Constraint on the sin^2 terms:
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
index_first = feature_names.index("x" + str(number_of_oscillators + j) + '^2')
index_first_transformed = index_first + int(num_basis_functions*i)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1, 'constant': 0})
type_symmetries = [('sin', 'cos'), ('cos'), ('sin')]
for symmetry in type_symmetries:
if(symmetry == ('sin', 'cos') or symmetry == ('cos', 'sin')):
#Symmetry cos-sin
#For i and j---> \xi_i(cos(x_i) sin(x_j)) = \xi_j(cos(x_j) sin(x_i))
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
if(i != j):
index_first = feature_names.index("x" + str(i) + ' x' + str(number_of_oscillators + j))
index_second = feature_names.index("x" + str(j) + ' x' + str(number_of_oscillators + i))
index_first_transformed = index_first + int(num_basis_functions*i)
index_second_transformed = index_second + int(num_basis_functions*j)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
#Symmetry sin-cos
#For i and j---> \xi_i(sin(x_i) cos(x_j)) = \xi_j(sin(x_j) cos(x_i))
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
if(i != j):
index_first = feature_names.index("x" + str(j) + ' x' + str(i + number_of_oscillators))
index_second = feature_names.index("x" + str(i) + ' x' + str(j + number_of_oscillators))
index_first_transformed = index_first + + int(num_basis_functions*i)
index_second_transformed = index_second + int(num_basis_functions*j)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
#Symmetry cos-cos
#For i and j---> \xi_i(cos(x_i) cos(x_j)) = \xi_j(cos(x_j) cos(x_i))
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
if(i != j):
if(i <= j):
index = feature_names.index("x" + str(i) + ' x' + str(j))
else:
index = feature_names.index("x" + str(j) + ' x' + str(i))
index_first_transformed = index + int(num_basis_functions*i)
index_second_transformed = index + int(num_basis_functions*j)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
#Symmetry sin-sin
#For i and j---> \xi_i(sin(x_i) sin(x_j)) = \xi_j(sin(x_j) sin(x_i))
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
if(i != j):
if(i <= j):
index = feature_names.index("x" + str(i + number_of_oscillators) + ' x' + str(j + number_of_oscillators))
else:
index = feature_names.index("x" + str(j + number_of_oscillators) + ' x' + str(i + number_of_oscillators))
index_first_transformed = index + int(num_basis_functions*i)
index_second_transformed = index + int(num_basis_functions*j)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
for symmetry in type_symmetries:
if(symmetry == ('cos')):
#Symmetry cos
#For i and j---> \xi_i(cos(x_i)) = \xi_j(cos(x_j))
for i in range(1, number_of_oscillators):
index_first = feature_names.index("x" + str(0))
index_second = feature_names.index("x" + str(i))
index_first_transformed = index_first
index_second_transformed = index_second + int(num_basis_functions*i)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
for symmetry in type_symmetries:
if(symmetry == ('cos')):
#Symmetry cos
#For i and j---> \xi_i(cos(x_j)) = \xi_j(cos(x_i))
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
if(i != j):
index_first = feature_names.index("x" + str(j))
index_second = feature_names.index("x" + str(i))
index_first_transformed = index_first + + int(num_basis_functions*i)
index_second_transformed = index_second + int(num_basis_functions*j)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
for symmetry in type_symmetries:
if(symmetry == ('sin')):
#Symmetry sin
#For i and j---> \xi_i(sin(x_i)) = \xi_j(sin(x_j))
for i in range(1, number_of_oscillators):
index_first = feature_names.index("x" + str(number_of_oscillators + 0))
index_second = feature_names.index("x" + str(number_of_oscillators + i))
index_first_transformed = index_first
index_second_transformed = index_second + int(num_basis_functions*i)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
for symmetry in type_symmetries:
if(symmetry == ('sin')):
#Symmetry sin
#For i and j---> \xi_i(sin(x_j)) = \xi_j(sin(x_i))
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
if(i != j):
index_first = feature_names.index("x" + str(j + number_of_oscillators))
index_second = feature_names.index("x" + str(i + number_of_oscillators))
index_first_transformed = index_first + + int(num_basis_functions*i)
index_second_transformed = index_second + int(num_basis_functions*j)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
# Remove duplicates before we output.
# pair_symmetries = [dict(t) for t in {tuple(d.keys()) for d in pair_symmetries}]
pair_symmetries =delete_dictionaries_with_duplicate_keys(pair_symmetries)
return pair_symmetries
def add_constraints_symmetry_kuramoto_backup(polinomial, number_of_oscillators, normalization_factors):
feature_names = polinomial.get_feature_names()
num_basis_functions = len(feature_names)
pair_symmetries = []
#Constraint on the cos^2 terms:
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
index_first = feature_names.index("x" + str(j) + '^2')
index_first_transformed = index_first + int(num_basis_functions*i)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1, 'constant': 0})
#Constraint on the sin^2 terms:
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
index_first = feature_names.index("x" + str(number_of_oscillators + j) + '^2')
index_first_transformed = index_first + int(num_basis_functions*i)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1, 'constant': 0})
type_symmetries = [('sin', 'cos'), ('cos'), ('sin')]
for symmetry in type_symmetries:
if(symmetry == ('sin', 'cos') or symmetry == ('cos', 'sin')):
#Symmetry cos-sin
#For i and j---> \xi_i(cos(x_i) sin(x_j)) = \xi_j(cos(x_j) sin(x_i))
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
if(i != j):
index_first = feature_names.index("x" + str(i) + ' x' + str(number_of_oscillators + j))
index_second = feature_names.index("x" + str(j) + ' x' + str(number_of_oscillators + i))
index_first_transformed = index_first + int(num_basis_functions*i)
index_second_transformed = index_second + int(num_basis_functions*j)
symmetry_pair = (min(index_first_transformed, index_second_transformed), max(index_first_transformed, index_second_transformed))
pair_symmetries.append({ "x" + str(symmetry_pair[0]) : normalization_factors[index_second % num_basis_functions][0], "x" + str(symmetry_pair[1]) : -normalization_factors[index_first % num_basis_functions][0], 'constant': 0})
#Symmetry sin-cos
#For i and j---> \xi_i(sin(x_i) cos(x_j)) = \xi_j(sin(x_j) cos(x_i))
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
if(i != j):
index_first = feature_names.index("x" + str(j) + ' x' + str(i + number_of_oscillators))
index_second = feature_names.index("x" + str(i) + ' x' + str(j + number_of_oscillators))
index_first_transformed = index_first + + int(num_basis_functions*i)
index_second_transformed = index_second + int(num_basis_functions*j)
symmetry_pair = (min(index_first_transformed, index_second_transformed), max(index_first_transformed, index_second_transformed))
pair_symmetries.append({ "x" + str(symmetry_pair[0]) : normalization_factors[index_second % num_basis_functions][0], "x" + str(symmetry_pair[1]) : -normalization_factors[index_first % num_basis_functions][0], 'constant': 0})
for symmetry in type_symmetries:
if(symmetry == ('cos')):
#Symmetry cos
#For i and j---> \xi_i(sin(x_i) cos(x_j)) = \xi_j(sin(x_j) cos(x_i))
for i in range(1, number_of_oscillators):
index_first = feature_names.index("x" + str(0))
index_second = feature_names.index("x" + str(i))
index_first_transformed = index_first
index_second_transformed = index_second + int(num_basis_functions*i)
symmetry_pair = (min(index_first_transformed, index_second_transformed), max(index_first_transformed, index_second_transformed))
pair_symmetries.append({ "x" + str(symmetry_pair[0]) : normalization_factors[index_second % num_basis_functions][0], "x" + str(symmetry_pair[1]) : -normalization_factors[index_first % num_basis_functions][0], 'constant': 0})
# for symmetry in type_symmetries:
# if(symmetry == ('cos')):
# #Symmetry cos
# #For i and j---> \xi_i(cos(x_j)) = \xi_j(cos(x_i))
# for i in range(number_of_oscillators):
# for j in range(number_of_oscillators):
# if(i != j):
# index_first = feature_names.index("x" + str(j))
# index_second = feature_names.index("x" + str(i))
# index_first_transformed = index_first + + int(num_basis_functions*i)
# index_second_transformed = index_second + int(num_basis_functions*j)
# symmetry_pair = (min(index_first_transformed, index_second_transformed), max(index_first_transformed, index_second_transformed))
# pair_symmetries.append({ "x" + str(symmetry_pair[0]) : normalization_factors[index_second % num_basis_functions][0], "x" + str(symmetry_pair[1]) : -normalization_factors[index_first % num_basis_functions][0], 'constant': 0})
for symmetry in type_symmetries:
if(symmetry == ('sin')):
#Symmetry sin
for i in range(1, number_of_oscillators):
index_first = feature_names.index("x" + str(number_of_oscillators + 0))
index_second = feature_names.index("x" + str(number_of_oscillators + i))
index_first_transformed = index_first
index_second_transformed = index_second + int(num_basis_functions*i)
symmetry_pair = (min(index_first_transformed, index_second_transformed), max(index_first_transformed, index_second_transformed))
pair_symmetries.append({ "x" + str(symmetry_pair[0]) : normalization_factors[index_second % num_basis_functions][0], "x" + str(symmetry_pair[1]) : -normalization_factors[index_first % num_basis_functions][0], 'constant': 0})
# for symmetry in type_symmetries:
# if(symmetry == ('sin')):
# #Symmetry cos
# #For i and j---> \xi_i(sin(x_j)) = \xi_j(sin(x_i))
# for i in range(number_of_oscillators):
# for j in range(number_of_oscillators):
# if(i != j):
# index_first = feature_names.index("x" + str(j + number_of_oscillators))
# index_second = feature_names.index("x" + str(i + number_of_oscillators))
# index_first_transformed = index_first + + int(num_basis_functions*i)
# index_second_transformed = index_second + int(num_basis_functions*j)
# symmetry_pair = (min(index_first_transformed, index_second_transformed), max(index_first_transformed, index_second_transformed))
# pair_symmetries.append({ "x" + str(symmetry_pair[0]) : normalization_factors[index_second % num_basis_functions][0], "x" + str(symmetry_pair[1]) : -normalization_factors[index_first % num_basis_functions][0], 'constant': 0})
# Remove duplicates before we output.
# pair_symmetries = [dict(t) for t in {tuple(d.keys()) for d in pair_symmetries}]
pair_symmetries =delete_dictionaries_with_duplicate_keys_backup(pair_symmetries)
return pair_symmetries
def add_constraints_symmetry_kuramoto_simple(polinomial, number_of_oscillators, normalization_factors):
feature_names = polinomial.get_feature_names()
num_basis_functions = len(feature_names)
pair_symmetries = []
#Constraint on the cos^2 terms:
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
index_first = feature_names.index("x" + str(j) + '^2')
index_first_transformed = index_first + int(num_basis_functions*i)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1, 'constant': 0})
#Constraint on the sin^2 terms:
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
index_first = feature_names.index("x" + str(number_of_oscillators + j) + '^2')
index_first_transformed = index_first + int(num_basis_functions*i)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1, 'constant': 0})
pair_symmetries = delete_dictionaries_with_duplicate_keys(pair_symmetries)
return pair_symmetries
def add_constraints_symmetry_fermi_pasta_ulam_tsingou(polinomial, number_of_oscillators, normalization_factors):
feature_names = polinomial.get_feature_names()
num_basis_functions = len(feature_names)
pair_symmetries = []
#First order monomials
#For x_i and x_{i+-1} ---> \xi_i(x_i) = \xi_{i+-1}(x_{i+-1})
for i in range(number_of_oscillators):
for j in [i-1, i+1]:
if j >= 0 and j < number_of_oscillators:
index_first = feature_names.index("x" + str(i))
index_first_transformed = index_first + int(num_basis_functions*i)
index_second = feature_names.index("x" + str(j))
index_second_transformed = index_second + int(num_basis_functions*j)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
#First order monomials
#For x_i and x_{i+-1} ---> \xi_i(x_{i+-1}) = \xi_{i+-1}(x_{i})
for i in range(number_of_oscillators):
for j in [i-1, i+1]:
if j >= 0 and j < number_of_oscillators:
index_first = feature_names.index("x" + str(i))
index_first_transformed = index_first + int(num_basis_functions*j)
index_second = feature_names.index("x" + str(j))
index_second_transformed = index_second + int(num_basis_functions*i)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
#Second order monomials
#For x_i and x_{i+-1} ---> \xi_i(x_i x_{i+-1}) = \xi_{i+-1}(x_i x_{i+-1})
for i in range(number_of_oscillators):
for j in [i-1, i+1]:
if j >= 0 and j < number_of_oscillators:
if(i <= j):
index = feature_names.index("x" + str(i) + " x" + str(j))
else:
index = feature_names.index("x" + str(j) + " x" + str(i))
index_first_transformed = index + int(num_basis_functions*i)
index_second_transformed = index + int(num_basis_functions*j)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
#Second order monomials
#For x_i and x_{i+-1} ---> \xi_i(x_i^2) = \xi_{i+-1}(x_{i+-1}^2)
for i in range(number_of_oscillators):
for j in [i-1, i+1]:
if j >= 0 and j < number_of_oscillators:
index_first = feature_names.index("x" + str(i) + "^2")
index_second = feature_names.index("x" + str(j) + "^2")
index_first_transformed = index_first + int(num_basis_functions*i)
index_second_transformed = index_second + int(num_basis_functions*j)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
#Second order monomials
#For x_i and x_{i+-1} ---> \xi_i(x_{i+-1}^2) = \xi_{i+-1}(x_i^2)
for i in range(number_of_oscillators):
for j in [i-1, i+1]:
if j >= 0 and j < number_of_oscillators:
index_first = feature_names.index("x" + str(i) + "^2")
index_second = feature_names.index("x" + str(j) + "^2")
index_first_transformed = index_first + int(num_basis_functions*j)
index_second_transformed = index_second + int(num_basis_functions*i)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
#Third order monomials
#For x_i and x_{i+-1} ---> \xi_i(x_i^2 x_{i+-1}) = \xi_{i+-1}(x_{i+-1}^2 x_i)
for i in range(number_of_oscillators):
for j in [i-1, i+1]:
if j >= 0 and j < number_of_oscillators:
if(i <= j):
index_first = feature_names.index("x" + str(i) + "^2 x" + str(j))
index_second = feature_names.index("x" + str(i) + " x" + str(j) + "^2")
else:
index_first = feature_names.index("x" + str(j) + " x" + str(i) + "^2")
index_second = feature_names.index("x" + str(j) + "^2 x" + str(i))
index_first_transformed = index_first + int(num_basis_functions*i)
index_second_transformed = index_second + int(num_basis_functions*j)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
#Third order monomials
#For x_i and x_{i+-1} ---> \xi_i(x_i x_{i+-1}^2) = \xi_{i+-1}(x_{i+-1} x_i^2)
for i in range(number_of_oscillators):
for j in [i-1, i+1]:
if j >= 0 and j < number_of_oscillators:
if(i <= j):
index_first = feature_names.index("x" + str(i) + " x" + str(j) + "^2")
index_second = feature_names.index("x" + str(i) + "^2 x" + str(j))
else:
index_first = feature_names.index("x" + str(j) + "^2 x" + str(i))
index_second = feature_names.index("x" + str(j) + " x" + str(i) + "^2")
index_first_transformed = index_first + int(num_basis_functions*i)
index_second_transformed = index_second + int(num_basis_functions*j)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
#Third order monomials
#For x_i and x_{i+-1} ---> \xi_i(x_i^3) = \xi_{i+-1}(x_{i+-1}^3)
for i in range(number_of_oscillators):
for j in [i-1, i+1]:
if j >= 0 and j < number_of_oscillators:
index_first = feature_names.index("x" + str(i) + "^3")
index_second = feature_names.index("x" + str(j) + "^3")
index_first_transformed = index_first + int(num_basis_functions*i)
index_second_transformed = index_second + int(num_basis_functions*j)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
#Third order monomials
#For x_i and x_{i+-1} ---> \xi_i(x_{i+-1}^3) = \xi_{i+-1}(x_{i}^3)
for i in range(number_of_oscillators):
for j in [i-1, i+1]:
if j >= 0 and j < number_of_oscillators:
index_first = feature_names.index("x" + str(i) + "^3")
index_second = feature_names.index("x" + str(j) + "^3")
index_first_transformed = index_first + int(num_basis_functions*j)
index_second_transformed = index_second + int(num_basis_functions*i)
pair_symmetries.append({ "x" + str(index_first_transformed) : 1.0/normalization_factors[index_first % num_basis_functions][0], "x" + str(index_second_transformed) : -1.0/normalization_factors[index_second % num_basis_functions][0], 'constant': 0})
pair_symmetries = delete_dictionaries_with_duplicate_keys(pair_symmetries)
return pair_symmetries
def delete_dictionaries_with_duplicate_keys(list_dictionaries):
seen = set()
new_dictionary = []
for d in list_dictionaries:
t = tuple(sorted(d.keys()))
if t not in seen:
seen.add(t)
new_dictionary.append(d)
return new_dictionary
def delete_dictionaries_with_duplicate_keys_backup(list_dictionaries):
seen = set()
new_dictionary = []
for d in list_dictionaries:
t = tuple(d.keys())
if t not in seen:
seen.add(t)
new_dictionary.append(d)
return new_dictionary
def add_constraints_symmetry_kuramoto_backup_v2(polinomial, number_of_oscillators):
type_symmetries = [('sin', 'cos'), ('cos'), ('sin')]
feature_names = polinomial.get_feature_names()
pair_symmetries = []
for symmetry in type_symmetries:
if(symmetry == ('sin', 'cos') or symmetry == ('cos', 'sin')):
#Symmetry cos-sin
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
if(i != j):
index_first = feature_names.index("x" + str(i) + ' x' + str(number_of_oscillators + j))
index_second = feature_names.index("x" + str(j) + ' x' + str(number_of_oscillators + i))
index_first_transformed = index_first + int(len(feature_names)*i)
index_second_transformed = index_second + int(len(feature_names)*j)
symmetry_pair = (min(index_first_transformed, index_second_transformed), max(index_first_transformed, index_second_transformed))
pair_symmetries.append(symmetry_pair)
#Symmetry sin-cos
for i in range(number_of_oscillators):
for j in range(number_of_oscillators):
if(i != j):
index_first = feature_names.index("x" + str(j) + ' x' + str(i + number_of_oscillators))
index_second = feature_names.index("x" + str(i) + ' x' + str(j + number_of_oscillators))
index_first_transformed = index_first + + int(len(feature_names)*i)
index_second_transformed = index_second + int(len(feature_names)*j)
symmetry_pair = (min(index_first_transformed, index_second_transformed), max(index_first_transformed, index_second_transformed))
pair_symmetries.append(symmetry_pair)
for symmetry in type_symmetries:
if(symmetry == ('cos')):
#Symmetry cos
for i in range(1, number_of_oscillators):
index_first = feature_names.index("x" + str(0))
index_second = feature_names.index("x" + str(i))
index_first_transformed = index_first
index_second_transformed = index_second + int(len(feature_names)*i)
symmetry_pair = (min(index_first_transformed, index_second_transformed), max(index_first_transformed, index_second_transformed))
pair_symmetries.append(symmetry_pair)
for symmetry in type_symmetries:
if(symmetry == ('sin')):
#Symmetry sin
for i in range(1, number_of_oscillators):
index_first = feature_names.index("x" + str(number_of_oscillators + 0))
index_second = feature_names.index("x" + str(number_of_oscillators + i))
index_first_transformed = index_first
index_second_transformed = index_second + int(len(feature_names)*i)
symmetry_pair = (min(index_first_transformed, index_second_transformed), max(index_first_transformed, index_second_transformed))
pair_symmetries.append(symmetry_pair)
#Remove duplicates before we output.
pair_symmetries = list(set([i for i in pair_symmetries]))
return pair_symmetries
def fermi_pasta_ulam_random(number_of_oscillators, number_of_snapshots, beta=0.7, x_min = - 0.1, x_max = 0.1):
"""Fermi–Pasta–Ulam problem.
Generate data for the Fermi–Pasta–Ulam problem represented by the differential equation
d^2/dt^2 x_i = (x_i+1 - 2x_i + x_i-1) + beta((x_i+1 - x_i)^3 - (x_i-x_i-1)^3).
See [1]_ for details.
Parameters
----------
number_of_oscillators: int
number of oscillators
number_of_snapshots: int
number of snapshots
Returns
-------
snapshots: ndarray(number_of_oscillators, number_of_snapshots)
snapshot matrix containing random displacements of the oscillators in [-0.1,0.1]
derivatives: ndarray(number_of_oscillators, number_of_snapshots)
matrix containing the corresponding derivatives
References
----------
.. [1] P. Gelß, S. Klus, J. Eisert, C. Schütte, "Multidimensional Approximation of Nonlinear Dynamical Systems",
arXiv:1809.02448, 2018
"""
# define random snapshot matrix
snapshots = (x_max - x_min) * (np.random.rand(number_of_oscillators, number_of_snapshots) - 0.5) + (x_max + x_min)/2.0
# compute derivatives
derivatives = np.zeros((number_of_oscillators, number_of_snapshots))
for j in range(number_of_snapshots):
derivatives[0, j] = (
snapshots[1, j]
- 2 * snapshots[0, j]
+ beta * ((snapshots[1, j] - snapshots[0, j]) ** 3 - snapshots[0, j] ** 3)
)
for i in range(1, number_of_oscillators - 1):
derivatives[i, j] = (
snapshots[i + 1, j]
- 2 * snapshots[i, j]
+ snapshots[i - 1, j]
+ beta
* (
(snapshots[i + 1, j] - snapshots[i, j]) ** 3
- (snapshots[i, j] - snapshots[i - 1, j]) ** 3
)
)
derivatives[-1, j] = (
-2 * snapshots[-1, j]
+ snapshots[-2, j]
+ beta
* (-snapshots[-1, j] ** 3 - (snapshots[-1, j] - snapshots[-2, j]) ** 3)
)
return snapshots, derivatives
def fermi_pasta_ulam_time(number_of_oscillators, exact_solution, polinomial, max_time, number_of_snapshots, number_of_experiments, relative_tolerance = 1.0e-10, absolute_tolerance = 1.0e-13):
from scipy.integrate import solve_ivp
# Plot the exact trajectory.
def fun_exact(t, y):
return np.dot(
exact_solution, polinomial.fit_transform(y.reshape(1, -1)).T
).squeeze()
list_snapshots = []
list_derivatives = []
list_times = []
for i in range(number_of_experiments):
initial_position = np.random.uniform(-0.1, 1.0, size=number_of_oscillators)
sol = solve_ivp(fun_exact, [0, max_time], initial_position, method='DOP853', t_eval=np.linspace(0, max_time, number_of_snapshots), rtol = relative_tolerance, atol = absolute_tolerance)
snapshots = sol.y
derivatives = np.zeros([number_of_oscillators, number_of_snapshots])
for i in range(number_of_snapshots):
derivatives[:, i] = fun_exact(0, snapshots[:, i])
list_derivatives.append(derivatives)
list_times.append(sol['t'])
list_snapshots.append(snapshots)
return list_snapshots, list_derivatives, list_times
def fermi_pasta_ulam_time_correct(number_of_oscillators, exact_solution, polinomial, max_time, number_of_snapshots, number_of_experiments, relative_tolerance = 1.0e-10, absolute_tolerance = 1.0e-13):
from scipy.integrate import solve_ivp
# Plot the exact trajectory.
# Assume that the
def fun_exact(t, y):
position = y[:number_of_oscillators]
velocity = y[number_of_oscillators:]
return np.append(velocity, np.dot(
exact_solution, polinomial.fit_transform(position.reshape(1, -1)).T
).squeeze())
list_snapshots = []
list_derivatives = []
list_second_derivatives = []
list_times = []
for i in range(number_of_experiments):
initial_position = np.random.uniform(-0.1, 0.1, size=number_of_oscillators)
initial_velocity = np.zeros(number_of_oscillators)
vector = np.append(initial_position, initial_velocity)
sol = solve_ivp(fun_exact, [0, max_time], vector, method='DOP853', t_eval=np.linspace(0, max_time, number_of_snapshots), rtol = relative_tolerance, atol = absolute_tolerance)
snapshots = sol.y
second_derivatives = np.zeros([number_of_oscillators, number_of_snapshots])
for i in range(number_of_snapshots):
second_derivatives[:, i] = fun_exact(0, snapshots[:number_of_oscillators, i])
list_second_derivatives.append(second_derivatives)
list_derivatives.append(snapshots[number_of_oscillators:, :])
list_snapshots.append(snapshots[:number_of_oscillators, :])
list_times.append(sol['t'])
return list_snapshots, list_derivatives, list_second_derivatives, list_times
# def kuramoto_time_individual(frequencies, initial_position, start_time, end_time, number_of_samples, K = 2.0, forcing_param = 0.2, relative_tolerance = 1.0e-10, absolute_tolerance = 1.0e-13):
# from scipy.integrate import solve_ivp
# number_of_oscillators = len(frequencies)
# def kuramoto_ode(_, theta):
# [theta_i, theta_j] = np.meshgrid(theta, theta)
# return frequencies + K / number_of_oscillators * np.sin(theta_j - theta_i).sum(0) + forcing_param * np.sin(theta)
# sol = solve_ivp(kuramoto_ode, [start_time, end_time], initial_position, method='DOP853', t_eval=np.linspace(start_time, end_time, number_of_samples), rtol = relative_tolerance, atol = absolute_tolerance)
# return sol.y, sol['t']
# # Perform an experiment where we let the system develop from an initial position.
# def fermi_pasta_ulam_time_individual(
# exact_solution, polinomial, initial_position, t_min, t_max, num_steps
# ):
# # Plot the exact trajectory.
# def fun_exact(t, y):
# return np.dot(
# exact_solution, polinomial.fit_transform(y.reshape(1, -1)).T
# ).squeeze()
# from scipy.integrate import solve_ivp
# sol_true = solve_ivp(
# fun_exact,
# [t_min, t_max],
# initial_position,
# t_eval=np.linspace(t_min, t_max, num_steps),
# vectorized=False,
# )
# assert (
# sol_true["status"] == 0
# ), "The integration of the initial value solver was not succesfull."
# return sol_true.y, sol_true['t']
# Perform an experiment where we let the system develop from an initial position.
def fermi_pasta_ulam_time_individual(
exact_solution, polinomial, initial_position, t_min, t_max, num_steps
):
# Plot the exact trajectory.
def fun_exact(t, y):
return np.dot(
exact_solution, polinomial.fit_transform(y.reshape(1, -1)).T
).squeeze()
from scipy.integrate import solve_ivp
sol_true = solve_ivp(
fun_exact,
[t_min, t_max],
initial_position,
t_eval=np.linspace(t_min, t_max, num_steps),
vectorized=False,
)
assert (
sol_true["status"] == 0
), "The integration of the initial value solver was not succesfull."
return sol_true.y, sol_true['t']
# Perform an experiment where we let the system develop from an initial position.
def fermi_pasta_ulam_time_individual_correct(
exact_solution, polinomial, initial_position, initial_velocity, t_min, t_max, num_steps
):
number_of_oscillators = exact_solution.shape[0]
# Plot the exact trajectory.
# Assume that the
def fun_exact(t, y):
position = y[:number_of_oscillators]
velocity = y[number_of_oscillators:]
return np.append(velocity, np.dot(
exact_solution, polinomial.fit_transform(position.reshape(1, -1)).T
).squeeze())
from scipy.integrate import solve_ivp
position_val = np.append(initial_position, initial_velocity)
sol_true = solve_ivp(
fun_exact,
[t_min, t_max],
position_val,
t_eval=np.linspace(t_min, t_max, num_steps),
vectorized=False,
)
assert (
sol_true["status"] == 0
), "The integration of the initial value solver was not succesfull."
return sol_true.y[:number_of_oscillators], sol_true.y[number_of_oscillators:], sol_true['t']
# Build the exact dynamics of the fermi_pasta_ulam if we have polynomials of order up to three.
def exact_solution_fermi_pasta_ulam(number_of_oscillators, polinomial, beta):
reference_polynomial = polinomial.get_feature_names()
num_basis_functions = len(reference_polynomial)
# Build the exact solution at the boundary
exact_solution = np.zeros((number_of_oscillators, num_basis_functions))
# First order terms.
exact_solution[0, reference_polynomial.index("x0")] = -2
exact_solution[0, reference_polynomial.index("x1")] = 1
# Third order terms
exact_solution[0, reference_polynomial.index("x0^3")] = -2 * beta
exact_solution[0, reference_polynomial.index("x1^3")] = beta
exact_solution[0, reference_polynomial.index("x0^2 x1")] = 3 * beta
exact_solution[0, reference_polynomial.index("x0 x1^2")] = -3 * beta
# Build the exact solution in the interior.
for i in range(1, number_of_oscillators - 1):
# First order terms.
exact_solution[i, reference_polynomial.index("x" + str(i))] = -2
exact_solution[i, reference_polynomial.index("x" + str(i - 1))] = 1
exact_solution[i, reference_polynomial.index("x" + str(i + 1))] = 1
# Third order terms
exact_solution[i, reference_polynomial.index("x" + str(i) + "^3")] = -2 * beta
exact_solution[i, reference_polynomial.index("x" + str(i - 1) + "^3")] = (
1 * beta
)
exact_solution[i, reference_polynomial.index("x" + str(i + 1) + "^3")] = (
1 * beta
)
exact_solution[
i, reference_polynomial.index("x" + str(i) + "^2 x" + str(i + 1))
] = (3 * beta)
exact_solution[
i, reference_polynomial.index("x" + str(i) + " x" + str(i + 1) + "^2")
] = (-3 * beta)
exact_solution[
i, reference_polynomial.index("x" + str(i - 1) + " x" + str(i) + "^2")
] = (3 * beta)
exact_solution[
i, reference_polynomial.index("x" + str(i - 1) + "^2 x" + str(i))
] = (-3 * beta)
# Equation for the end point.
exact_solution[
number_of_oscillators - 1,
reference_polynomial.index("x" + str(number_of_oscillators - 1)),
] = -2
exact_solution[
number_of_oscillators - 1,
reference_polynomial.index("x" + str(number_of_oscillators - 2)),
] = 1
# Third order terms
exact_solution[
number_of_oscillators - 1,
reference_polynomial.index("x" + str(number_of_oscillators - 1) + "^3"),
] = (-2 * beta)
exact_solution[
number_of_oscillators - 1,
reference_polynomial.index("x" + str(number_of_oscillators - 2) + "^3"),
] = beta
exact_solution[
number_of_oscillators - 1,
reference_polynomial.index(
"x"
+ str(number_of_oscillators - 2)
+ "^2 x"
+ str(number_of_oscillators - 1)
),
] = (-3 * beta)
exact_solution[
number_of_oscillators - 1,
reference_polynomial.index(
"x"
+ str(number_of_oscillators - 2)
+ " x"
+ str(number_of_oscillators - 1)
+ "^2"
),
] = (3 * beta)
return exact_solution
def brusselator_time(initial_position, t_min = 0.0, t_max = 10.0, num_steps = 1000, r_coefficients = [1, 3, 1, 1]):
#% rate constants:
#r1 = 1; % 0 -> A
#r2 = 3; % A -> B
#r3 = 1; % 2A + B -> 3A
#r4 = 1; % A -> 0
r1, r2, r3, r4 = r_coefficients
def fun_exact(t, y):
return np.array([r1-r2*y[0]+r3*y[0]**2*y[1]-r4*y[0], r2*y[0]-r3*y[0]**2*y[1]])
from scipy.integrate import solve_ivp
sol_true = solve_ivp(
fun_exact,
[0, t_max],
initial_position,
t_eval=np.linspace(t_min, t_max, num_steps),
vectorized=False,
)
assert (
sol_true["status"] == 0
), "The integration of the initial value solver was not succesfull."
return sol_true, fun_exact(sol_true['t'], sol_true['y'])
# Build the exact dynamics of the fermi_pasta_ulam if we have polynomials of order up to three.
def exact_solution_brusselator(dimension, polinomial, r_coefficients):
r1, r2, r3, r4 = r_coefficients
reference_polynomial = polinomial.get_feature_names()
num_basis_functions = len(reference_polynomial)
# Build the exact solution at the boundary
exact_solution = np.zeros((dimension, num_basis_functions))
# Solution for the first species
exact_solution[0, reference_polynomial.index("1")] = r1
exact_solution[0, reference_polynomial.index("x0")] = - r2 - r4
exact_solution[0, reference_polynomial.index("x0")] = - r2 - r4
exact_solution[0, reference_polynomial.index("x0^2 x1")] = r3
# Solution for the first species
exact_solution[1, reference_polynomial.index("x0")] = r2
exact_solution[1, reference_polynomial.index("x0^2 x1")] = -r3
return exact_solution
def lutka_volterra_time(initial_position, t_min = 0.0, t_max = 10.0, num_steps = 1000, r_coefficients = [1, 1, 1, 1]):
#% rate constants:
#r1 = 1; % reproduction of prey: A -> 2A
#r2 = 1; % death of predator: B -> 0
#r3 = 1; % consumption: A + B -> B
#r4 = 1; % reproduction of predator: A + B -> A + 2B
r1, r2, r3, r4 = r_coefficients
def fun_exact(t, y):
return np.array([y[0]*(r1-r3*y[1]), -y[1]*(r2-r4*y[0])])
from scipy.integrate import solve_ivp
sol_true = solve_ivp(
fun_exact,
[0, t_max],
initial_position,
t_eval=np.linspace(t_min, t_max, num_steps),
vectorized=False,
)
assert (
sol_true["status"] == 0
), "The integration of the initial value solver was not succesfull."
return sol_true, fun_exact(sol_true['t'], sol_true['y'])
# Build the exact dynamics of the fermi_pasta_ulam if we have polynomials of order up to three.
def exact_solution_lutka_volterra(dimension, polinomial, r_coefficients):
r1, r2, r3, r4 = r_coefficients
reference_polynomial = polinomial.get_feature_names()
num_basis_functions = len(reference_polynomial)
# Build the exact solution at the boundary
exact_solution = np.zeros((dimension, num_basis_functions))
# Solution for the first species
exact_solution[0, reference_polynomial.index("x0")] = r1
exact_solution[0, reference_polynomial.index("x0 x1")] = - r3
# Solution for the first species
exact_solution[1, reference_polynomial.index("x1")] = - r2
exact_solution[1, reference_polynomial.index("x0 x1")] = r4
return exact_solution
def michaelis_menten_time(dimension, number_of_snapshots, number_of_experiments = 10, t_min = 0.0, t_max = 10.0, coefficients = [0.01, 1, 1]):
#Expressions
#d/dt C_1 = - k_{1}*C_{1}*C_{2} + k_{-1}* C_{3}
#d/dt C_2 = - k_{1}*C_{1}*C_{2} + (k_{-1} + k_{2})* C_{3}
#d/dt C_3 = k_{1}*C_{1}*C_{2} - (k_{-1} + k_{2})* C_{3}
#d/dt C_4 = k_{2}*C_{3}
# initial_position = np.array([1.0, 0.7, 0.0, 0.0])
k_1, k_2, k_minus1 = coefficients
def fun_exact(t, y):
return np.array([-k_1*y[0]*y[1] + k_minus1*y[2], -k_1*y[0]*y[1] + (k_minus1 + k_2)*y[2], k_1*y[0]*y[1] - (k_minus1 + k_2)*y[2], k_2*y[2]])
from scipy.integrate import solve_ivp
list_snapshots = []
list_derivatives = []
list_times = []
for i in range(number_of_experiments):
initial_position = np.random.rand(4)
sol = solve_ivp(
fun_exact,
[0, t_max],
initial_position,
t_eval=np.linspace(t_min, t_max, number_of_snapshots),
vectorized=False,
)
assert (
sol["status"] == 0
), "The integration of the initial value solver was not succesfull."
snapshots = sol.y
derivatives = np.zeros([dimension, number_of_snapshots])
for i in range(number_of_snapshots):
derivatives[:, i] = fun_exact(0, snapshots[:, i])
list_derivatives.append(derivatives)
list_times.append(sol['t'])
list_snapshots.append(snapshots)
return list_snapshots, list_derivatives, list_times
def michaelis_menten_time_individual(initial_position, t_min = 0.0, t_max = 10.0, num_steps = 1000, coefficients = [0.01, 1, 1]):
#Expressions
#d/dt C_1 = - k_{1}*C_{1}*C_{2} + k_{-1}* C_{3}
#d/dt C_2 = - k_{1}*C_{1}*C_{2} + (k_{-1} + k_{2})* C_{3}
#d/dt C_3 = k_{1}*C_{1}*C_{2} - (k_{-1} + k_{2})* C_{3}
#d/dt C_4 = k_{2}*C_{3}
k_1, k_2, k_minus1 = coefficients
def fun_exact(t, y):
return np.array([-k_1*y[0]*y[1] + k_minus1*y[2], -k_1*y[0]*y[1] + (k_minus1 + k_2)*y[2], k_1*y[0]*y[1] - (k_minus1 + k_2)*y[2], k_2*y[2]])
from scipy.integrate import solve_ivp
sol_true = solve_ivp(
fun_exact,
[0, t_max],
initial_position,
t_eval=np.linspace(t_min, t_max, num_steps),
vectorized=False,
)
assert (
sol_true["status"] == 0
), "The integration of the initial value solver was not succesfull."
return sol_true, fun_exact(sol_true['t'], sol_true['y'])
# Build the exact dynamics of the fermi_pasta_ulam if we have polynomials of order up to three.
def exact_solution_michaelis_menten(dimension, polinomial, coefficients):
assert dimension == 4, "The dimension of the michaelis-menten dynamic should be 4."
k_1, k_2, k_minus1 = coefficients
reference_polynomial = polinomial.get_feature_names()