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Van_der_Pol.py
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#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Wed Jun 28 13:55:18 2023
@author: maslyaev
"""
import numpy as np
import time
import torch
import matplotlib.pyplot as plt
import matplotlib
SMALL_SIZE = 12
matplotlib.rc('font', size=SMALL_SIZE)
matplotlib.rc('axes', titlesize=SMALL_SIZE)
from functools import reduce
import pysindy as ps
import epde.interface.interface as epde_alg
from epde.interface.solver_integration import BOPElement
from epde.interface.logger import Logger
from epde.preprocessing.preprocessor_setups import PreprocessorSetup
from epde.preprocessing.preprocessor import ConcretePrepBuilder
from epde.interface.equation_translator import translate_equation
def second_order_ODE_by_RK(initial: tuple, timestep: float, steps: int, epsilon: float):
res = np.full(shape = (steps, 2), fill_value = initial, dtype=np.float64)
for step in range(steps-1):
t = step*timestep
k1 = res[step, 1] ; x1 = res[step, 0] + timestep/2. * k1
l1 = - epsilon*(res[step, 0]**2 - 1)*res[step, 1] - res[step, 0]; y1 = res[step, 1] + timestep/2. * l1
k2 = y1; x2 = res[step, 0] + timestep/2. * k2
l2 = - epsilon*(x1**2 - 1)*y1 - x1; y2 = res[step, 1] + timestep/2. * l2
k3 = y2
l3 = - epsilon*(x2**2 - 1)*y2 - x2
x3 = res[step, 0] + timestep * k1 - 2 * timestep * k2 + 2 * timestep * k3
y3 = res[step, 1] + timestep * l1 - 2 * timestep * l2 + 2 * timestep * l3
k4 = y3
l4 = - epsilon*(x3**2 - 1)*y3 - x3
res[step+1, 0] = res[step, 0] + timestep / 6. * (k1 + 2 * k2 + 2 * k3 + k4)
res[step+1, 1] = res[step, 1] + timestep / 6. * (l1 + 2 * l2 + 2 * l3 + l4)
return res
def prepare_data(initial = (np.sqrt(3)/2., 1./2.), step = 0.05, steps_num = 640, epsilon = 0.2):
t = np.arange(start = 0., stop = step * steps_num, step = step)
solution = second_order_ODE_by_RK(initial=initial, timestep=step, steps=steps_num,
epsilon=epsilon)
return t, solution
def translate_sindy_eq(equation):
print(equation)
correspondence = {"0" : "u{power: 1.0}",
"0_1" : "du/dx1{power: 1.0}",
"1" : "v{power: 1.0}",
"1_1" : "dv/dx1{power: 1.0}",}
def replace(term):
term = term.replace(' ', '').split('x')
for idx, factor in enumerate(term[1:]):
try:
if '^' in factor:
factor = factor.split('^')
term[idx+1] = correspondence[factor[0]].replace('{power: 1.0}', '{power: '+str(factor[1])+'.0}')
else:
term[idx+1] = correspondence[factor]
except KeyError:
print(f'Key of term {factor} is missing')
raise KeyError()
return term
if isinstance(equation, str):
terms = []
split_eq = equation.split('+')
const = split_eq[0][:-2]
if 'x' in const:
for term in split_eq:
print('To replace:', term, replace(term))
terms.append(reduce(lambda x, y: x + ' * ' + y, replace(term)))
terms_comb = reduce(lambda x, y: x + ' + ' + y, terms) + ' + 0.0 = du/dx1{power: 1.0}'
else:
for term in split_eq[1:]:
print('To replace:', term, replace(term))
terms.append(reduce(lambda x, y: x + ' * ' + y, replace(term)))
terms_comb = reduce(lambda x, y: x + ' + ' + y, terms) + ' + ' + const + ' = du/dx1{power: 1.0}'
return terms_comb
elif isinstance(equation, list):
assert len(equation) == 2
var_list = ['u', 'v', 'w']
eqs_tr = []
for idx, eq in enumerate(equation):
terms = []
split_eq = eq.split('+')
const = split_eq[0][:-2]
if 'x' in const:
for term in split_eq:
print('To replace:', term, replace(term))
terms.append(reduce(lambda x, y: x + ' * ' + y, replace(term)))
terms_comb = reduce(lambda x, y: x + ' + ' + y, terms) + ' + 0.0 = d' + var_list[idx] + '/dx1{power: 1.0}'
else:
for term in split_eq[1:]:
print('To replace:', term, replace(term))
terms.append(reduce(lambda x, y: x + ' * ' + y, replace(term)))
terms_comb = reduce(lambda x, y: x + ' + ' + y, terms) + ' + ' + const + ' = d' + var_list[idx] + '/dx1{power: 1.0}'
eqs_tr.append(terms_comb)
print('Translated system:', eqs_tr)
return eqs_tr
else:
raise NotImplementedError()
def get_epde_pool(t, x, y, use_ann = False):
dimensionality = x.ndim - 1
'''
Подбираем Парето-множество систем дифф. уравнений.
'''
epde_search_obj = epde_alg.EpdeSearch(use_solver = False, dimensionality = dimensionality, boundary = 10,
coordinate_tensors = [t,])
if use_ann:
epde_search_obj.set_preprocessor(default_preprocessor_type='ANN',
preprocessor_kwargs={'epochs_max' : 35000})
else:
epde_search_obj.set_preprocessor(default_preprocessor_type='poly',
preprocessor_kwargs={'use_smoothing' : True, 'sigma' : 1,
'polynomial_window' : 3, 'poly_order' : 3})
popsize = 12
epde_search_obj.set_moeadd_params(population_size = popsize, training_epochs=85)
epde_search_obj.create_pool(data=[x, y], variable_names=['u', 'v'], max_deriv_order=(1,))
return epde_search_obj.pool
def sindy_discovery(t, x, y, sparsity = 0.05):
poly_order = 4
x_train = np.array([x, y]).T
print(x_train.shape)
model = ps.SINDy(
optimizer=ps.STLSQ(alpha=sparsity),
feature_library=ps.PolynomialLibrary(degree=poly_order),
)
model.fit(
x_train,
t=t[1] - t[0],
quiet=True,
)
return model
def epde_discovery_as_system(t, x, y, use_ann = False):
dimensionality = x.ndim - 1
'''
Подбираем Парето-множество систем дифф. уравнений.
'''
epde_search_obj = epde_alg.EpdeSearch(use_solver = False, dimensionality = dimensionality, boundary = 10,
coordinate_tensors = [t,])
if use_ann:
epde_search_obj.set_preprocessor(default_preprocessor_type='ANN',
preprocessor_kwargs={'epochs_max' : 35000})
else:
epde_search_obj.set_preprocessor(default_preprocessor_type='poly',
preprocessor_kwargs={'use_smoothing' : True, 'sigma' : 1,
'polynomial_window' : 3, 'poly_order' : 3})
popsize = 12
epde_search_obj.set_moeadd_params(population_size = popsize, training_epochs=85)
factors_max_number = {'factors_num' : [1, 2, 3], 'probas' : [0.4, 0.3, 0.3]}
epde_search_obj.fit(data=[x, y], variable_names=['u', 'v'], max_deriv_order=(1,),
equation_terms_max_number=6, data_fun_pow = 3,
equation_factors_max_number=factors_max_number,
eq_sparsity_interval=(1e-12, 1e-4))
'''
Смотрим на найденное Парето-множество,
"идеальное уравнение" имеет вид:
/ -1.3048888807580532 * u{power: 1.0} * v{power: 1.0} + 0.3922851274813135 * u{power: 1.0} + -0.0003917278536547386 = du/dx1{power: 1.0}
\ -0.9740492564964498 * v{power: 1.0} + 0.9717873909925121 * u{power: 1.0} * v{power: 1.0} + 0.0003500773115704403 = dv/dx1{power: 1.0}
{'terms_number': {'optimizable': False, 'value': 3}, 'max_factors_in_term': {'optimizable': False, 'value': {'factors_num': [1, 2], 'probas': [0.8, 0.2]}},
('sparsity', 'u'): {'optimizable': True, 'value': 0.00027172388370453704}, ('sparsity', 'v'): {'optimizable': True, 'value': 0.00019292375116125682}} , with objective function values of [0.2800438 0.18041074 4. 4. ]
'''
epde_search_obj.equations(only_print = True, num = 1)
equation_obtained = False; compl = [5,]; attempt = 0
while not equation_obtained:
try:
sys = epde_search_obj.get_equations_by_complexity(compl)
res = sys[0]
except IndexError:
compl[attempt % 2] += 1
attempt += 1
continue
equation_obtained = True
return epde_search_obj, res
def epde_discovery_as_ode(t, x, y, use_ann = False):
dimensionality = x.ndim - 1
'''
Подбираем Парето-множество систем дифф. уравнений.
'''
epde_search_obj = epde_alg.EpdeSearch(use_solver = False, dimensionality = dimensionality, boundary = 50,
coordinate_tensors = [t,])
if use_ann:
epde_search_obj.set_preprocessor(default_preprocessor_type='ANN',
preprocessor_kwargs={'epochs_max' : 35000})#
else:
epde_search_obj.set_preprocessor(default_preprocessor_type='poly',
preprocessor_kwargs={'use_smoothing' : True, 'sigma' : 1,
'polynomial_window' : 3, 'poly_order' : 3}) # 'epochs_max' : 10000})#
popsize = 12
epde_search_obj.set_moeadd_params(population_size = popsize, training_epochs=100)
factors_max_number = {'factors_num' : [1, 2, 3], 'probas' : [0.4, 0.3, 0.3]}
epde_search_obj.fit(data=[x,], variable_names=['u',], max_deriv_order=(2,),
equation_terms_max_number=6, data_fun_pow = 2,
equation_factors_max_number=factors_max_number,
eq_sparsity_interval=(1e-12, 1e-3))
epde_search_obj.equations(only_print = True, num = 1)
equation_obtained = False; compl = [5,]; attempt = 0
while not equation_obtained:
if attempt > 5:
res = epde_search_obj.equations(False)[0][0]
try:
sys = epde_search_obj.get_equations_by_complexity(compl)
res = sys[0]
except IndexError:
compl[0] += 0.5
attempt += 1
continue
equation_obtained = True
return epde_search_obj, res
if __name__ == "__main__":
as_system = False
t, x_stacked = prepare_data(steps_num=640)
t_max = 320
x, y = x_stacked[:, 0], x_stacked[:, 1]
t_train, t_test = t[:t_max], t[:t_max]
x_train, x_test = x[:t_max], x[:t_max]
y_train, y_test = y[:t_max], y[:t_max]
aux_preprocessor_type = 'poly'
aux_preprocessor_kwargs = {'use_smoothing' : False,
'include_time' : True}
setup = PreprocessorSetup()
builder = ConcretePrepBuilder()
setup.builder = builder
if aux_preprocessor_type == 'ANN':
setup.build_ANN_preprocessing(**aux_preprocessor_kwargs)
elif aux_preprocessor_type == 'poly':
setup.build_poly_diff_preprocessing(**aux_preprocessor_kwargs)
elif aux_preprocessor_type == 'spectral':
setup.build_spectral_preprocessing(**aux_preprocessor_kwargs)
else:
raise NotImplementedError('Incorrect default preprocessor type. Only ANN or poly are allowed.')
aux_preprocessor_pipeline = setup.builder.prep_pipeline
if 'max_order' not in aux_preprocessor_pipeline.deriv_calculator_kwargs.keys():
aux_preprocessor_pipeline.deriv_calculator_kwargs['max_order'] = None
run_epde = True
run_sindy = True
pool = None
pred = False
exps = {}
magnitudes = [0, 0.5*1e-2, 1.*1e-2, 2.5*1e-2, 5.*1e-2]# 10.*1e-2, 15.*1e-2] # #
for magnitude in magnitudes:
x_train_n = x_train + np.random.normal(scale = np.abs(magnitude*x_train),
size = x_train.shape)
_, dx_train_n = aux_preprocessor_pipeline.run(x_train_n, grid=[t_test,],
max_order=(1,))
dx_train_n = dx_train_n.reshape(-1)
plt.plot(t_train, x_train, color = 'k', label = 'Initial data')
plt.plot(t_train, x_train_n, color = 'r', label = 'Corrupted data')
plt.grid()
plt.legend()
plt.show()
errs_epde = []
models_epde = []
calc_epde = []
if run_epde:
test_launches = 10
for idx in range(test_launches):
t1 = time.time()
if as_system:
epde_search_obj = epde_discovery_as_system(t_train, x_train_n, y_train, True)
else:
epde_search_obj, sys = epde_discovery_as_ode(t_train, x_train_n, y_train, True)
t2 = time.time()
def get_ode_bop(key, grid_loc, value, var = 0, term = [None]):
bop = BOPElement(axis = 0, key = key, term = term, power = 1, var = var)
bop_grd_np = np.array([[grid_loc,]])
bop.set_grid(torch.from_numpy(bop_grd_np).type(torch.FloatTensor))
bop.values = torch.from_numpy(np.array([[value,]])).float()
return bop
if pred:
plt.plot(t_train[50:-50], x_train[50:-50], color = 'k', label = "x, input")
plt.plot(t_train[50:-50], epde_search_obj.cache[1].get(('u', (1.0,)))[50:-50],
color = 'r', label = "x, calculated")
plt.grid()
plt.legend()
plt.show()
plt.plot(t_train[50:-50], y_train[50:-50], color = 'k', label = "x', input")
plt.plot(t_train[50:-50], epde_search_obj.saved_derivaties['u'][50:-50, 0],
color = 'r', label = "x', calculated")
plt.grid()
plt.legend()
plt.show()
bop_u = get_ode_bop('u', t_test[0], x_test[0], term = [None])
bop_dudt = get_ode_bop('dudt', t_test[0], y_test[0], term = [0])
pred_u_v = epde_search_obj.predict(system=sys, boundary_conditions=[bop_u(), bop_dudt()],
grid = [t_test,], strategy='autograd')
pred_u_v = pred_u_v.reshape(x_test.shape)
_, pred_derivatives = aux_preprocessor_pipeline.run(pred_u_v, grid=[t_test,],
max_order=(1,))
plt.figure(figsize=(11, 6))
plt.plot(t_test, x_test, '+', color = 'b', label = 'x, test data')
plt.plot(t_test, y_test, '*', color = 'r', label = "x', test data")
plt.plot(t_test, pred_u_v, color = 'b', label='x, solution')
plt.plot(t_test, pred_derivatives[:, 0], color = 'r', label="x', solution")
plt.xlabel('Time')
plt.ylabel('x')
plt.grid()
plt.legend(loc='upper right')
plt.savefig(f'/home/maslyaev/epde/EPDE_main/projects/wSINDy/figs/van_der_pol_ANN_{magnitude}_attempt_{idx}.png', dpi = 300)
plt.show()
plt.figure(figsize=(7, 6))
plt.plot(pred_u_v, pred_derivatives[:, 0], color = 'r')
plt.scatter(x_test, y_test, s = 3, color = 'k')
plt.xlabel("x")
plt.ylabel("x'")
plt.savefig(f'/home/maslyaev/epde/EPDE_main/projects/wSINDy/figs/van_der_pol_ANN_{magnitude}_attempt_{idx}_traj.png', dpi = 300)
plt.show()
else:
pred_u_v = 0
models_epde.append(epde_search_obj)
errs_epde.append(np.mean(np.abs(x_test - pred_u_v)))
calc_epde.append(pred_u_v)
try:
logger.add_log(key = f'Van_der_Pol_noise_{magnitude}_attempt_{idx}', entry = sys, aggregation_key = ('epde', magnitude),
error_pred = np.mean(np.abs(x_test - pred_u_v)), time = t2 - t1)
except NameError:
logger = Logger(name = 'logs/Van_der_Pol_time_check.json', referential_equation = '-0.2 * du/dx1{power: 1.0} * u{power: 2.0} + 0.2 * du/dx1{power: 1.0} + -1.000 * u{power: 1.0} + 0.0 * u{power: 1.0} * d^2u/dx1^2{power: 2.0} + 0.0 = d^2u/dx1^2{power: 1.0}',
pool = epde_search_obj.pool)
logger.add_log(key = f'Van_der_Pol_noise_{magnitude}_attempt_{idx}', entry = sys, aggregation_key = ('epde', magnitude),
error_pred = np.mean(np.abs(x_test - pred_u_v)), time = t2 - t1)
errs_SINDy = []
models_SINDy = []
calc_SINDy = []
if run_sindy:
test_launches = 1
for idx in range(test_launches):
model_quality = np.inf; model_container = (None, None, None)
for sparsity_thr in [50.,]: # 7., 12., ]:
pool = get_epde_pool(t_train, x_train_n, dx_train_n)
t1 = time.time()
model_base = sindy_discovery(t_train, x_train_n, dx_train_n, sparsity=sparsity_thr)
t2 = time.time()
if pred:
print('Initial conditions', np.array([x_test[0], y_test[0]]))
eq_translated = translate_sindy_eq(model_base.equations())
sys = translate_equation({'u': eq_translated[0],
'v': eq_translated[1]}, pool)
print(sys.text_form)
pred_u_v = model_base.simulate(np.array([x_test[0], y_test[0]]), t_test)
plt.plot(t_test, x_test, '+', label = 'preys_odeint')
plt.plot(t_test, y_test, '*', label = "predators_odeint")
plt.plot(t_test, pred_u_v[:, 0], color = 'b', label='preys_NN')
plt.plot(t_test, pred_u_v[:, 1], color = 'r', label='predators_NN')
plt.xlabel('Time t, [days]')
plt.ylabel('Population')
plt.grid()
plt.legend(loc='upper right')
plt.title(f'Basic SINDy {magnitude}')
plt.show()
err_u, err_v = np.mean(np.abs(x_test - pred_u_v[:, 0])), np.mean(np.abs(y_test - pred_u_v[:, 1]))
else:
pred_u_v = 0
err_u, err_v = 0, 0
if err_u + err_v < model_quality:
model_quality = err_u + err_v
model_container = (model_base, (err_u, err_v), pred_u_v)
models_SINDy.append(model_container[0])
errs_SINDy.append(model_container[1])
calc_SINDy.append(model_container[2])
print('Discovered by SINDy:', sys.text_form)
try:
logger.add_log(key = f'VdP_SINDy_noise_{magnitude}_attempt_{idx}', entry = sys, aggregation_key = ('sindy', magnitude),
error_pred = (err_u, err_v), time = t2 - t1)
except NameError:
logger = Logger(name = 'logs/Van_der_Pol_time_check.json', referential_equation = {'u' : '20.0 * u{power: 1.0} + -20.0 * u{power: 1.0} * v{power: 1.0} + 0.0 = du/dx1{power: 1.0}',
'v' : '-20.0 * v{power: 1.0} + 20.0 * u{power: 1.0} * v{power: 1.0} + 0.0 = dv/dx1{power: 1.0}'},
pool = pool)
logger.add_log(key = f'VdP_noise_{magnitude}_attempt_{idx}', entry = sys, aggregation_key = ('sindy', magnitude),
error_pred = (err_u, err_v), time = t2 - t1)
exps[magnitude] = {'epde' : (models_epde, errs_epde, calc_epde),
'sindy' : (models_SINDy, errs_SINDy, calc_SINDy)}
logger.dump()