This example uses a Cahn-Hilliard model to compare two different bases representations to discretize the microstructure. One basis representaion uses the primitive (or hat) basis and the other uses Legendre polynomials. The example includes the background theory about using Legendre polynomials as a basis in MKS. The MKS with two different bases are compared with the standard spectral solution for the Cahn-Hilliard solution at both the calibration domain size and a scaled domain size.
The Cahn-Hilliard equation is used to simulate microstructure evolution during spinodial decomposition and has the following form,
$$ \dot{\phi} = \nabla^2 \left( \phi^3 - \phi \right) - \gamma \nabla^4 \phi $$where $\phi$ is a conserved ordered parameter and $\sqrt{\gamma}$ represents the width of the interface. In this example, the Cahn-Hilliard equation is solved using a semi-implicit spectral scheme with periodic boundary conditions, see Chang and Rutenberg for more details.
In this example, we will explore the differences when using the Legendre polynomials as the basis function compared to the primitive (or hat) basis for the microstructure function and the influence coefficients.
For more information about both of these basis please see the theory section.
In [1]:
%matplotlib inline
%load_ext autoreload
%autoreload 2
import numpy as np
import matplotlib.pyplot as plt
Because the microstructure is a continuous field that can have a range of values and changes over time, the first order influence coefficients cannot be calibrated with delta microstructures. Instead, a large number of simulations with random initial conditions will be used to calibrate the first order influence coefficients using linear regression. Let's show how this is done.
The function make_cahnHilliard from pymks.datasets provides a nice interface to generate calibration datasets for the influence coefficients. The function make_cahnHilliard requires the number of calibration samples, given by n_samples, and the size and shape of the domain, given by size.
In [2]:
import pymks
from pymks.datasets import make_cahn_hilliard
length = 41
n_samples = 400
dt = 1e-2
np.random.seed(101)
size=(length, length)
X, y = make_cahn_hilliard(n_samples=n_samples, size=size, dt=dt)
The function make_cahnHilliard has generated n_samples number of random microstructures, X, and returned the same microstructures after they have evolved for one time step, given by y. Let's take a look at one of them.
In [3]:
from pymks.tools import draw_concentrations
draw_concentrations((X[0], y[0]),('Calibration Input', 'Calibration Output'))
In this example, we compare the difference between using the primitive (or hat) basis and the Legendre polynomial basis to represent the microstructure function. As mentioned above, the microstructures (concentration fields) are not discrete phases. This leaves the number of local states in local state space n_states as a free hyperparameter. In the next section, we look to see what a practical number of local states for bases would be.
Below, we compare the difference in performance, as we vary the local state, when we choose the primitive basis and the Legendre polynomial basis.
The (X, y) sample data is split into training and test data. The code then optimizes n_states between 2 and 11 and the two basis with the parameters_to_tune variable. The GridSearchCV takes an MKSLocalizationModel instance, a scoring function (figure of merit) and the parameters_to_tune and then finds the optimal parameters with a grid search.
In [4]:
from pymks.bases import PrimitiveBasis
from sklearn.grid_search import GridSearchCV
from sklearn import metrics
mse = metrics.mean_squared_error
from pymks.bases import LegendreBasis
from pymks import MKSLocalizationModel
from sklearn.cross_validation import train_test_split
train_split_shape = (X.shape[0],) + (np.prod(X.shape[1:]),)
X_train, X_test, y_train, y_test = train_test_split(X.reshape(train_split_shape),
y.reshape(train_split_shape),
test_size=0.5, random_state=3)
prim_basis = PrimitiveBasis(2, [-1, 1])
leg_basis = LegendreBasis(2, [-1, 1])
params_to_tune = {'n_states': np.arange(2, 11),
'basis': [prim_basis, leg_basis]}
Model = MKSLocalizationModel(prim_basis)
scoring = metrics.make_scorer(lambda a, b: -mse(a, b))
fit_params = {'size': size}
gs = GridSearchCV(Model, params_to_tune, cv=5, fit_params=fit_params, n_jobs=3).fit(X_train, y_train)
The optimal parameters are the LegendreBasis with only 4 local states. More terms don't improve the R-squared value.
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print(gs.best_estimator_)
print(gs.score(X_test, y_test))
In [6]:
from pymks.tools import draw_gridscores
lgs = [x for x in gs.grid_scores_ \
if type(x.parameters['basis']) is type(leg_basis)]
pgs = [x for x in gs.grid_scores_ \
if type(x.parameters['basis']) is type(prim_basis)]
draw_gridscores([lgs, pgs], 'n_states', data_labels=['Legendre', 'Primitve'],
colors=['#f46d43', '#1a9641'], score_label='R-Squared',
param_label = 'L - Total Number of Local States')
As you can see the LegendreBasis converges faster than the PrimitiveBasis. In order to further compare performance between the two models, lets select 4 local states for both bases.
In [7]:
prim_basis = PrimitiveBasis(n_states=4, domain=[-1, 1])
prim_model = MKSLocalizationModel(basis=prim_basis)
prim_model.fit(X, y)
leg_basis = LegendreBasis(4, [-1, 1])
leg_model = MKSLocalizationModel(basis=leg_basis)
leg_model.fit(X, y)
Now let's look at the influence coefficients for both bases.
First, the PrimitiveBasis influence coefficients:
In [8]:
from pymks.tools import draw_coeff
draw_coeff(prim_model.coeff)
Now, the LegendreBasis influence coefficients:
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draw_coeff(leg_model.coeff)
Now, let's do some simulations with both sets of coefficients and compare the results.
In order to compare the difference between the two bases, we need to have the Cahn-Hilliard simulation and the two MKS models start with the same initial concentration phi0 and evolve in time. In order to do the Cahn-Hilliard simulation, we need an instance of the class CahnHilliardSimulation.
In [10]:
from pymks.datasets.cahn_hilliard_simulation import CahnHilliardSimulation
np.random.seed(66)
phi0 = np.random.normal(0, 1e-9, ((1,) + size))
ch_sim = CahnHilliardSimulation(dt=dt)
phi_sim = phi0.copy()
phi_prim = phi0.copy()
phi_legendre = phi0.copy()
Let's look at the inital concentration field.
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draw_concentrations([phi0[0]], ['Initial Concentration'])
In order to move forward in time, we need to feed the concentration back into the Cahn-Hilliard simulation and the MKS models.
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time_steps = 50
for steps in range(time_steps):
ch_sim.run(phi_sim)
phi_sim = ch_sim.response
phi_prim = prim_model.predict(phi_prim)
phi_legendre = leg_model.predict(phi_legendre)
Let's take a look at the concentration fields.
In [13]:
from pymks.tools import draw_concentrations
draw_concentrations((phi_sim[0], phi_prim[0], phi_legendre[0]),
('Simulation', 'Primative', 'Legendre'))
By just looking at the three microstructures is it difficult to see any differences. Below, we plot the difference between the two MKS models and the simulation.
In [14]:
from sklearn import metrics
mse = metrics.mean_squared_error
from pymks.tools import draw_differences
draw_differences([(phi_sim[0] - phi_prim[0]), (phi_sim[0] - phi_legendre[0])],
['Simulaiton - Prmitive', 'Simulation - Legendre'])
print 'Primative mse =', mse(phi_sim[0], phi_prim[0])
print 'Legendre mse =', mse(phi_sim[0], phi_legendre[0])
The LegendreBasis basis clearly outperforms the PrimitiveBasis for the same value of n_states.
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big_length = 3 * length
big_size = (big_length, big_length)
prim_model.resize_coeff(big_size)
leg_model.resize_coeff(big_size)
phi0 = np.random.normal(0, 1e-9, (1,) + big_size)
phi_sim = phi0.copy()
phi_prim = phi0.copy()
phi_legendre = phi0.copy()
Let's take a look at the initial large concentration field.
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draw_concentrations([phi0[0]], ['Initial Concentration'])
Let's look at the resized coefficients.
First, the influence coefficients from the PrimitiveBasis.
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draw_coeff(prim_model.coeff)
Now, the influence coefficients from the LegendreBases.
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draw_coeff(leg_model.coeff)
Once again, we are going to march forward in time by feeding the concentration fields back into the Cahn-Hilliard simulation and the MKS models.
In [19]:
for steps in range(time_steps):
ch_sim.run(phi_sim)
phi_sim = ch_sim.response
phi_prim = prim_model.predict(phi_prim)
phi_legendre = leg_model.predict(phi_legendre)
In [20]:
draw_concentrations((phi_sim[0], phi_prim[0], phi_legendre[0]), ('Simulation', 'Primiative', 'Legendre'))
Both MKS models seem to predict the concentration faily well. However, the Legendre polynomial basis looks to be better. Again, let's look at the difference between the simulation and the MKS models.
In [21]:
draw_differences([(phi_sim[0] - phi_prim[0]), (phi_sim[0] - phi_legendre[0])],
['Simulaiton - Primiative','Simulation - Legendre'])
print 'Primative mse =', mse(phi_sim[0], phi_prim[0])
print 'Legendre mse =', mse(phi_sim[0], phi_legendre[0])
With the resized influence coefficients, the LegendreBasis outperforms the PrimitiveBasis for the same value of n_states. The value of n_states does not necessarily guarantee a fair comparison between the two basis in terms of floating point calculations and memory used.
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