Cahn-Hilliard with Primtive and Legendre Bases

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.

Cahn-Hilliard Equation

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.

Basis Functions for the Microstructure Function and Influence Function

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.

%matplotlib inline
%load_ext autoreload
%autoreload 2

import numpy as np
import matplotlib.pyplot as plt

Modeling with MKS

Generating Calibration Datasets

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 funcion make_cahnHilliard requires the number of calibration samples given by n_samples and the size and shape of the domain given by size.

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.

from pymks.tools import draw_concentrations

draw_concentrations((X[0], y[0]),('Calibration Input', 'Calibration Output'))
../_images/cahn_hilliard_Legendre_7_0.png

Calibrate Influence Coefficients

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 hyper parameter. 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.

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.

print(gs.best_estimator_)
print(gs.score(X_test, y_test))
MKSLocalizationModel(basis=<pymks.bases.legendre.LegendreBasis object at 0x7fa6f49e4210>,
           n_states=4)
1.0
from pymks.tools import draw_gridscores

lgs = [x for x in gs.grid_scores_ \
       if type(x.parameters['basis']) is type(leg_basis)]
cgs = [x for x in gs.grid_scores_ \
       if type(x.parameters['basis']) is type(prim_basis)]

draw_gridscores([lgs, cgs], 'n_states', data_labels=['Legendre', 'Primitve'],
                colors=['#f46d43', '#1a9641'], score_label='R-Squared',
                param_label = 'L - Total Number of Local States')
../_images/cahn_hilliard_Legendre_12_0.png

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.

Comparing the Bases for n_states=4

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

from pymks.tools import draw_coeff

draw_coeff(prim_model.coeff)
../_images/cahn_hilliard_Legendre_17_0.png

Now for the LegendreBasis influence coefficients.

draw_coeff(leg_model.coeff)
../_images/cahn_hilliard_Legendre_19_0.png

Now let’s do some simulations with both sets of coefficients and compare the results.

Predict Microstructure Evolution

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.

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.

draw_concentrations([phi0[0]], ['Initial Concentration'])
../_images/cahn_hilliard_Legendre_24_0.png

In order to move forward in time, we need to feed the concentration back into the Cahn-Hilliard simulation and the MKS models.

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.

from pymks.tools import draw_concentrations

draw_concentrations((phi_sim[0], phi_prim[0], phi_legendre[0]),
                    ('Simulation', 'Primative', 'Legendre'))
../_images/cahn_hilliard_Legendre_28_0.png

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.

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])
../_images/cahn_hilliard_Legendre_30_0.png
Primative mse = 5.28702717916e-23
Legendre mse = 4.35706317904e-28

The LegendreBasis basis clearly out performs the PrimitiveBasis for the same value of n_states.

Resizing the Coefficients to use on Larger Systems

Below we compare the bases after the coefficients are resized.

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.

draw_concentrations([phi0[0]], ['Initial Concentration'])
../_images/cahn_hilliard_Legendre_35_0.png

Let’s look at the resized coefficients.

First the influence coefficients from the PrimitiveBasis.

draw_coeff(prim_model.coeff)
../_images/cahn_hilliard_Legendre_37_0.png

Now the influence coefficients from the LegendreBases.

draw_coeff(leg_model.coeff)
../_images/cahn_hilliard_Legendre_39_0.png

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.

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)
draw_concentrations((phi_sim[0], phi_prim[0], phi_legendre[0]), ('Simulation', 'Primiative', 'Legendre'))
../_images/cahn_hilliard_Legendre_42_0.png

Both the 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.

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])
../_images/cahn_hilliard_Legendre_44_0.png
Primative mse = 4.43202082745e-23
Legendre mse = 4.45272856882e-28

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.