In this short introduction, we will demonstrate the functionality in PyMKS. We will quantify microstructures using 2-point statistics, predict effective properties using homogenization and predict local properties using localization. If you would like more technical details about any of these methods please see the theory section.

```
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```

```
import pymks
import warnings
warnings.filterwarnings('ignore')
%matplotlib inline
%load_ext autoreload
%autoreload 2
```

Lets make two dual-phase microstructures with different morphologies.

```
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```

```
from pymks.datasets import make_microstructure
import numpy as np
X_1 = make_microstructure(n_samples=1, grain_size=(25, 25))
X_2 = make_microstructure(n_samples=1, grain_size=(95, 15))
X = np.concatenate((X_1, X_2))
```

Throughout PyMKS `X`

is used to represent microstructures. Now that we
have made the two microstructures, lets take a look at them.

```
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```

```
from pymks.tools import draw_microstructures
draw_microstructures(X)
```

We can compute the 2-point statistics for these two periodic
microstructures using the `correlate`

function from `pymks.stats`

.
This function computes all of the autocorrelations and
cross-correlation(s) for a microstructure. Before we compute the 2-point
statistics, we will discretize them using the `PrimitiveBasis`

function.

```
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```

```
from pymks import PrimitiveBasis
from pymks.stats import correlate
p_basis = PrimitiveBasis(n_states=2, domain=[0, 1])
X_corr = correlate(X, p_basis, periodic_axes=[0, 1])
```

Let’s take a look at the two autocorrelations and the cross-correlation for these two microstructures.

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```

```
from pymks.tools import draw_correlations
print(X_corr[0].shape)
draw_correlations(X_corr[0])
```

```
(101, 101, 3)
```

```
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```
draw_correlations(X_corr[1])
```

2-Point statistics provide an object way to compare microstructures, and have been shown as an effective input to machine learning methods.

In this section of the intro, we are going to predict the effective
stiffness for two-phase microstructures using the
`MKSHomogenizationModel`

, but we could have chosen any other effective
material property.

First we need to make some microstructures and their effective stress values to fit our model. Let’s create 200 random instances 3 different types of microstructures, totaling to 600 microstructures.

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```
from pymks.datasets import make_elastic_stress_random
grain_size = [(47, 6), (4, 49), (14, 14)]
n_samples = [200, 200, 200]
X_train, y_train = make_elastic_stress_random(n_samples=n_samples, size=(51, 51),
grain_size=grain_size, seed=0)
```

Once again, `X_train`

is our microstructures. Throughout PyMKS `y`

is used as either the property, or the field we would like to predict.
In this case `y_train`

is the effective stress values for `X_train`

.
Let’s look at one of each of the three different types of
microstructures.

```
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```
draw_microstructures(X_train[::200])
```

The `MKSHomogenizationModel`

uses 2-point statistics, so we need to
provide a discretization method for the microstructures by providing a
basis function. We will also specify which correlations we want.

```
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```

```
from pymks import MKSHomogenizationModel
p_basis = PrimitiveBasis(n_states=2, domain=[0, 1])
homogenize_model = MKSHomogenizationModel(basis=p_basis, periodic_axes=[0, 1],
correlations=[(0, 0), (1, 1), (0, 1)])
```

Let’s fit our model with the data we created.

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```
homogenize_model.fit(X_train, y_train)
```

Now let’s make some new data to see how good our model is.

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```

```
n_samples = [10, 10, 10]
X_test, y_test = make_elastic_stress_random(n_samples=n_samples, size=(51, 51),
grain_size=grain_size, seed=100)
```

We will try and predict the effective stress of our `X_test`

microstructures.

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```

```
y_pred = homogenize_model.predict(X_test)
```

The `MKSHomogenizationModel`

generates low dimensional representations
of microstructures and regression methods to predict effective
properties. Take a look at the low-dimensional representations.

```
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```
from pymks.tools import draw_components_scatter
draw_components_scatter([homogenize_model.reduced_fit_data[:,:2],
homogenize_model.reduced_predict_data[:,:2]],
['Training Data', 'Test Data'])
```

Now let’s look at a goodness of fit plot for our
`MKSHomogenizationModel`

.

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```

```
from pymks.tools import draw_goodness_of_fit
fit_data = np.array([y_train, homogenize_model.predict(X_train)])
pred_data = np.array([y_test, y_pred])
draw_goodness_of_fit(fit_data, pred_data, ['Training Data', 'Test Data'])
```

Looks good.

The `MKSHomogenizationModel`

can be used to predict effective
properties and processing-structure evolutions.

In this section of the intro, we are going to predict the local strain
field in a microstructure using `MKSLocalizationModel`

, but we could
have predicted another local property.

First we need some data, so let’s make some.

```
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```
from pymks.datasets import make_elastic_FE_strain_delta
X_delta, y_delta = make_elastic_FE_strain_delta()
```

Once again, `X_delta`

is our microstructures and `y_delta`

is our
local strain fields. We need to discretize the microstructure again, so
we will also use the same basis function.

```
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```
from pymks import MKSLocalizationModel
p_basis = PrimitiveBasis(n_states=2)
localize_model = MKSLocalizationModel(basis=p_basis)
```

Let’s use the data to fit our `MKSLocalizationModel`

.

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```
localize_model.fit(X_delta, y_delta)
```

Now that we have fit our model, we will create a random microstructure and compute its local strain field, using finite element analysis. We will then try and reproduce the same strain field with our model.

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```
from pymks.datasets import make_elastic_FE_strain_random
X_test, y_test = make_elastic_FE_strain_random()
```

Let’s look at the microstructure and its local strain field.

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```

```
from pymks.tools import draw_microstructure_strain
draw_microstructure_strain(X_test[0], y_test[0])
```

Now let’s pass that same microstructure to our `MKSLocalizationModel`

and compare the predicted and computed local strain field.

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```
from pymks.tools import draw_strains_compare
y_pred = localize_model.predict(X_test)
draw_strains_compare(y_test[0], y_pred[0])
```

Not bad.

The `MKSLocalizationModel`

can be used to predict local properties and
local processing-structure evolutions.

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```
```