PyDMD

Tutorial 1: Dynamic Mode Decomposition on a toy dataset

In this tutorial we will show the typical use case, applying the dynamic mode decomposition on the snapshots collected during the evolution of a generic system. We present a very simple system since the main purpose of this tutorial is to show the capabilities of the algorithm and the package interface.

First of all we import the DMD class from the pydmd package, we set matplotlib for the notebook and we import numpy.

In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np

from pydmd import DMD

We create the input data by summing two different functions:
$f_1(x,t) = \text{sech}(x+3)\exp(i2.3t)$
$f_2(x,t) = 2\text{sech}(x)\tanh(x)\exp(i2.8t)$.

In [2]:
def f1(x,t): 
    return 1./np.cosh(x+3)*np.exp(2.3j*t)

def f2(x,t):
    return 2./np.cosh(x)*np.tanh(x)*np.exp(2.8j*t)

x = np.linspace(-5, 5, 128)
t = np.linspace(0, 4*np.pi, 256)

xgrid, tgrid = np.meshgrid(x, t)

X1 = f1(xgrid, tgrid)
X2 = f2(xgrid, tgrid)
X = X1 + X2

The plots below represent these functions and the dataset.

In [3]:
titles = ['$f_1(x,t)$', '$f_2(x,t)$', '$f$']
data = [X1, X2, X]

fig = plt.figure(figsize=(17,6))
for n, title, d in zip(range(131,134), titles, data):
    plt.subplot(n)
    plt.pcolor(xgrid, tgrid, d.real)
    plt.title(title)
plt.colorbar()
plt.show()

Now we have the temporal snapshots in the input matrix rows: we can easily create a new DMD instance and exploit it in order to compute the decomposition on the data. Since the snapshots must be arranged by columns, in this case we need to transpose the matrix.

In [4]:
dmd = DMD(svd_rank=2)
dmd.fit(X.T)
Out[4]:
<pydmd.dmd.DMD at 0x105f720d0>

The dmd object contains the principal information about the decomposition:

  • the attribute modes is a 2D numpy array where the columns are the low-rank structures individuated;
  • the attribute dynamics is a 2D numpy array where the rows refer to the time evolution of each mode;
  • the attribute eigs refers to the eigenvalues of the low dimensional operator;
  • the attribute reconstructed_data refers to the approximated system evolution.

Moreover, some helpful methods for the graphical representation are provided.

Thanks to the eigenvalues, we can check if the modes are stable or not: if an eigenvalue is on the unit circle, the corresponding mode will be stable; while if an eigenvalue is inside or outside the unit circle, the mode will converge or diverge, respectively. From the following plot, we can note that the two modes are stable.

In [5]:
for eig in dmd.eigs:
    print('Eigenvalue {}: distance from unit circle {}'.format(eig, np.abs(eig.imag**2+eig.real**2 - 1)))

dmd.plot_eigs(show_axes=True, show_unit_circle=True)

Eigenvalue (0.990495347078+0.13754623738j): distance from unit circle 1.55431223448e-15
Eigenvalue (0.99358347261+0.1131012067j): distance from unit circle 7.77156117238e-16

We can plot the modes and the dynamics:

In [6]:
for mode in dmd.modes.T:
    plt.plot(x, mode.real)
    plt.title('Modes')
plt.show()

for dynamic in dmd.dynamics:
    plt.plot(t, dynamic.real)
    plt.title('Dynamics')
plt.show()

Finally, we can reconstruct the original dataset as the product of modes and dynamics. We plot the evolution of each mode to emphasize their similarity with the input functions and we plot the reconstructed data.

In [7]:
fig = plt.figure(figsize=(17,6))

for n, mode, dynamic in zip(range(131, 133), dmd.modes.T, dmd.dynamics):
    plt.subplot(n)
    plt.pcolor(xgrid, tgrid, (mode.reshape(-1, 1).dot(dynamic.reshape(1, -1))).real.T)
    
plt.subplot(133)
plt.pcolor(xgrid, tgrid, dmd.reconstructed_data.T.real)
plt.colorbar()

plt.show()

We can also plot the absolute error between the approximated data and the original one.

In [8]:
plt.pcolor(xgrid, tgrid, (X-dmd.reconstructed_data.T).real)
fig = plt.colorbar()

The reconstructed system looks almost equal to the original one: the dynamic mode decomposition made possible the identification of the meaningful structures and the complete reconstruction of the system using only the collected snapshots.