Note
Click here to download the full example code
Self-Organizing Map¶
In this example, we illustrate the use of the SOM
plot
on the Sobieski’s SSBJ problem.
from gemseo.api import configure_logger
from gemseo.api import create_discipline
from gemseo.api import create_scenario
from gemseo.problems.sobieski.core.problem import SobieskiProblem
from matplotlib import pyplot as plt
Import¶
The first step is to import some functions from the API and a method to get the design space.
configure_logger()
Out:
<RootLogger root (INFO)>
Description¶
The SOM
post-processing performs a Self Organizing Map
clustering on the optimization history.
A SOM
is a 2D representation of a design of experiments
which requires dimensionality reduction since it may be in a very high dimension.
A SOM is built by using an unsupervised artificial neural network
[KSH01].
A map of size n_x.n_y
is generated, where
n_x
is the number of neurons in the \(x\) direction and n_y
is the number of neurons in the \(y\) direction. The design space
(whatever the dimension) is reduced to a 2D representation based on
n_x.n_y
neurons. Samples are clustered to a neuron when their design
variables are close in terms of their L2 norm. A neuron is always located at the
same place on a map. Each neuron is colored according to the average value for
a given criterion. This helps to qualitatively analyze whether parts of the design
space are good according to some criteria and not for others, and where
compromises should be made. A white neuron has no sample associated with
it: not enough evaluations were provided to train the SOM.
SOM’s provide a qualitative view of the objective function, the constraints, and of their relative behaviors.
Create disciplines¶
At this point, we instantiate the disciplines of Sobieski’s SSBJ problem: Propulsion, Aerodynamics, Structure and Mission
disciplines = create_discipline(
[
"SobieskiPropulsion",
"SobieskiAerodynamics",
"SobieskiStructure",
"SobieskiMission",
]
)
Create design space¶
We also read the design space from the SobieskiProblem
.
design_space = SobieskiProblem().design_space
Create and execute scenario¶
The next step is to build an MDO scenario in order to maximize the range, encoded ‘y_4’, with respect to the design parameters, while satisfying the inequality constraints ‘g_1’, ‘g_2’ and ‘g_3’. We can use the MDF formulation, the Monte Carlo DOE algorithm and 30 samples.
scenario = create_scenario(
disciplines,
formulation="MDF",
objective_name="y_4",
maximize_objective=True,
design_space=design_space,
scenario_type="DOE",
)
scenario.set_differentiation_method("user")
for constraint in ["g_1", "g_2", "g_3"]:
scenario.add_constraint(constraint, "ineq")
scenario.execute({"algo": "OT_MONTE_CARLO", "n_samples": 30})
Out:
INFO - 10:03:11:
INFO - 10:03:11: *** Start DOEScenario execution ***
INFO - 10:03:11: DOEScenario
INFO - 10:03:11: Disciplines: SobieskiPropulsion SobieskiAerodynamics SobieskiStructure SobieskiMission
INFO - 10:03:11: MDO formulation: MDF
INFO - 10:03:11: Optimization problem:
INFO - 10:03:11: minimize -y_4(x_shared, x_1, x_2, x_3)
INFO - 10:03:11: with respect to x_1, x_2, x_3, x_shared
INFO - 10:03:11: subject to constraints:
INFO - 10:03:11: g_1(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 10:03:11: g_2(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 10:03:11: g_3(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 10:03:11: over the design space:
INFO - 10:03:11: +----------+-------------+-------+-------------+-------+
INFO - 10:03:11: | name | lower_bound | value | upper_bound | type |
INFO - 10:03:11: +----------+-------------+-------+-------------+-------+
INFO - 10:03:11: | x_shared | 0.01 | 0.05 | 0.09 | float |
INFO - 10:03:11: | x_shared | 30000 | 45000 | 60000 | float |
INFO - 10:03:11: | x_shared | 1.4 | 1.6 | 1.8 | float |
INFO - 10:03:11: | x_shared | 2.5 | 5.5 | 8.5 | float |
INFO - 10:03:11: | x_shared | 40 | 55 | 70 | float |
INFO - 10:03:11: | x_shared | 500 | 1000 | 1500 | float |
INFO - 10:03:11: | x_1 | 0.1 | 0.25 | 0.4 | float |
INFO - 10:03:11: | x_1 | 0.75 | 1 | 1.25 | float |
INFO - 10:03:11: | x_2 | 0.75 | 1 | 1.25 | float |
INFO - 10:03:11: | x_3 | 0.1 | 0.5 | 1 | float |
INFO - 10:03:11: +----------+-------------+-------+-------------+-------+
INFO - 10:03:11: Solving optimization problem with algorithm OT_MONTE_CARLO:
INFO - 10:03:11: Generation of OT_MONTE_CARLO DOE with OpenTURNS
INFO - 10:03:11: ... 0%| | 0/30 [00:00<?, ?it]
INFO - 10:03:11: ... 3%|▎ | 1/30 [00:00<00:00, 273.45 it/sec]
INFO - 10:03:11: ... 13%|█▎ | 4/30 [00:00<00:00, 125.55 it/sec]
INFO - 10:03:11: ... 23%|██▎ | 7/30 [00:00<00:00, 81.68 it/sec]
INFO - 10:03:12: ... 33%|███▎ | 10/30 [00:00<00:00, 58.15 it/sec]
INFO - 10:03:12: ... 43%|████▎ | 13/30 [00:00<00:00, 42.80 it/sec]
INFO - 10:03:12: ... 50%|█████ | 15/30 [00:00<00:00, 36.32 it/sec]
INFO - 10:03:12: ... 57%|█████▋ | 17/30 [00:00<00:00, 32.07 it/sec]
INFO - 10:03:12: ... 63%|██████▎ | 19/30 [00:01<00:00, 28.45 it/sec]
INFO - 10:03:12: ... 73%|███████▎ | 22/30 [00:01<00:00, 25.09 it/sec]
INFO - 10:03:12: ... 80%|████████ | 24/30 [00:01<00:00, 22.87 it/sec]
INFO - 10:03:13: ... 90%|█████████ | 27/30 [00:01<00:00, 20.48 it/sec]
INFO - 10:03:13: ... 100%|██████████| 30/30 [00:01<00:00, 18.83 it/sec]
INFO - 10:03:13: ... 100%|██████████| 30/30 [00:01<00:00, 18.80 it/sec]
INFO - 10:03:13: Optimization result:
INFO - 10:03:13: Optimizer info:
INFO - 10:03:13: Status: None
INFO - 10:03:13: Message: None
INFO - 10:03:13: Number of calls to the objective function by the optimizer: 30
INFO - 10:03:13: Solution:
INFO - 10:03:13: The solution is feasible.
INFO - 10:03:13: Objective: -367.45739115001027
INFO - 10:03:13: Standardized constraints:
INFO - 10:03:13: g_1 = [-0.02478574 -0.00310924 -0.00855146 -0.01702654 -0.02484732 -0.04764585
INFO - 10:03:13: -0.19235415]
INFO - 10:03:13: g_2 = -0.09000000000000008
INFO - 10:03:13: g_3 = [-0.98722984 -0.01277016 -0.60760341 -0.0557087 ]
INFO - 10:03:13: Design space:
INFO - 10:03:13: +----------+-------------+---------------------+-------------+-------+
INFO - 10:03:13: | name | lower_bound | value | upper_bound | type |
INFO - 10:03:13: +----------+-------------+---------------------+-------------+-------+
INFO - 10:03:13: | x_shared | 0.01 | 0.01230934749207792 | 0.09 | float |
INFO - 10:03:13: | x_shared | 30000 | 43456.87364611478 | 60000 | float |
INFO - 10:03:13: | x_shared | 1.4 | 1.731884935123487 | 1.8 | float |
INFO - 10:03:13: | x_shared | 2.5 | 3.894765253193514 | 8.5 | float |
INFO - 10:03:13: | x_shared | 40 | 57.92631048228255 | 70 | float |
INFO - 10:03:13: | x_shared | 500 | 520.4048463450415 | 1500 | float |
INFO - 10:03:13: | x_1 | 0.1 | 0.3994784918586811 | 0.4 | float |
INFO - 10:03:13: | x_1 | 0.75 | 0.9500312867674923 | 1.25 | float |
INFO - 10:03:13: | x_2 | 0.75 | 1.205851870260564 | 1.25 | float |
INFO - 10:03:13: | x_3 | 0.1 | 0.2108042391973412 | 1 | float |
INFO - 10:03:13: +----------+-------------+---------------------+-------------+-------+
INFO - 10:03:13: *** End DOEScenario execution (time: 0:00:01.609221) ***
{'eval_jac': False, 'algo': 'OT_MONTE_CARLO', 'n_samples': 30}
Post-process scenario¶
Lastly, we post-process the scenario by means of the
SOM
plot which performs a self organizing map
clustering on optimization history.
Tip
Each post-processing method requires different inputs and offers a variety
of customization options. Use the API function
get_post_processing_options_schema()
to print a table with
the options for any post-processing algorithm.
Or refer to our dedicated page:
Post-processing algorithms.
scenario.post_process("SOM", save=False, show=False)
# Workaround for HTML rendering, instead of ``show=True``
plt.show()
Out:
INFO - 10:03:13: Building Self Organizing Map from optimization history:
INFO - 10:03:13: Number of neurons in x direction = 4
INFO - 10:03:13: Number of neurons in y direction = 4
Figure SOM example on the Sobieski problem. illustrates another SOM on the Sobieski
use case. The optimization method is a (costly) derivative free algorithm
(NLOPT_COBYLA
), indeed all the relevant information for the optimization
is obtained at the cost of numerous evaluations of the functions. For
more details, please read the paper by
[KJO+06] on wing MDO post-processing
using SOM.
A DOE may also be a good way to produce SOM maps. Figure SOM example on the Sobieski problem with a 10 000 samples DOE. shows an example with 10000 points on the same test case. This produces more relevant SOM plots.
Total running time of the script: ( 0 minutes 2.407 seconds)