.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_meta_modeling/fields_metamodels/plot_viscous_fall_metamodel.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_meta_modeling_fields_metamodels_plot_viscous_fall_metamodel.py: Viscous free fall: metamodel of a field function ================================================ .. GENERATED FROM PYTHON SOURCE LINES 6-7 In this example, we present how to create the metamodel of a field function. This examples considers the :ref:`free fall model `. We first compute the Karhunen-Loève decomposition of a sample of trajectories. Then we create a create a polynomial chaos which takes the inputs and returns the KL decomposition modes as outputs. Finally, we create a metamodel by combining the KL decomposition and the polynomial chaos. .. GENERATED FROM PYTHON SOURCE LINES 10-12 Define the model ---------------- .. GENERATED FROM PYTHON SOURCE LINES 14-20 .. code-block:: default import openturns as ot import numpy as np import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) .. GENERATED FROM PYTHON SOURCE LINES 21-22 We first define the time grid associated with the model. .. GENERATED FROM PYTHON SOURCE LINES 24-29 .. code-block:: default tmin = 0.0 # Minimum time tmax = 12. # Maximum time gridsize = 100 # Number of time steps mesh = ot.IntervalMesher([gridsize-1]).build(ot.Interval(tmin, tmax)) .. GENERATED FROM PYTHON SOURCE LINES 30-32 .. code-block:: default vertices = mesh.getVertices() .. GENERATED FROM PYTHON SOURCE LINES 33-34 Creation of the input distribution. .. GENERATED FROM PYTHON SOURCE LINES 36-42 .. code-block:: default distZ0 = ot.Uniform(100.0, 150.0) distV0 = ot.Normal(55.0, 10.0) distM = ot.Normal(80.0, 8.0) distC = ot.Uniform(0.0, 30.0) distribution = ot.ComposedDistribution([distZ0, distV0, distM, distC]) .. GENERATED FROM PYTHON SOURCE LINES 43-47 .. code-block:: default dimension = distribution.getDimension() dimension .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 4 .. GENERATED FROM PYTHON SOURCE LINES 48-49 Then we define the Python function which computes the altitude at each time value. In order to compute all altitudes with a vectorized evaluation, we first convert the vertices into a Numpy `array` and use the Numpy functions`exp` and `maximum`: this increases the evaluation performance of the script. .. GENERATED FROM PYTHON SOURCE LINES 51-65 .. code-block:: default def AltiFunc(X): g = 9.81 z0 = X[0] v0 = X[1] m = X[2] c = X[3] tau = m / c vinf = - m * g / c t = np.array(vertices) z = z0 + vinf * t + tau * (v0 - vinf) * (1 - np.exp(- t / tau)) z = np.maximum(z, 0.) return [[zeta[0]] for zeta in z] .. GENERATED FROM PYTHON SOURCE LINES 66-67 In order to create a `Function` from this Python function, we use the `PythonPointToFieldFunction` class. Since the altitude is the only output field, the third argument `outputDimension` is equal to `1`. If we had computed the speed as an extra output field, we would have set `2` instead. .. GENERATED FROM PYTHON SOURCE LINES 69-73 .. code-block:: default outputDimension = 1 alti = ot.PythonPointToFieldFunction( dimension, mesh, outputDimension, AltiFunc) .. GENERATED FROM PYTHON SOURCE LINES 74-75 Compute a training sample. .. GENERATED FROM PYTHON SOURCE LINES 77-82 .. code-block:: default size = 2000 ot.RandomGenerator.SetSeed(0) inputSample = distribution.getSample(size) outputSample = alti(inputSample) .. GENERATED FROM PYTHON SOURCE LINES 83-85 Compute the KL decomposition of the output ------------------------------------------ .. GENERATED FROM PYTHON SOURCE LINES 87-92 .. code-block:: default algo = ot.KarhunenLoeveSVDAlgorithm(outputSample, 1.0e-6) algo.run() KLResult = algo.getResult() scaledModes = KLResult.getScaledModesAsProcessSample() .. GENERATED FROM PYTHON SOURCE LINES 93-99 .. code-block:: default graph = scaledModes.drawMarginal(0) graph.setTitle('KL modes') graph.setXTitle(r'$t$') graph.setYTitle(r'$z$') view = viewer.View(graph) .. image-sg:: /auto_meta_modeling/fields_metamodels/images/sphx_glr_plot_viscous_fall_metamodel_001.png :alt: KL modes :srcset: /auto_meta_modeling/fields_metamodels/images/sphx_glr_plot_viscous_fall_metamodel_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 100-101 We create the `postProcessingKL` function which takes coefficients of the the K.-L. modes as inputs and returns the trajectories. .. GENERATED FROM PYTHON SOURCE LINES 103-105 .. code-block:: default karhunenLoeveLiftingFunction = ot.KarhunenLoeveLifting(KLResult) .. GENERATED FROM PYTHON SOURCE LINES 106-107 The `project` method computes the projection of the output sample (i.e. the trajectories) onto the K.-L. modes. .. GENERATED FROM PYTHON SOURCE LINES 109-111 .. code-block:: default outputSampleChaos = KLResult.project(outputSample) .. GENERATED FROM PYTHON SOURCE LINES 112-113 We limit the sampling size of the Lilliefors selection in order to reduce the computational burden. .. GENERATED FROM PYTHON SOURCE LINES 115-118 .. code-block:: default ot.ResourceMap.SetAsUnsignedInteger( "FittingTest-LillieforsMaximumSamplingSize", 1) .. GENERATED FROM PYTHON SOURCE LINES 119-120 We create a polynomial chaos metamodel which takes the input sample and returns the K.-L. modes. .. GENERATED FROM PYTHON SOURCE LINES 122-126 .. code-block:: default algo = ot.FunctionalChaosAlgorithm(inputSample, outputSampleChaos) algo.run() chaosMetamodel = algo.getResult().getMetaModel() .. GENERATED FROM PYTHON SOURCE LINES 127-128 The final metamodel is a composition of the KL lifting function and the polynomial chaos metamodel. In order to combine these two functions, we use the `PointToFieldConnection` class. .. GENERATED FROM PYTHON SOURCE LINES 130-133 .. code-block:: default metaModel = ot.PointToFieldConnection( karhunenLoeveLiftingFunction, chaosMetamodel) .. GENERATED FROM PYTHON SOURCE LINES 134-136 Validate the metamodel ---------------------- .. GENERATED FROM PYTHON SOURCE LINES 138-139 Create a validation sample. .. GENERATED FROM PYTHON SOURCE LINES 141-145 .. code-block:: default size = 10 validationInputSample = distribution.getSample(size) validationOutputSample = alti(validationInputSample) .. GENERATED FROM PYTHON SOURCE LINES 146-157 .. code-block:: default graph = validationOutputSample.drawMarginal(0) graph.setColors(['red']) graph2 = metaModel(validationInputSample).drawMarginal(0) graph2.setColors(['blue']) graph.add(graph2) graph.setTitle('Model/metamodel comparison') graph.setXTitle(r'$t$') graph.setYTitle(r'$z$') view = viewer.View(graph) plt.show() .. image-sg:: /auto_meta_modeling/fields_metamodels/images/sphx_glr_plot_viscous_fall_metamodel_002.png :alt: Model/metamodel comparison :srcset: /auto_meta_modeling/fields_metamodels/images/sphx_glr_plot_viscous_fall_metamodel_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 158-159 We see that the blue trajectories (i.e. the metamodel) are close to the red trajectories (i.e. the validation sample). This shows that the metamodel is quite accurate. However, we observe that the trajectory singularity that occurs when the object touches the ground (i.e. when :math:`z` is equal to zero), makes the metamodel less accurate. .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 7.762 seconds) .. _sphx_glr_download_auto_meta_modeling_fields_metamodels_plot_viscous_fall_metamodel.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_viscous_fall_metamodel.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_viscous_fall_metamodel.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_