.. mlp_summary .. _ref_to_mlp_summary: Multilayer Perceptron ===================== Multi-layer Perceptron (MLP) is a supervised learning algorithm that learns a function by training on a dataset. It can have multiple input variables and create multiple output values. Given a set of features and a target , it can learn a non-linear function approximator. See the discussion on `Wikipedia `_ and the `sklearn `_ implementation at `Multi-layer Perceptron regressor `_. Monte Carlo Method ------------------ Many of the calculated parameters required by **RocketIsp** are best calculated by large complex numerical models. Since including those models in **RocketIsp** is not feasible, correlations of those models' output is used. **RocketIsp** uses `Multi-layer Perceptron regressors `_ to correlate data generated by complex numerical models. Each model is treated as a `Black Box `_ and a large random sampling of inputs to the model are used to generate a large sampling of results. The results are then correlated with a `Multi-layer Perceptron regressor `_. Such a data-generation approach is known as a `Monte Carlo `_ method. Example ------- In order to demonstrate the method, the large complex numerical model `Black Box `_ will be replaced by a very simple model, `Monte Carlo `_ data will be generated from that simple model and that data will be fitted with a `Multi-layer Perceptron regressor `_. For this example the `Black Box `_ will be just a single equation taken from the `Expanded Liquid Engine Simulation (ELES) <./_static/ELES_Paper_AIAA-85-1385.pdf>`_ computer program. ELES was created in 1984 by Aerojet Techsystems Company (now `Aerojet Rocketdyne `_) to model the size, weight and performance of liquid propellant stages. An `AIAA paper <./_static/ELES_Paper_AIAA-85-1385.pdf>`_ describes ELES in overview. On page 13 of the `ELES Technical Information Manual <./_static/ELES_Technical_Information_Manual.pdf>`_ is a rough approximation equation for boundary layer efficiency, :math:`\large{\eta_{bl}}`. This example will use that equation to simulate a large complex model. .. image:: ./_static/etabl_from_eles_tech_manual.jpg The above :math:`\large{\eta_{bl}}` equation can be implemented in python code as the following function. .. code-block:: python def eles_effbl( Pc=100.0, Fvac=1000.0, eps=20.0): """ Return effBL from ELES boundary layer correlation. where: Pc=psia, Fvac=lbf, eps=dimensionless """ logPF = log(0.01*Pc*Fvac) effBL = 0.997 - (log(eps) * 0.01 * (1-0.065*logPF + 0.001*logPF**2)) return effBL Create Dataset ~~~~~~~~~~~~~~ One of the considerations when generating data for a `Monte Carlo `_ analysis is the selection of distribution type for each input parameter. In general a `continuous uniform distribution `_ is good for this application, however, when an input can cover `orders of magnitude `_ , it is sometimes better to use a uniform distribution of the `logarithm `_ of that input. For this example, distributions of the `logarithm `_ of chamber pressure, vacuum thrust and area ratio were used, A small python code (expand section below) was used to make a `pandas `_ dataframe of Monte Carlo data and save it to a `CSV file `_. This is the data that will be fitted with a `Multi-layer Perceptron regressor `_. .. raw:: html
Click HERE to view python code .. literalinclude:: ./_static/example_scripts/mlp_example_make_csv.py :language: python .. raw:: html
A display of the `pandas `_ dataframe description is shown below. Notice that **timeOfDay** is included to act as an uncorrelated input. .. code-block:: python effBL log10(Pc) log10(Fvac) log10(eps) timeOfDay count 10000.000000 10000.000000 10000.000000 10000.000000 10000.000000 mean 0.981543 2.740944 3.066303 1.296131 12.077017 std 0.008608 0.422945 1.379687 0.580750 6.920006 min 0.951005 2.000074 0.699824 0.301047 0.003217 25% 0.975976 2.380903 1.857840 0.789792 6.126944 50% 0.982975 2.748261 3.061789 1.297682 12.051867 75% 0.988453 3.103367 4.261623 1.803856 18.014434 max 0.995327 3.477062 5.476351 2.300927 23.999201 Select Independent Inputs ~~~~~~~~~~~~~~~~~~~~~~~~~ While it is possible to simply use all of the complex model's inputs as inputs to the `Multi-layer Perceptron regressor `_, it is probably better to select only those inputs that make a difference in the output, :math:`\large{\eta_{bl}}`. One way to learn, not only which inputs matter, but their order of importance, is the following: One by one, fit each individual independent parameter to the dataset with a `Multi-layer Perceptron regressor `_. Select the input with the best correlation coefficient as the most important input. Next combine the most important input with the remaining inputs, one by one, to find the second most important input. Continue the process until all possible inputs are in order of importance. The chart below shows the results of this process on the example dataset. It indicates that AreaRatio and Thrust are both very important. Pc adds some improvement and timeOfDay adds nothing. As a result, it seems reasonable to use **AreaRatio, Thrust and Pc** as inputs to the final `Multi-layer Perceptron regressor `_. (knowing what is in the `Black Box `_ we should have expected those 3 inputs, but we may not have known their order of importance) .. image:: ./_static/mlp_example_plot_dep.png :width: 49% Fit MLP to Dataset ~~~~~~~~~~~~~~~~~~ A small python script was used to fit the monte carlo data to a `Multi-layer Perceptron regressor `_ .. raw:: html
Click HERE to view python code .. literalinclude:: ./_static/example_scripts/mlp_example_fit.py .. raw:: html
The script outputs a summary of the scaled inputs and output. Note that all of the inputs and the output are scaled to between 0 and 1. Also note that the correlation coefficient is excellent... it is greater than 0.999 (will vary slightly with each run). The script saves the MLP regressor to a file named "mlp_example.joblib" that can be reloaded in order to make use of the correlation.:: fitting eff with indepL = ['Pc', 'Fvac', 'eps'] Pc Fvac eps count 10000.000000 10000.000000 10000.000000 mean 0.493963 0.493261 0.498065 std 0.281963 0.275937 0.290375 min 0.000049 0.019965 0.000523 25% 0.253935 0.251568 0.244896 50% 0.498841 0.492358 0.498841 75% 0.735578 0.732325 0.751928 max 0.984708 0.975270 1.000463 -------------------------------------------- count 10000.000000 mean 0.630870 std 0.172157 min 0.020108 25% 0.519511 50% 0.659495 75% 0.769058 max 0.906547 Name: eff, dtype: float64 -------------------------------------------- correlation coefficient: 0.9998603014551914 Use The Regressor ~~~~~~~~~~~~~~~~~ Our black box was very well behaved and the MLP fit was excellent. The charts below demonstrate an excellent agreement between the original black box data and the MLP prediction. In general, if the black box being examined is as well behaved as this example, similarly good results will likely be realized. Complex numerical models may have much more numerical noise due to things like iteration termination criteria, local optima, calculation uncertainties, numerically induced oscillations, internal discontinuities, etc. In those cases, the MLP fit will still likely reflect the character of the underlying data, however, the correlation coefficient of the fit will likely be below the value of this example The following charts compare the MLP regressor values to the original values of the `Black Box `_. .. raw:: html
MLP Fit at Pc=100 psia MLP Fit at Pc=1000 psia

Click image to see full size
The following python script reads the MLP regressor from its saved file and uses it to predict the original values of the `Black Box `_. .. literalinclude:: ./_static/example_scripts/mlp_example_predict.py