The noise generated by an aircraft is an efficiency and environmental matter for the aerospace industry.

An important component of the total airframe noise is the airfoil self-noise, which is due to the interaction between an airfoil blade and the turbulence produced in its own boundary layer and near wake.

Performance optimization can be applied to understand the behaviour of airfoils and make designs with reduced noise.

The self-noise data set used in this example was processed by NASA. It was obtained from a series of aerodynamic and acoustic tests of two and three-dimensional airfoil blade sections conducted in an anechoic wind tunnel.

The NASA data set comprises different size NACA 0012 airfoils at various wind tunnel speeds and angles of attack. The span of the airfoil and the observer position were the same in all of the experiments.

This is an approximation project, since the variable to be predicted is continuous (sound pressure level).

The basic goal here is to model the sound pressure level, as a function of the airfoil features and air speed.

The first step is to prepare the data set , which is the source of information for the approximation problem. It is composed of:

- Data source.
- Variables.
- Instances.

The file airfoil_self_noise.csv contains the data for this example. Here the number of variables (columns) is 6 and the number of instances (rows) is 1503.

In that way, this problem has the 6 following variables:

**frequency**, in Hertzs, used as input.**angle_of_attack**, in degrees, used as input.**chord_length**, in meters, used as input.**free_stream_velocity**, in meters per second, used as input.**suction_side_displacement_thickness**, in meters, used as input.**scaled_sound_pressure_level**, in decibels, used as target.

On the other hand, the NASA data set contains 1503 instances. They are divided at random into training, selection and testing subsets, containing 60%, 20% and 20% of the instances, respectively. More specifically, 753 samples are used here for training, 375 for validation and 375 for testing.

Once all the data set information has been set, we are ready to perform some analytics, to check the quality of the data.

For instance, we can calculate the data distribution. The next figure depicts the histogram for the target variable.

As we can see, the scaled sound pressure level has a normal distribution.

The next figure depicts the inputs-targets correlations. This might help us to see the influence of the different inputs on the sound level. As we can see, the frequency of the wave has the greatest impact on the noise.

The above chart shows that the frequency of the wave has the greatest impact on the noise.

We can also plot a scatter chart with the scaled sound pressure level versus the frequency.

In general, the more frequency the less scaled sound pressure level. However, the scaled sound pressure level depends on all the inputs at the same time.

The neural network will output the scaled sound pressure level as a function of the frequency, angle of attack, chord length, free stream velocity and suction side displacement thickness.

For this approximation example, the neural network is composed by:

- Scaling layer.
- Perceptron layers.
- Unscaling layer.

The scaling layer transforms the original inputs to normalized values. Here the mean and standard deviation scaling method is set so that the input values have mean 0 and standard deviation 1.

Here two perceptron layers are added to the neural network. This number of layers is enough for most applications. The first layer has 5 inputs and 3 neurons. The second layer has 3 inputs and 1 neuron.

The unscaling layer transforms the normalized values from the neural network into original outputs. Here the mean and standard deviation unscaling method will also be used.

The next figure shows the resulting network architecture.

This neural network represents a function containing 22 adjustable parameters.

The next step is to select an appropriate training strategy, which defines what the neural network will learn. A general training strategy is composed of two concepts:

- A loss index.
- An optimization algorithm.

The loss index chosen is the normalized squared error with L2 regularization. This loss index is the default in approximation applications.

The optimization algorithm chosen is the quasi-Newton method. This optimization algorithm is the default for medium sized applications like this one.

Once the strategy has been set, we can train the neural network. The following chart shows how the training (blue) and selection (orange) errors decrease with the training epoch during the training process.

The most important training result is the final selection error.
Indeed, this a measure of the generalization capabilities of the neural network.
Here the final selection error is **selection error = 0.112 NSE**.

The objective of model selection is to find the network architecture with best generalization properties. That is, we want to improve the final selection error obtained before (0.112 NSE).

The best selection error is achieved by using a model whose complexity is the most appropriate to produce and adequate fit of the data. Order selection algorithms are responsible to find the optimal number of perceptrons in the neural network.

The following chart shows the results of the incremental order algorithm. The blue line plots the final training error as a function of the number of neurons. The orange line plots the final selection error as a function of the number of neurons.

As we can see, the final training error always decreases with the number of neurons.
However, the final selection error takes a minimum value at some point.
Here, the optimal number of neurons is 13, which corresponds to a selection error of **0.100 NSE**.

The following figure shows the optimal network architecture for this application.

The objective of testing analysis is to validate the generalization performance of the trained neural network. Testing compares the values provided by this technique to the actually observed values.

A standard testing technique in approximation problems is to perform a linear regression analysis between the predicted and the real values, using an independent testing set. The next figure illustrates a graphical output provided by this testing analysis.

From the above chart, we can see that the neural network is predicting well the entire range of sound level data.
The correlation value is **R2 = 0.952**, which is very close to 1.

The model is now ready to estimate the self-noise of new airfoils with satisfactory quality over the same range of data.

We can plot a directional output of the neural network to see hoy the sound level varies with a given input, for all other inputs fixed. The next plot shows the sound level as a function of the frequency, through the following point:

**angle_of_attack**: 6.782 degrees.**chord_length**: 0.136 meters.**free_stream_velocity**: 50.860 meters per second.**suction_side_displacement_thickness**: 0.011 meters.

The file airfoil_self_noise.py contains the Python code for the scaled sound pressure level.

- UCI Machine Learning Repository. Airfoil Self-Noise Data Set.
- T.F. Brooks, D.S. Pope, and A.M. Marcolini. Airfoil self-noise and prediction. Technical report, NASA RP-1218, July 1989.