This is an example designed to show how the scientist, researcher or engineer can use neural networks to analyze and model data. A scientist gathers experimental data for some process or phenomenon in an attempt to predict and understand its behavior. Traditional methods of using the experimental data to predict a future events usually involves implementing slow, complex multi-variable, table lookup techniques combined with various interpolation and data fitting schemes. These traditional techniques often fail when confronted with nonlinear behavior, data discontinuities, and non-uniform data sampling. Neural networks easily handle such modeling tasks and can reduce modeling time, increase prediction accuracy and reduce data storage requirements.

For this example, a model is needed to predict the aerodynamic drag on a sphere for a given range of speeds and atmospheric conditions. Let’s assume the aerodynamic drag of a sphere is needed for air speeds ranging from Mach .3 to Mach 4.5 and for Reynolds numbers from 200000 to 600000. From fluid dynamics theory, it is known these are the two critical factors that influence aerodynamic drag. The researcher runs a series of wind tunnel tests to gather the data. For a sphere, the experimental drag data contains peaks, valleys and discontinuities (due to the onset of turbulence at the critical Reynolds numbers and Mach number effects). Using a neural network to model and predict sphere drag will allow the researcher to accurately capture these irregular features and utilize experimental data taken at non-uniform test conditions.

The example network is contained in "SphereDrag.net" (the untrained network) and "SphereDragTrained.net" (the trained network). The network model has 6 fully connected layers utilizing multiple transfer function types. The input layer has 2 nodes, one for the Reynolds number (divided by 10^5) and the other for the Mach number. The 4 hidden layers contain 10 nodes, 8 nodes, 6 nodes and 2 nodes respectively. The output layer has 1 node that represents the aerodynamic drag on the sphere. A total of six layers were used in this network in an attempt to properly model the nonlinear behavior of the experimental data. There are 102 experimental points to train the network. Instead of incorporating a test set, the resulting network will be interrogated by passing a large number of test cases through the network and visually checking the results.

The results obtained from the network are excellent. The sphere drag data is accurately predicted for the training cases. The average error of the fit is less than 1%. The figure shown is the result of passing 378 Mach/Reynolds number conditions through the network in recall mode. The resulting 3D surface (see below) models the drag trends extremely well, including the discontinuity that represents the onset of turbulence. Convergence of the untrained network takes several hundred thousand iterations. This is from the use of a large, multi-layered hidded structure. Using hybrid layers in this network considerably improved convergence time over Qnet's default sigmoid construction.

Neural networks provide the perfect tool to model scientific data. They provide the researcher with a fast, accurate tool for utilizing experimental data in research and design activities.

Figure - Sphere drag computed from the trained model.

Figure - Sphere drag from Qnet's Color Contour graphing option