Other Multi-Dimensional Approximation Methods

 

In Predictive Systems Lab, it is possible in to approximate any multi-dimensional surface by the use of global neural networks. Though the resulting neural network will never truly interpolate the interpolation points, you can minimize the approximation error to several decimal magnitudes. To do so, define a model for the surface data, choosing the independent variables (inputs) as the nodes at which you want to approximate the neural network, and the functional values at these nodes as the dependent variables (outputs).  Note, that there are no restriction to the number of dependent variables (outputs) that the model can have. In other words, you can create a surface in the form:

 

,

 

where is the trained neural network, the are the dependent variables, , and the are the independent variables, .

 

Once the model is defined, create the neural network and train it on the known data to an acceptable error level. After this, generate a series of independent variable values (see the Populate Range of the Data Operations command) and feed them to the neural network solution through the Solution Run command. The result of this operation, is an approximated interpolation to the surface in question.

 

 

Figure 1. The XOR function approximated by a neural network which was ran with 1000 3-dimensional random points in the range [0, 1].