Nonparametric assessment of multiple predictors

What statistical analysis should I use when I have two or more continuous independent predictor variables and two or more dependent variables to be predicted? I cannot assume normal distribution.
I want to describe in impact of the predictor variables on the dependent variables.

Some more info:
I have created a computer model simulating the flow in a hydraulic system where there are 3-5 input parameters for which I examine their impact on several output parameters in this model. The output data from the model is not necessarily normal distributed. The input parameters are set to certain levels according to an experimental matrix. The number of simulations typically are 120-250.