In many practical situations like weather prediction, we are interested in large-scale (averaged) value of the predicted quantities. For example, it is impossible to predict the exact future temperature at different spatial locations, but we can reasonably well predict average temperature over a region. Traditionally, to obtain such large-scale predictions, we first perform a detailed integration of the corresponding differential equation, and then average the resulting detailed solution. This procedure is often very time-consuming, since we need to process all the details of the original data.
In our previous papers, we have shown that similar quality large-scale prediction results can be obtained if instead, we apply a much faster procedure -- first average the inputs (by applying an appropriate fuzzy transform), and then use these averaged inputs to solve the corresponding (discretization of the) differential equation.
In this paper, we provide a general theoretical explanation of why our semi-heuristic method works, i.e., why fuzzy transforms are efficient in large-scale predictions.