Model Transition Region: Neural Net

The constraints on the above models ensure that there is a region in which neither model is accurate. Specifically, the Fresnel model requirement that $\vert\alpha\vert N^2 \geq 50\pi$ and the Taylor model requirement that $\vert\alpha\vert (\ensuremath{\frac{N-1}{2}})^2 \leq \ensuremath{\frac{\pi}{16}}$ leave what may appear a large transition region. (In reality, however, these constraints are conservative, as can be seen in section 4.) To obtain an estimator in the transition region, we use the outputs of each of the three estimators as the inputs to a 3-4-1 neural net with `bias' (in neural net terms). We train the net with normalized data, taken from estimator outputs in the transition region. When choosing momentum of 0.9 for the input-to-hidden and hidden-to-output weights and a learning rate of 0.04, the net converges after approximately 1000 iterations.

Figure 2: Estimators without noise.