Seite - 231 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

A.2. DerivationofBackpropagationAlgorithm By applying the chain rule, the local gradient δLj (k) can be calculated as δLj (k) = N∑ p=1 ∂J(k) ∂fo(zop(k)) ∂fo(z o p(k)) ∂zop(k) ∂zop(k) ∂g(zLj (k)) ∂g(zLj (k)) ∂zLj (k) . (B.25) whereN is thenumberofnodes in theoutput layer. According to the notations shown in figure 3.9 and the definitions in B.22, the terms in theaboveequation B.25areexpressedas ∂J(k) ∂fo(zop(k)) =δop(k), ∂fo(z o p(k)) ∂zop(k) =1, ∂zop(k) ∂g(zLj (k)) =wop,j(k), ∂g(zLj (k)) ∂zLj (k) =g′(zLj (k)). (B.26) Substituting these expressions into the above equation B.25, it is rewrittenas δLj (k) = N∑ p=1 δop(k)w o p,j(k)g ′(zLj (k)) =g′(zLj (k)) N∑ p=1 δop(k)w o p,j(k). (B.27) Equations B.24 and B.27 can be extended into a more general case where thenode j is in formerhiddenlayers [Hay98], suchas ∆wlj,i(k) =−η ·δlj(k)xli(k) =−η ·   g′(zl+1j (k))Nl+1∑ p=1 δl+1p (k)w l+1 p,j (k),   xli(k). (B.28) 231