Seite - 229 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

A.2. DerivationofBackpropagationAlgorithm wherexLi (k) is the input from the i-th node of theL-th hidden layer. Then theoutput of this node is YNN,j(k) =fo(z o j(k)), (B.20) wherefo represents the activationfunction of the output layer. The corresponding update rule of any weightwoj,i(k)connecting with this node is woj,i(k+1) = w o j,i(k)+∆w o j,i(k), where ∆woj,i(k) =−η ∂J(k) ∂woj,i(k) =−η ∂J(k) ∂YNN,j(k) ∂YNN,j(k) ∂fo(zoj(k)) ∂fo(z o j(k)) ∂zoj(k) ∂zoj(k) ∂woj,i(k) . (B.21) Substituting following expressions into the above equation (for each node in this layer) ∂J(k) ∂YNN,j(k) =YNN,j(k)−Yd,j(k), ∂YNN,j(k) ∂fo(zoj(k)) = 1, ∂fo(z o j(k)) ∂zoj(k) =f′o(z o j(k)), ∂zoj(k) ∂woj,i(k) =xLi (k), then equation B.21will be transformed into ∆woj,i(k) =−η · ( YNN,j(k)−Yd,j(k) ) f′o(z o j(k))x L i (k) =−η ·δoj(k)xLi (k), δoj(k) = ( YNN,j(k)−Yd,j(k) ) · f′o(zoj(k)), (B.22) where δoj(k) is the local gradient equivalent to ∂J(k)/∂z o j(k). When the activation function in the output layer is the identity function as 229