  The Fisher Information Matrix (FIM) provides a way for quantifying the
information content of an observable random variable concerning unknown
parameters within a model that characterizes the variable. When parameters in a
model are directly linked to individual features, the diagonal elements of the
FIM can signify the relative importance of each feature. However, in scenarios
where feature interactions may exist, a comprehensive exploration of the full
FIM is necessary rather than focusing solely on its diagonal elements. This
paper presents an end-to-end black box system identification approach that
integrates the FIM into the training process to gain insights into dynamic
importance and overall model structure. A decision module is added to the first
layer of the network to determine the relevance scores using the entire FIM as
input. The forward propagation is then performed on element-wise multiplication
of inputs and relevance scores. Simulation results demonstrate that the
proposed methodology effectively captures various types of interactions between
dynamics, outperforming existing methods limited to polynomial interactions.
Moreover, the effectiveness of this novel approach is confirmed through its
application in identifying a real-world industrial system, specifically the PH
neutralization process.
