Modeling with Neural Networks: Principles and Model Design Methodology - Neural Networks: Methodology and Applications - page 186

Fig. 2.59. Canonical form of the network resulting from the discretization by an
explicit scheme

Explicit vs. Implicit Discretization Schemes: Impact on Neural Network
Implementation and Training

The explicit discretization of a knowledge-based model provides equations
that are readily put in neural network form, as shown in the above illustrative
example: one has

K − 1 [ x ( kT )] Ψ [ x ( kT ) , u ( kT ) ,T ] , x [( k +1) T ]= −

which is the canonical form of a recurrent neural network, as shown on
Fig. 2.59, where the neural network is an approximation of function K − 1 Ψ .

The didactic example discussed above is an example of the design of a semi-
physical model from a knowledge-based model discretized by an explicit
scheme.
When an implicit discretization method must be used for stability reasons,
the neural implementation of the resulting equations is less straightforward,
but still feasible. A detailed description of that technique can be found in
[Oussar et al. 2001].

−

2.9 Conclusion: What Tools?

This chapter gave a presentation of the basic concepts of modeling with neural
networks. Elements of statistics were first provided, then a complete design
methodology of nonlinear models, including but not limited to neural net-
works, was described. Static and dynamic models were discussed (the latter
being considered in a deeper fashion in Chap. 4). Finally, the design method-
ology of semiphysical models was described.