  Recent advances in learning or identification of nonlinear dynamics focus on
learning a suitable model within a pre-specified model class. However, a key
difficulty that remains is the choice of the model class from which the
dynamics will be learned. The fundamental challenge is trading the richness of
the model class with the learnability within the model class. Toward addressing
the so-called model selection problem, we introduce a novel notion of
Structural Risk Minimization (SRM) for learning nonlinear dynamics. Inspired by
classical SRM for classification, we minimize a bound on the true prediction
error over hierarchies of model classes. The class selected by our SRM scheme
is shown to achieve a nearly optimal learning guarantee among all model classes
contained in the hierarchy. Employing the proposed scheme along with computable
model class complexity bounds, we derive explicit SRM schemes for learning
nonlinear dynamics under hierarchies of: i) norm-constrained Reproducing Kernel
Hilbert Spaces, and ii) norm-constrained Neural Network classes. We empirically
show that even though too loose to be used as absolute estimates, our SRM
bounds on the true prediction error are able to track its relative behavior
across different model classes of the hierarchy.
