  A common goal in the study of high dimensional and complex system is to model
the system by a low order representation. In this letter we propose a general
approach for assessing the quality of a reduced order model for high
dimensional chaotic systems. The key of this approach is the use of optimal
shadowing, combined with dimensionality reduction techniques. Rather than
quantify the quality of a model based on the quality of predictions, which can
be irrelevant for chaotic systems since even excellent models can do poorly, we
suggest that a good model should allow shadowing by modeled data for long
times; this principle leads directly to an optimal shadowing criterion of model
reduction. This approach overcomes the usual difficulties encountered by
traditional methods which either compare systems of the same size by
normed-distance in the functional space, or measure how close an orbit
generated by a model is to the observed data. Examples include interval
arithmetic computations to validate the optimal shadowing.
