Nonlinear model reduction in the general case is a challenge. The most popular method is Proper Orthogonal Decomposition. It can be generalized by means of empirical gramians. There is a generalization of balancing for a nonlinear model.An interesting new approach is trajectory piecewise model reduction. Links below will give you some introduction to these methods.
D. J. Lucia, P. S. Beran, and W. A. Silva,
Reduced-order modeling: new approaches for computational physics.
Progress in Aerospace Sciences, vol. 40, N 1/2, pp. 51-117, 2004.
Paper at ScienceDirect.
Evgenii B. Rudnyi