  Nonlinear balanced truncation is a model order reduction technique that
reduces the dimension of nonlinear systems in a manner that accounts for either
open- or closed-loop observability and controllability aspects of the system.
Two computational challenges have so far prevented its deployment on
large-scale systems: (a) the energy functions required for characterization of
controllability and observability are solutions of high-dimensional
Hamilton-Jacobi-(Bellman) equations, which have been computationally
intractable and (b) the transformations to construct the reduced-order models
(ROMs) are potentially ill-conditioned and the resulting ROMs are difficult to
simulate on the nonlinear balanced manifolds. Part~1 of this two-part article
addressed challenge (a) via a scalable tensor-based method to solve for
polynomial approximations of the open- and closed-loop energy functions. This
article, (Part~2), addresses challenge (b) by presenting a novel and scalable
method to reduce the dimensionality of the full-order model via model reduction
on polynomially-nonlinear balanced manifolds. The associated nonlinear state
transformation simultaneously 'diagonalizes' relevant energy functions in the
new coordinates. Since this nonlinear balancing transformation can be
ill-conditioned and expensive to evaluate, inspired by the linear case we
develop a computationally efficient balance-and-reduce strategy, resulting in a
scalable and better conditioned truncated transformation to produce balanced
nonlinear ROMs. The algorithm is demonstrated on a semi-discretized partial
differential equation, namely Burgers equation, which illustrates that
higher-degree transformations can improve the accuracy of ROM outputs.
