  When complex systems with nonlinear dynamics achieve an output performance
objective, only a fraction of the state dynamics significantly impacts that
output. Those minimal state dynamics can be identified using the differential
geometric approach to the observability of nonlinear systems, but the theory is
limited to only analytical systems. In this paper, we extend the notion of
nonlinear observable decomposition to the more general class of data-informed
systems. We employ Koopman operator theory, which encapsulates nonlinear
dynamics in linear models, allowing us to bridge the gap between linear and
nonlinear observability notions. We propose a new algorithm to learn Koopman
operator representations that capture the system dynamics while ensuring that
the output performance measure is in the span of its observables. We show that
a transformation of this linear, output-inclusive Koopman model renders a new
minimum Koopman representation. This representation embodies only the
observable portion of the nonlinear observable decomposition of the original
system. A prime application of this theory is to identify genes in biological
systems that correspond to specific phenotypes, the performance measure. We
simulate two biological gene networks and demonstrate that the observability of
Koopman operators can successfully identify genes that drive each phenotype. We
anticipate our novel system identification tool will effectively discover
reduced gene networks that drive complex behaviors in biological systems.
