  We present a novel data-driven approach for learning linear representations
of a class of stable nonlinear systems using Koopman eigenfunctions. By
learning the conjugacy map between a nonlinear system and its Jacobian
linearization through a Normalizing Flow one can guarantee the learned function
is a diffeomorphism. Using this diffeomorphism, we construct eigenfunctions of
the nonlinear system via the spectral equivalence of conjugate systems -
allowing the construction of linear predictors for nonlinear systems. The
universality of the diffeomorphism learner leads to the universal approximation
of the nonlinear system's Koopman eigenfunctions. The developed method is also
safe as it guarantees the model is asymptotically stable regardless of the
representation accuracy. To our best knowledge, this is the first work to close
the gap between the operator, system and learning theories. The efficacy of our
approach is shown through simulation examples.
