  Nonlinear-wave mixing in optical microresonators offers new perspectives to
generate compact optical-frequency microcombs, which enable an ever-growing
number of applications. Microcombs exhibit a spectral profile that is primarily
determined by their microresonator's dispersion; an example is the $
\operatorname{sech}^2 $ spectrum of dissipative Kerr solitons under anomalous
group-velocity dispersion. Here, we introduce an inverse-design approach to
spectrally shape microcombs, by optimizing an arbitrary meta-dispersion in a
resonator. By incorporating the system's governing equation into a genetic
algorithm, we are able to efficiently identify a dispersion profile that
produces a microcomb closely matching a user-defined target spectrum, such as
spectrally-flat combs or near-Gaussian pulses. We show a concrete
implementation of these intricate optimized dispersion profiles, using
selective bidirectional-mode hybridization in photonic-crystal resonators.
Moreover, we fabricate and explore several microcomb generators with such
flexible `meta' dispersion control. Their dispersion is not only controlled by
the waveguide composing the resonator, but also by a corrugation inside the
resonator, which geometrically controls the spectral distribution of the
bidirectional coupling in the resonator. This approach provides programmable
mode-by-mode frequency splitting and thus greatly increases the design space
for controlling the nonlinear dynamics of optical states such as Kerr solitons.
