  In the development of locally resonant metamaterials, the physical resonator
design is often omitted and replaced by an idealized mass-spring system. This
paper presents a novel approach for designing multimodal resonant structures,
which give rise to multi-bandgap metamaterials with predefined band gaps. Our
method uses a conditional variational autoencoder to identify nontrivial
patterns between design variables of complex-shaped resonators and their modal
effective parameters. After training, the cost of generating designs satisfying
arbitrary criteria - frequency and mass of multiple modes - becomes negligible.
An example of a resonator family with six geometric variables and two targeted
modes is further elaborated. We find that the autoencoder performs well even
when trained with a limited dataset, resulting from a few hundred numerical
modal analyses. The method generates several designs that very closely
approximate the desired modal characteristics. The accuracy of the best
designs, proposed by the auto-encoder, is confirmed in tests of 3D-printed
resonator prototypes. Further experiments demonstrate the close agreement
between the measured and desired dispersion relation of a sample metamaterial
beam.
