  We demonstrate the application of machine learning for rapid and accurate
extraction of plasmonic particles cluster geometries from hyperspectral image
data via a dual variational autoencoder (dual-VAE). In this approach, the
information is shared between the latent spaces of two VAEs acting on the
particle shape data and spectral data, respectively, but enforcing a common
encoding on the shape-spectra pairs. We show that this approach can establish
the relationship between the geometric characteristics of nanoparticles and
their far-field photonic responses, demonstrating that we can use hyperspectral
darkfield microscopy to accurately predict the geometry (number of particles,
arrangement) of a multiparticle assemblies below the diffraction limit in an
automated fashion with high fidelity (for monomers (0.96), dimers (0.86), and
trimers (0.58). This approach of building structure-property relationships via
shared encoding is universal and should have applications to a broader range of
materials science and physics problems in imaging of both molecular and
nanomaterial systems.
