  The PHANGS survey uses ALMA, HST, VLT, and JWST to obtain an unprecedented
high-resolution view of nearby galaxies, covering millions of spatially
independent regions. The high dimensionality of such a diverse multi-wavelength
dataset makes it challenging to identify new trends, particularly when they
connect observables from different wavelengths. Here we use unsupervised
machine learning algorithms to mine this information-rich dataset to identify
novel patterns. We focus on three of the PHANGS-JWST galaxies, for which we
extract properties pertaining to their stellar populations; warm ionized and
cold molecular gas; and Polycyclic Aromatic Hydrocarbons (PAHs), as measured
over 150 pc-scale regions. We show that we can divide the regions into groups
with distinct multiphase gas and PAH properties. In the process, we identify
previously-unknown galaxy-wide correlations between PAH band and optical line
ratios and use our identified groups to interpret them. The correlations we
measure can be naturally explained in a scenario where the PAHs and the ionized
gas are exposed to different parts of the same radiation field that varies
spatially across the galaxies. This scenario has several implications for
nearby galaxies: (i) The uniform PAH ionized fraction on 150 pc scales suggests
significant self-regulation in the ISM, (ii) the PAH 11.3/7.7 \mic~ band ratio
may be used to constrain the shape of the non-ionizing far-ultraviolet to
optical part of the radiation field, and (iii) the varying radiation field
affects line ratios that are commonly used as PAH size diagnostics. Neglecting
this effect leads to incorrect or biased PAH sizes.
