  Recent multi-dimensional (multi-D) core-collapse supernova (CCSN) simulations
characterize gravitational waves (GWs) and neutrino signals, offering insight
into universal properties of CCSN independent of progenitor. Neutrino analysis
in real observations, however, will be complicated due to the ambiguity of
self-induced neutrino flavor conversion (NFC), which poses an obstacle to
extracting detailed physical information. In this paper, we propose a novel
approach to place a constraint on NFC from observed quantities of GWs and
neutrinos based on correlation analysis from recent, detailed multi-D CCSN
simulations. The proposed method can be used even in cases with low
significance - or no detection of GWs. We also discuss how we can utilize
electro-magnetic observations to complement the proposed method. Although our
proposed method has uncertainties associated with CCSN modeling, the present
result will serve as a base for more detailed studies. Reducing the systematic
errors involved in CCSN models is a key to success in this multi-messenger
analysis that needs to be done in collaboration with different theoretical
groups.
