Neural ordinary differential equations for mapping the magnetic QCD phase diagram via holography

The quantum chromodynamics (QCD) phase diagram is crucial for understanding strongly interacting matter under extreme conditions, with major implications for cosmology, neutron stars, and heavy-ion collisions. We present a novel holographic QCD model utilizing neural ordinary differential equations (ODEs) to map the QCD phase diagram under magnetic field $B$, baryon chemical potential $\mu_B$, and temperature $T$. By solving the inverse problem of constructing a gravitational theory from Lattice QCD data, we reveal an unprecedentedly rich phase structure at finite $B$, including multiple critical endpoints (CEPs) in strong magnetic fields. Specifically, for $B~=~1.618~\,~\mathrm{GeV}^2=2.592~\times~10^{19}$ Gauss we identify two distinct CEPs at $(T_C~=~87.3~\,~\mathrm{MeV},~\,~\mu_C~=~115.9~\,~\mathrm{MeV})$ and $(T_C~=~78.9~\,~\mathrm{MeV},~\,~\mu_C~=~244.0~\,~\mathrm{MeV})$. Notably, the critical exponents vary depending on the CEP's location, and the conventional scaling relations can be violated in the presence of strong magnetic fields. These findings significantly advance our understanding of the QCD phase structure and provide concrete predictions for experimental validation at upcoming facilities such as FAIR, JPARC-HI, and NICA.