Probing the Hidden Sector: Neural Lagrangian Detection of Dark Matter Couplings from Visible Field Dynamics

The detection of Dark Matter remains one of the most profound unsolved problems in modern physics and cosmology. Standard detection approaches rely on specific theoretical assumptions about the nature of dark matter interactions. This paper proposes a model-agnostic variational framework for identifying hidden sector interactions by analyzing the Euler-Lagrange residuals of observable fields. We simulate a coupled system of visible and dark matter scalar fields and demonstrate that a specialized InteractionNet can detect the presence and recover the functional form of unobservable field interactions by minimizing only the Euler-Lagrange residual of the visible sector. Our results show that the neural framework accurately recovers the effective potential energy contribution of the hidden interaction term g*phi_vis^2*phi_dm^2, providing a neural blueprint for model-agnostic discovery of new particles and forces from astrophysical field observations. This constitutes the fourth paper in the Neural Lagrangian Series, applying the variational discovery framework to the problem of hidden sector physics.