Symmetry-Aware Residual Learning of Few-Body Molecular Wave Functions

Ultracold polar molecules are a highly controllable platform for engineering quantum matter, but modeling their few-body properties remains computationally challenging. Their anisotropic long-range interactions and rich internal structure lead to high-dimensional eigenvalue problems for which conventional grid-based solvers are largely restricted to two-molecule systems. To address this limitation, we introduce a physics-informed neural variational method that exploits known physical structure of the molecular wave function. Our key idea is to factor the wave function into analytic and neural components. The analytic component encodes rotational symmetry and asymptotic decay, while the neural residual learns the remaining intermediate-range behavior. This decomposition reduces the effective dimensionality of the problem without relying on ad hoc approximations or problem-specific discretization grids. We benchmark our method against high-accuracy two-molecule solvers and recover both ground and excited bound states with relative errors at the percent level. We also validate the numerical building blocks required for extension to three-body systems. Specifically, we test the angular-channel decomposition and two-coordinate radial representation in settings where reliable benchmarks are available. This provides a systematic route to larger molecular systems and opens opportunities to study correlations and dynamics in interacting quantum matter.