Compact and stable representation of real-frequency spectral functions for machine learning

We introduce a compact and stable moment representation for real-frequency Green's functions, hybridization functions, and self-energies for machine-learning applications, avoiding the inefficiency of dense frequency grids as well as the ill-posed analytic continuation of Matsubara approaches. The representation is constructed from Cayley-mapped trigonometric moments with the Jacobian included, which preserve spectral-weight normalization, tie the moment sequence to a positive matrix-valued spectral measure, and admit a systematic route to a pole representation via ESPRIT. This provides a fixed-dimensional learning target in which physical constraints such as normalization and positivity can be imposed directly. Using a graph-attention neural network with FiLM conditioning, we benchmark the representation on single-orbital DMFT, antiferromagnetic DMFT, and a two-orbital impurity model. The results demonstrate accuracy matching or exceeding that of direct frequency-domain learning, reliable reproduction of the density and staggered magnetization, stable self-energy reconstruction through Dyson equation inversion, and accurate recovery of matrix-valued spectra with orbital mixing.