Data-efficient surrogate modeling of spectral functions using Gaussian processes: An application to the t-t′-t′′-J model

Spectral functions encode key information on one- and many-body excitations but are costly to compute with high fidelity. Machine-learning surrogates have recently emerged as powerful alternative, yet most require large training datasets. We present a data-efficient surrogate for spectral functions by taking a case study of the t-t′-t′′-J model which describes the motion of a hole in a quantum antiferromagnet. Using ∼10⁵ self-consistent Born approximation-based spectra generated in prior work of Lee, Carbone and Yin (Phys. Rev. B 107, 205132 (2023)), we train a deep-kernel stochastic Gaussian process (GP) model on only 10% of the data and benchmark against feed-forward neural networks (FFNN) trained on the same reduced budget and on the full dataset. The GP consistently outperforms the FFNN under limited data. In addition, despite using only 10% of the data, it achieves the same order-of-magnitude accuracy as the full-data FFNN baseline for both peak positions and peak heights, while providing calibrated predictive uncertainties by construction. These findings highlight GP-based surrogates as promising and previously unexplored approach for spectral function prediction in scarce-data regimes. The method integrates naturally with active-learning workflows for both forward prediction and inverse parameter inference, paving the way towards data-efficient, uncertainty-aware spectral solvers.