Data-efficient surrogate modeling of spectral functions using Gaussian processes: An application to the $t$-$t'$-$t''$-$J$ model

Abstract

Spectral functions encode key many-body information but are costly to compute with high fidelity. Machine-learning surrogates have emerged as a powerful alternative, yet many approaches require large training datasets. We develop a data-efficient surrogate for spectral functions using the $t$-$t'$-$t''$-$J$ model, which describes the motion of a hole in a quantum antiferromagnet. Using $\sim$ 10$^5$ self-consistent Born approximation-based spectra from Lee, Carbone and Yin (Phys. Rev. B 107, 205132 (2023)), we train a deep-kernel Gaussian process surrogate model with sparse variational inference (DKL-SVGP) using only 10% of the available training spectra. We benchmark against feed-forward neural networks (FFNN) trained on the same reduced subset and on the full dataset. The proposed DKL-SVGP model consistently outperforms the reduced-data FFNN and, despite using only 10% of the training spectra, achieves spectrum-wise errors within the same order-of-magnitude as the full-data FFNN baseline. Worst-tail diagnostics show improved fidelity on difficult spectra, while peak-level analysis indicates that DKL-SVGP recovers dominant peak heights with comparable accuracy and improves peak-location agreement under a matched-peak evaluation that mitigates rare peak-swapping cases. Overall, these results highlight GP-based surrogates as a competitive and data-efficient approach for spectral-function prediction in scarce-data regimes.

Figures (6)

• Figure 1: Illustration of our deep kernel Gaussian process surrogate model trained with stochastic variational inference and applied to the forward problem of predicting a DOS given Hamiltonian parameters $x=(t', t", J)$ and energy-grid points $\omega$. We first standardize $x\to \widetilde{x}$ and rescale $\omega \to \widetilde{\omega} \in [-1,1]$. Then, we pass $(\widetilde{x},\widetilde{\omega})$ through a feature network $(\phi_\theta$) consisting of parameterization network $h_x(\widetilde{x})$ and Fourier trunk $h_\omega(\widetilde{\omega})$, their outputs and element-wise interaction $(h_x\odot h_\omega)$ are concatenated and linearly mapped to form learned features $\mathbf{z} = \phi_\theta(\widetilde{x}, \widetilde{\omega})$. Finally, a stochastic variational GP is applied to the joint input $\mathbf{z}_{joint}=[\mathbf{z},\widetilde{\omega}]$ using a flexible kernel and $M$ inducing points, producing the surrogate prediction which is GP mean -- $\widehat{A}(\omega)$.
• Figure 2: Worst-tail diagnostic on the held-out test dataset. Test spectra are ranked from worst to best by a reference error score (here, the FFNN row-RMSE over test dataset), and representative spectra at percentiles $\{0,2,4,6,8,10\}\%$ within this worst tail are shown. Each panel overlays the ground-truth DOS with predictions from the full-data FFNN baseline, the FFNN trained on a 10% random subset (FFNN Subset), and our DKL-SVGP model trained on the same 10% subset; the corresponding $(t',t",J)$ values are annotated in each subplot.
• Figure 3: Peak-level accuracy on the test dataset with top-5 peak matching. Scatter plots compare predicted vs. ground-truth peak height (top row) and peak location (bottom row) for the full-data FFNN baseline, the FFNN trained on a uniformly random 10% subset, and DKL-SVGP trained on the same 10% subset. For each spectrum, peak errors are computed after matching the ground-truth dominant peak to the closest peak among the top-5 peaks of each model prediction. Each panel reports RMSE, MAE, and Pearson correlation with respect to the ground truth.
• Figure 4: Peak-level accuracy on the test dataset. Scatter plots compare predicted vs. ground-truth (top row) dominant peak height $A_{\max}$ and (bottom row) peak location $\omega_{\max}$ for the full-data FFNN baseline, the FFNN trained on a 10% random subset, and DKL-SVGP trained on the same 10% subset. Each panel reports RMSE, MAE, and correlation of predictions with respect to the ground truth. We use consistent axis limits across panels for easier visual comparison; see Fig. \ref{['fig:peaks_scatter_extended']} for an extended version showing additional outliers in DKL-SVGP peak location predictions.
• Figure 5: Peak-level accuracy on the test dataset. Scatter plots compare predicted vs. ground-truth (top row) dominant peak height $A_{\max}$ and (bottom row) peak location $\omega_{\max}$ for the full-data FFNN baseline, the FFNN trained on a 10% random subset, and DKL-SVGP trained on the same 10% subset. Each panel reports RMSE, MAE, and correlation of predictions with respect to the ground truth. Compared to Fig. \ref{['fig:peaks_scatter']}, we extend the axis limits in the DKL-SVGP peak-location panel (c.ii) to highlight extreme outliers.