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

Sanket Jantre, Nathan M. Urban+2 more|arXiv (Cornell University)|Mar 13, 2026

Machine Learning in Materials Science0 citations

TL;DR

A data-efficient forward surrogate for spectral functions in the t–t′–t″–J model using a deep kernel Gaussian process with sparse variational inference, outperforming reduced-data FFNN baselines and approaching full-data performance.

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.

Motivation & Objective

• Motivate spectral-function surrogates that work well with scarce training data.
• Develop a data-efficient forward model linking Hamiltonian parameters to density of states A(ω).
• Demonstrate that a deep-kernel SVGP surrogate outperforms reduced-data neural baselines.
• Assess spectrum-level and peak-level fidelity, including worst-tail and matched-peak diagnostics.
• Highlight implications for active learning and inverse modeling in spectral problems.

Proposed method

• Treat A(ω) as a function of Hamiltonian parameters x=(t', t'', J) and energy ω; train on spectra on a fixed ω-grid.
• Use a deep kernel learning (DKL) with a sparse variational Gaussian process (SVGP) surrogate (DKL-SVGP) with inducing points M=1536.
• Employ a lightweight feature network φθ that combines a parameter network hx and a Fourier ω-trunk, feeding into a GP with a flexible kernel.
• Model the joint input zjoint=[φθ(s), ω̃] with a mixed kernel k(s,z') that couples learned features and ω via a Matérn-3/2 kernel in the learned space and a Spectral Mixture kernel for ω.
• Preprocess: standardize x, rescale ω, log-standardize A(ω) and train on 10% of LCY's dataset; evaluate predictive mean as the surrogate output.
• Compare against FFNN baselines trained on the same 10% subset and on full data; include KRR and kNN baselines in appendix.

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 $x=(t^{\prime},t^{\prime\prime},J)$ and energy-grid points $\omega$ . We first standardize $x\t

Experimental results

Research questions

• RQ1 Can a GP-based surrogate with deep kernel learning achieve competitive accuracy for A(ω) with only 10% of the training data?
• RQ2 How does DKL-SVGP compare to reduced-data FFNN and full-data FFNN in spectrum-wise and peak-wise fidelity?
• RQ3 Does the surrogate maintain fidelity on difficult spectra (worst-tail) and better recover peak heights and locations under a matched-peak evaluation?

Key findings

• DKL-SVGP trained on 10% of data reduces mean row-RMSE on test by ~42% compared to FFNN trained on the same subset.
• Mean test row-RMSE for FFNN (full) vs DKL-SVGP is 1.93e-3 vs 4.54e-3 (DKL-SVGP within ~2.4× of full-data FFNN).
• Normalized RMSE (NRMSE) on test: FFNN (full) 1.365e-2; DKL-SVGP 3.369e-2 (subset) indicating substantial data-efficiency gains.
• Worst-tail diagnostics show DKL-SVGP better preserves dominant spectral structure on difficult spectra than reduced-data FFNN.
• Peak-level fidelity under matched-peak evaluation: peak heights comparable, peak locations in DKL-SVGP are slightly more accurate (RMSE 0.021 vs 0.028 for FFNN subset).
• Overall, GP-based surrogates offer strong accuracy-with-data budgets for spectral modeling in scarce-data regimes.

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 gr