High-precision relativistic atomic-structure calculations for many-electron heavy atoms and highly charged ions are often limited by the rapid growth of the configuration space. Selected configuration-interaction methods alleviate this problem by retaining only configuration state functions (CSFs) that contribute appreciably to the target wave function, but their efficiency depends on how rapidly important configurations can be identified. The previously developed variational Monte Carlo configuration-interaction method (VMCCI) improves the stability of configuration selection through a variational rejection-acceptance sampling mechanism. However, as the configuration space expands, Monte Carlo sampling becomes increasingly inefficient and converges slowly because the probability of sampling physically important configurations is governed by their typically small fraction of the candidate space. To address this bottleneck, we develop a machine-learning-accelerated VMCCI method (ML-VMCCI), in which model-guided sampling replaces random Monte Carlo sampling.
In ML-VMCCI, each CSF is represented by an atomic-structure descriptor containing orbital occupations, orbital angular momenta, and coupled angular momenta. Importance labels are assigned from CI coefficients obtained with GRASP2018 and updated during the iterative screening process. We first evaluate eight machine-learning models in the Lu I system, whose reference configuration space contains about 2.1×10⁶ CSFs. Under identical screening parameters, all ML-VMCCI variants converge to energies consistent with VMCCI, with a maximum deviation below 0.000514 Hartree. The number of iterations decreases from 28 to 7-18, the total convergence time is reduced from 1050.60 s to 298.14-901.74 s, and the average sampling efficiency increases by a factor of 1.7-6.3. ANN, CNN, and RF show the best balance among sampling efficiency, machine-learning cost, and iterative stability.
We then compare binary classification, configuration-importance ranking, and a hybrid strategy in the odd-parity J = 2 subspace of Ge-like Nd²⁸⁺, which contains about 5.9×10⁵ CSFs. The results indicate that ranking enriches high-contribution configurations in early iterations, whereas classification improves coverage near the importance threshold. Their combination gives the most balanced sampling behavior. In this benchmark, VMCCI requires 45 iterations and 150.18 min to converge. By contrast, ANN-VMCCI, CNN-VMCCI, and RF-VMCCI, all employing the hybrid sampling strategy, converge in 7, 6, and 9 iterations, respectively, yielding speedups of 4.04, 3.29, and 2.61 relative to VMCCI.
Finally, we apply the ML-VMCCI method with the hybrid sampling strategy to J = 0-5 excited-state calculations of Ge-like Nd²⁸⁺ in a reference configuration space containing 7.14×10⁶ CSFs. For even-parity states, ANN-M, CNN-M, and RF-M reduce the convergence time from 1549.55 min to 253.79, 314.69, and 386.62 min, respectively. For odd-parity states, the corresponding time decreases from 390.26 min to 84.74, 98.54, and 120.21 min. Overall, ML-VMCCI accelerates configuration screening by about 3-6 times and improves important-configuration sampling by about 6-10 times.
The acceleration does not introduce a noticeable loss of accuracy. Relative to experimental energies and direct MCDHF calculations in the reference configuration space, the ML-VMCCI error distributions remain comparable to VMCCI, with no systematic increase in bias or dispersion. These results show that ML-VMCCI connects configuration-space compression with sampling-process acceleration, substantially reducing the cost of high-precision excited-state calculations while maintaining reference-space accuracy.