Self-consistency error correction for accurate machine learning potentials from variational Monte Carlo

TL;DR

This work tackles the self-consistency error that arises when training machine-learning interatomic potentials with variational Monte Carlo data obtained from frozen orbitals. By applying a recently developed SCE correction, the authors generate unbiased training data from a Jastrow-correlated single-determinant WF, enabling MLIPs that approach the accuracy of models trained on fully optimized WFs. Across a high-pressure hydrogen Hugoniot test case, corrected MLIPs show improved forces, pressures, and thermodynamic observables, and MD simulations confirm closer agreement with reference results. The framework demonstrates a practical path to high-quality QMC-based training sets and is readily extendable to larger systems and other MLIP architectures.

Abstract

Variational Monte Carlo (VMC) can be used to train accurate machine learning interatomic potentials (MLIPs), enabling molecular dynamics (MD) simulations of complex materials on time scales and for system sizes previously unattainable. VMC training sets are often based on partially optimized wave functions (WFs) to circumvent expensive energy optimizations of the whole set of WF parameters. However, frozen variational parameters lead to VMC forces and pressures not consistent with the underlying potential energy surface, a bias called the self-consistency error (SCE). Here, we demonstrate how the SCE can spoil the accuracy of MLIPs trained on these data, taking high-pressure hydrogen as test case. We then apply a recently introduced SCE correction [ Phys. Rev. B 109, 205151 (2024)] to generate unbiased VMC training sets based on a Jastrow-correlated single determinant WF with frozen Kohn-Sham orbitals. The MLIPs generated within this framework are significantly improved and can approach in quality those trained on datasets built with fully optimized WFs. Our conclusions are further supported by MD simulations, which show how MLIPs trained on SCE-corrected datasets systematically yield more reliable physical observables. Our framework opens the possibility of constructing extended high-quality training sets with VMC.

Figures (9)

• Figure 1: (a) Comparison among the biased pressure evaluated with an expression equivalent to Eq. \ref{['eq: VMC force']} (red diamonds), the corrected pressure obtained by applying Eq. \ref{['eq: pressure correction stochastic average']} (violet square), and the pressure obtained by fitting the PES (dashed green line and green diamond) for a selected configuration in the Hugoniot datasetTenti2024. The PES of the system is also shown (green dots and dash-dotted line). (b) Comparison among the biased force evaluated with Eq. \ref{['eq: VMC force']} (red diamonds), the corrected force obtained by applying Eq. \ref{['eq: force correction stochastic average']} (violet square), and the force calculated by fitting the PES (dashed green line and green diamond) for the $z$-component of the second hydrogen atom belonging to the same configuration as in (a). The PES of the system along the atomic displacement is also shown (green dots and dash-dotted line).
• Figure 2: (a) Value of the biased (red markers) and corrected (violet markers) pressure for $6$ different $128$-atom configurations as a function of the numerical pressure estimated from the derivative of the PES with respect to the volume. (b) Values of the biased (red markers) and corrected (violet markers) force components as a function of the numerical force estimated by the PES for one of the configurations belonging to the Hugoniot datasetTenti2024. The dashed line is a reference for perfect consistency.
• Figure 3: Relative RMSE variation $\Delta_{X}$ (Eq. \ref{['eq: Delta models']}) for energy, forces, and pressure, as a function of the force/energy weight ratio in the loss function of Eq. \ref{['eq.5: loss function']} (expressed in atomic units) for MLIPs trained on the biased, corrected and reference datasets, respectively. The weight on the pressure is set to zero for all models.
• Figure 4: $\Delta_{X}$ (Eq. \ref{['eq: Delta models']}) for energy, forces, and pressure, as a function of the pressure/energy weight ratio in the loss function of Eq. \ref{['eq.5: loss function']} used for training (expressed in atomic units), for the different models. Solid lines correspond to $W_E = 1$ and $W_F = 3 / 128$, while dashed lines correspond to $W_E = 1$ and $W_F = 15 / 128$. The arrows highlight the "best" MLIPs for each of the three datasets, which are used for the MD simulations described in Sec. \ref{['sec: MD results']}.
• Figure 5: Radial distribution function $g(r)$ for densities and temperatures close to the Hugoniot curve, obtained with MLIPs trained on different datasets. The difference $\Delta g(r)$ with respect to the model trained on the reference dataset is also shown for both the "biased" and "corrected" models. The shaded area indicates the statistical uncertainty in $\Delta g(r)$.