Scalar machine learning of tensorial quantities -- Born effective charges from monopole models

TL;DR

This work demonstrates that the Born effective charge tensor $Z_{j,\alpha\beta}^*$ can be accurately predicted using invariant scalar descriptors by decomposing the response into a local monopole term and a charge-redistribution term, thus avoiding explicit tensorial equivariance. The authors compare kernel-based monopole, dipole, and combined models with a neural-network MACE implementation, showing that a monopole–dipole combination often yields the best performance among simple models, while MACE can surpass kernel methods with sufficient data. Across diverse materials, the scalar monopole formulation reproduces both BECs and finite-temperature infrared spectra, highlighting its practical utility for large-scale simulations and electrostatic embedding. Interpretability of individual monopole charges remains model-dependent, but the approach offers a scalable, plug-in alternative to tensorial ML frameworks for predicting electrostatic and spectroscopic properties.

Abstract

Predicting tensorial properties with machine learning models typically requires carefully designed tensorial descriptors. In this work, we introduce an alternative strategy for learning tensorial quantities based on scalar descriptors. We apply this approach to the Born effective charge tensor, showing that scalar (monopole) kernel models can successfully capture its tensorial nature by exploiting the definition of the Born effective charge tensor as the derivative of the polarisation with respect to atomic displacements. We compare this method with tensorial (dipole) kernel models, as established in our previous work, in which the tensorial structure of the Born effective charge is encoded directly in the kernel and obtained via its derivative. Both approaches are then used for charge partitioning, enabling the separation of monopole and dipole contributions. Finally, we demonstrate the effectiveness of the framework by computing finite-temperature infrared spectra for a range of complex materials.

Figures (4)

• Figure 1: Root mean square error (RMSE), left axes and normalised root mean square error (RMSE divided by the standard deviation of the values in the reference DFT data set, right axes) on a test set (10% of the total data) as a function of the number of training configurations for three models: $q$--monopoles (circle), $p$--dipoles (square), $q+p$--combined monopoles and dipoles (triangle), and MACE (star). Results are shown for data sets on liquid water at room temperature (H2O), the orthorhombic, tetragonal, and cubic phases of MAPbI3, liquid NaCl between [range-phrase = and ]11001400, and solid ZrO2. Broken lines indicate training set errors, while the solid lines are the test set errors (for MACE only test set errors are shown, while the number of configurations refers to the combined training and validation set).
• Figure 2: Root mean square error (RMSE), mean atomic charge of Na ($\mathrm{mean}(q_{\ce{Na}})$), maximum absolute atomic charge ($\mathrm{max}(|q|)$), and maximum absolute atomic polarisation over all Cartesian components ($\mathrm{max}(|\mathbf{p}|)$) as a function of different regularisation schemes for the liquid NaCl dataset for the combined model ($q+p$). Predicted values are reported alongside colour coding of that value. Two types of regularisation are shown: Tikhonov regularisation (x-axes $\omega$-regularisation), which penalises large fitting parameters $\omega$, and a charge regularisation that constrains the atomic charges $q$ to reproduce $\mathrm{trace}(\mathbf{Z}^*)/3$ (y-axes $p$-regularisation).
• Figure 3: Experimental and computational IR spectra of liquid water. The computational spectra were obtained using three different models: $q$--monopoles (dashed), $p$--dipoles (dotted), $q+p$--combined monopoles and dipoles (dash-dot). Experimental reference data are taken from Ref. bertie1996infrared. Statistical uncertainties of the simulations are indicated by the shaded regions surrounding the calculated spectra, corresponding to the 95% confidence interval, i.e., $\pm 2 \sigma$, where $\sigma$ is the standard error of the sample mean.
• Figure 4: Experimental and computational IR spectra for the orthorhombic and tetragonal phases of MAPbI3. The computational spectra were obtained using three different models: $q$--monopoles (dashed), $p$--dipoles (dotted), and the combined $q+p$ monopole--dipole model (dash--dot). The vibrational frequencies of the computed spectra have been uniformly redshifted by 1.5% to align with the experimental data. Experimental reference spectra are taken from Ref. schuck2018infrared and are shown in arbitrary units, as only relative intensities are reported. Statistical uncertainties of the simulations are indicated by the shaded regions surrounding the calculated spectra, corresponding to the 95% confidence interval, i.e., $\pm 2\sigma$, where $\sigma$ is the standard error of the sample mean.