Benchmarking Data Efficiency in $Δ$-ML and Multifidelity Models for Quantum Chemistry

TL;DR

This work benchmarks data-generation costs for Δ-ML, MFML, o-MFML, and introduces MF$\Delta$ML on the QeMFi QC dataset to predict $E_{\text{gs}}$, $E_{(1)}$, $E_{(2)}$, and $|\boldsymbol{\mu}_e|$ across five fidelities. It shows that multifidelity approaches generally yield better data efficiency than pure Δ-ML, with MFML and MF$\Delta$ML delivering the strongest performance for large prediction sets, while MF$\Delta$ML offers advantages when only a few evaluations are needed. The study also clarifies the cost structure: Δ-ML incurs baseline QC costs, MFML avoids those costs by predicting the baseline, and MF$\Delta$ML combines Δ-ML with MFML to further improve efficiency. Overall, the results provide practical guidance on selecting MF- and Δ-ML strategies to minimize training data cost while achieving target accuracy for QC properties.

Abstract

The development of machine learning (ML) methods has made quantum chemistry (QC) calculations more accessible by reducing the compute cost incurred in conventional QC methods. This has since been translated into the overhead cost of generating training data. Increased work in reducing the cost of generating training data resulted in the development of $Δ$-ML and multifidelity machine learning methods which use data at more than one QC level of accuracy, or fidelity. This work compares the data costs associated with $Δ$-ML, multifidelity machine learning (MFML), and optimized MFML (o-MFML) in contrast with a newly introduced Multifidelity$Δ$-Machine Learning (MF$Δ$ML) method for the prediction of ground state energies, vertical excitation energies, and the magnitude of electronic contribution of molecular dipole moments from the multifidelity benchmark dataset QeMFi. This assessment is made on the basis of training data generation cost associated with each model and is compared with the single fidelity kernel ridge regression (KRR) case. The results indicate that the use of multifidelity methods surpasses the standard $Δ$-ML approaches in cases of a large number of predictions. For applications which require only a few evaluations to be made using ML models, while the $Δ$-ML method might be favored, the MF$Δ$ML method is shown to be more efficient.

Figures (13)

• Figure 1: A visual depiction of the different ML methods benchmarked in this work. The MFML and o-MFML models do not need any further QC-calculations for the unseen test set, once they have been trained. In contrast, $\Delta$-ML method and the MF$\Delta$ML method that is introduced herein, both require additional QC computations at the QC-baseline that is used in these models. This work benchmarks these different models to understand the time-cost versus model accuracy efficiency.
• Figure 2: Distribution of training, validation, and test sets used in this work. All the nine molecules of the QeMFi dataset are evenly present in each of the sets. The same train/test/validation split is used for all QC properties studied in this work.
• Figure 3: Learning curves for $\Delta$-ML with varying $QC_b$. These are shown for the prediction of ground state energies, first vertical excitation energies ($E_{(1)}$), second vertical excitation energies ($E_{(2)}$), and the magnitude of electronic contribution to molecular dipole moments ($\lvert\boldsymbol{\mu}_e\rvert$). Across the QC properties, it is observed that the closer the $QC_b$ is in hierarchy to the target fidelity, the better the model accuracy, as also observed in ref. Ramakrishnan2015.
• Figure 4: MFML and o-MFML learning curves for different QC properties studied in this work. The different baselines used in the MFML and o-MFML models are reported in the legend. It is seen that the o-MFML model does not provide a significant improvement over the conventional MFML model in terms of MAE. This could indicate that the MFML combination of sub-models is already sufficiently optimized.
• Figure 5: Learning curves for the newly introduced MF$\Delta$ML and o-MF$\Delta$ML with differing baseline fidelities.