Assessing Non-Nested Configurations of Multifidelity Machine Learning for Quantum-Chemical Properties

TL;DR

The MFML method still requires a nested structure of training data across the fidelities, however, the o-MFML method shows promising results for non-nested multifidelity training data with model errors comparable to the nested configurations.

Abstract

Multifidelity machine learning (MFML) for quantum chemical (QC) properties has seen strong development in the recent years. The method has been shown to reduce the cost of generating training data for high-accuracy low-cost ML models. In such a set-up, the ML models are trained on molecular geometries and some property of interest computed at various computational chemistry accuracies, or fidelities. These are then combined in training the MFML models. In some multifidelity models, the training data is required to be nested, that is the same molecular geometries are included to calculate the property across all the fidelities. In these multifidelity models, the requirement of a nested configuration restricts the kind of sampling that can be performed while selection training samples at different fidelities. This work assesses the use of non-nested training data for two of these multifidelity methods, namely MFML and optimized MFML (o-MFML). The assessment is carried out for the prediction of ground state energies and first vertical excitation energies of a diverse collection of molecules of the CheMFi dataset. Results indicate that the MFML method still requires a nested structure of training data across the fidelities. However, the o-MFML method shows promising results for non-nested multifidelity training data with model errors comparable to the nested configurations.

Figures (7)

• Figure 1: Comparison of the use of unsorted and row-norm sorted CM Rup12CM and SLATM Huang2020slatm representations for the prediction of ground state and excitation energies with single fidelity KRR at the TZVP fidelity. For both ground state and excitation energies, the unsorted CM outperforms the other representations.
• Figure 2: Preliminary analysis for the multifidelity structure of ground state energies. The distribution of the ground state energies shows that it covers a wider range of values. The absolute difference of the various fidelities to the target fidelity of TZVP shows that for the most part this decreases with increasing fidelity. A scatter plot of the various fidelity energies with respect to TZVP shows a systematic distribution of the energies as can be seen in the inset image.
• Figure 3: The multifidelity structure of the first vertical excitation energies are analyzed to confirm the assumption of hierarchy. The distribution of the energies on the left-most plot shows distinct peaks which correspond to the different molecules. The difference in fidelities to the TZVP fidelity decreases on average with STO3G having a large standard deviation as can be seen in the plot in the center. This is confirmed in the scatter-plot from the left-most plot as well where the STO3G energies show a wide distribution with respect to TZVP energies.
• Figure 4: Learning curves of the MFML and o-MFML models built for ground state energies. The top row corresponds to nested training set case while the bottom row shows the results when non-nested training sets are used to build multifidelity models. Both conventional MFML and o-MFML are assessed here with the help of learning curves. The reference single fidelity KRR is also shown.
• Figure 5: A study of the optimized coefficient values of o-MFML for both the nested and non-nested cases in predicting the ground state energies. The default coefficients of MFML are shown on the right-most plot for comparison.