Neural network-based first-principles method for predicting heat of formation (HOF) was previously demonstrated to be able to achieve chemical accuracy in a broad spectrum of target molecules [L. H. Hu et al., J. Chem. Phys. 119, 11501 (2003)]. However, its accuracy deteriorates with the increase in molecular size. A closer inspection reveals a systematic correlation between the prediction error and the molecular size, which appears correctable by further statistical analysis, calling for a more sophisticated machine learning algorithm. Despite the apparent difference between simple and complex molecules, all the essential physical information is already present in a carefully selected set of small molecule representatives. A model that can capture the fundamental physics would be able to predict large and complex molecules from information extracted only from a small molecules database. To this end, a size-independent, multi-step multi-variable linear regression-neural network-B3LYP method is developed in this work, which successfully improves the overall prediction accuracy by training with smaller molecules only. And in particular, the calculation errors for larger molecules are drastically reduced to the same magnitudes as those of the smaller molecules. Specifically, the method is based on a 164-molecule database that consists of molecules made of hydrogen and carbon elements. 4 molecular descriptors were selected to encode molecule's characteristics, among which raw HOF calculated from B3LYP and the molecular size are also included. Upon the size-independent machine learning correction, the mean absolute deviation (MAD) of the B3LYP/6-311+G(3df,2p)-calculated HOF is reduced from 16.58 to 1.43 kcal/mol and from 17.33 to 1.69 kcal/mol for the training and testing sets (small molecules), respectively. Furthermore, the MAD of the testing set (large molecules) is reduced from 28.75 to 1.67 kcal/mol.

When it comes to predicting experimental values of molecular properties with deep learning, the key problem is the lack of sufficient experimental data for training. We propose a method that consists of pretraining a graph neural network that aims to reproduce first-principles quantum mechanical results, followed by fine-tuning of a fully connected neural network against experimental results. The combined pretraining and fine-tuning model is expected to yield molecular properties close to experimental accuracy. This is made possible because first-principles quantum mechanical methods are often qualitatively correct or semiquantitatively accurate; thus, a calibration of the calculation results against high-precision but limited experiment data can improve accuracy greatly. Moreover, the method is highly efficient, as first-principles quantum mechanical calculation is bypassed. To demonstrate this, we apply the combined model to determine the experimental heats of formation of organic molecules made of H, C, O, N, or F atoms (up to 30 atoms), where mere 405 experimental data are used. The overall mean absolute error is 1.8 kcal/mol for these molecules.