Quantum-chemical insights from deep tensor neural networks

Abstract

Learning from data has led to paradigm shifts in a multitude of disciplines, including web, text and image search, speech recognition, as well as bioinformatics. Can machine learning enable similar breakthroughs in understanding quantum many-body systems? Here we develop an efficient deep learning approach that enables spatially and chemically resolved insights into quantum-mechanical observables of molecular systems. We unify concepts from many-body Hamiltonians with purpose-designed deep tensor neural networks, which leads to size-extensive and uniformly accurate (1 kcal mol^-1) predictions in compositional and configurational chemical space for molecules of intermediate size. As an example of chemical relevance, the model reveals a classification of aromatic rings with respect to their stability. Further applications of our model for predicting atomic energies and local chemical potentials in molecules, reliable isomer energies, and molecules with peculiar electronic structure demonstrate the potential of machine learning for revealing insights into complex quantum-chemical systems.

Figure 1. Prediction and explanation of molecular energies with a deep tensor neural network.

(a) Molecules are encoded as input for the neural network by a vector of nuclear charges and an inter-atomic distance matrix. This description is complete and invariant to rotation and translation. (b) Illustration of the network architecture. Each atom type corresponds to a vector of coefficients , which is repeatedly refined by interactions v_ij. The interactions depend on the current representation , as well as the distance D_ij to an atom j. After T iterations, an energy contribution E_i is predicted for the final coefficient vector . The molecular energy E is the sum over these atomic contributions. (c) Mean absolute errors of predictions for the GDB-9 dataset of 133,885 molecules as a function of the number of atoms. The employed neural network uses two interaction passes (T=2) and 50,000 reference calculation during training. The inset shows the error of an equivalent network trained on 5,000 GDB-9 molecules with 20 or more atoms, as small molecules with 15 or less atoms are added to the training set. (d) Extract from the calculated (black) and predicted (orange) molecular dynamics trajectory of toluene. The curve on the right shows the agreement of the predicted and calculated energy distributions. (e) Energy contribution E_probe (or local chemical potential , see text) of a hydrogen test charge on a isosurface for various molecules from the GDB-9 dataset for a DTNN model with T=2.

Figure 2. Chemical potentials for A={C, N, O, H} atoms.

The isosurface was generated for =3.8 Å⁻² (the index i is used to sum over all atoms of the corresponding molecule). The molecules shown are (in order from top to bottom of the figure): benzene, toluene, salicylic acid and malondehyde. Atom colouring: carbon=black, hydrogen=white, oxygen=red.

Figure 3. Classification of molecular carbon ring stability.

Shown are 20 molecules (10 most stable and 10 least stable) with respect to the energy of the carbon ring predicted by the DTNN model. Atom colouring: carbon=black; hydrogen=white; oxygen=red; nitrogen=blue; fluorine=yellow.

Figure 4. Isomer energies with chemical formula C₇O₂H₁₀.

DTNN trained on the GDB-9 database is able to acurately discriminate between 6,095 different isomers of C₇O₂H₁₀, which exhibit a non-trivial spectrum of relative energies.