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 (DTNN), which leads to size-extensive and uniformly accurate (1 kcal/mol) predictions in compositional and configurational chemical space for molecules of intermediate size. As an example of chemical relevance, the DTNN model reveals a classification of aromatic rings with respect to their stability -a useful property that is not contained as such in the training dataset. Further applications of DTNN for predicting atomic energies and local chemical potentials in molecules, reliable isomer energies, and molecules with peculiar electronic structure demonstrate the high potential of machine learning for revealing novel insights into complex quantum-chemical systems. Chemistry permeates all aspects of our life, from the development of new drugs to the food that we consume and materials we use on a daily basis. Chemists rely on empirical observations based on creative and painstaking experimentation that leads to eventual discoveries of molecules and materials with desired properties and mechanisms to synthesize them. Many discoveries in chemistry can be guided by searching large databases of experimental or computational molecular structures and properties by using concepts based on chemical similarity. Because the structure and properties of molecules are determined by the laws of quantum mechanics, ultimately chemical discovery must be based on fundamental quantum principles. Indeed, electronic structure calculations and intelligent data analysis (machine learning, ML) have recently been combined aiming towards the goal of accelerated discovery of chemicals with desired properties [1][2][3][4][5][6][7][8]. However, so far the majority of these pioneering efforts have focused on the construction of reduced models trained on large datasets of density-functional theory calculations. In this work, we develop an efficient deep learning approach that enables spatially and chemically resolved insights into quantum-mechanical properties of molecular systems beyond those trivially contained in the training dataset. Obviously, computational models are not predictive if they lack accuracy. In addition to being interpretable, size extensive and efficient, our deep tensor neural netwok (DTNN) approach is uniformly accurate (1 kcal/mol) throughout compositional and configurational chemical space. On the more fundamental side, the mathematical construction of the DTNN model provides statistically rigorous partitioning of extensive molecular properties into atomic contributions -a long-standing challenge for quantum-mechanical calculations of molecules. It is common to use a carefully chosen representation of the problem at hand as a basis for machine learning [9][10][11]. For example, molecules can be represented as Coulomb matrices [7, 12, 13], scattering transforms [14], bags of bonds (BoB) [15], smooth overlap of atomic positions (SOAP) [16, 17], or generalized symmetry functions [18, 19]. Kernelbased learning of molecular properties transforms these representations non-linearly by virtue of kernel functions. In contrast, deep neural networks [20] are able to infer the underlying regularities and learn an efficient representation in a layerwise fashion [21]. Molecular properties are governed by the laws of quantum mechanics, which yield the remarkable flexibility of chemical systems, but also impose constraints on the behavior of bonding in molecules. The approach presented here utilizes the many-body Hamiltonian concept for the construction of the DTNN architecture (see Fig. 1 ), embracing the principles of quantum chemistry, while maintaining the full flexibility of a complex data-driven learning machine. DTNN receives molecular structures through a vector of nuclear charges Z and a matrix of atomic distances D ensuring rotational and translational invariance by construction (Fig. 1A) . The distances are expanded in a Gaussian basis, yielding a feature vectord i j , which accounts for the different nature of interactions at various distance regimes. The total energy E M for the molecule M composed of N atoms is written as a sum over N atomic energy contributions E i , thus satisfying permutational invariance with respect to atom indexing. Each atom i is represented by a coefficient vector c ∈ R B , where B is the number of basis functions, or features. Motivated by quantum-chemical atomic basis set expansions, we assign an atom type-specific descriptor vector c Z i to these coefficients c (0) i . Subsequently, this atomic expansion is repeatedly refined by pairwise interactions with the arXiv:1609.08259v2 [physics.chem-ph]
The Deep Tensor Neural Network (DTNN) model REF uses as input a vector of nuclear charges and an inter-atomic distance matrix, and introduces a sequence of interaction passes where "the atom representations influence each other in a pair-wise fashion".
Quantum-Chemical Insights from Deep Tensor Neural Networks
Nature Comm. 8, 13890 (2017)
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