Fast and Accurate Modeling of Molecular Atomization Energies with Machine Learning

Abstract

We introduce a machine learning model to predict atomization energies of a diverse set of organic molecules, based on nuclear charges and atomic positions only. The problem of solving the molecular Schrödinger equation is mapped onto a non-linear statistical regression problem of reduced complexity. Regression models are trained on and compared to atomization energies computed with hybrid density-functional theory. Cross-validation over more than seven thousand small organic molecules yields a mean absolute error of ∼10 kcal/mol. Applicability is demonstrated for the prediction of molecular atomization potential energy curves.Solving the Schrödinger equation (SE), HΨ = EΨ, for assemblies of atoms is a fundamental problem in quantum mechanics. Alas, solutions that are exact up to numerical precision are intractable for all but the smallest systems with very few atoms. Hierarchies of approximations have evolved, usually trading accuracy for computational efficiency [1]. Conventionally, the external potential, defined by a set of nuclear charges {Z I } and atomic positions {R I }, uniquely determines the Hamiltonian H of any system, and thereby the potential energy by optimizingFor a diverse set of organic molecules, we show that one can use machine learning (ML) instead, {Z I , R I } ML −→ E. Thus, we circumvent the task of explicitly solving the SE by training once a machine on a finite subset of known solutions. Since many interesting questions in physics require to repeatedly solve the SE, the highly competitive performance of our ML approach may pave the way to large scale exploration of molecular energies in chemical compound space [3,4]. ML techniques have recently been used with success to map the problem of solving complex physical differential equations to statistical models. Successful attempts include solving Fokker-Planck stochastic differential equations [5], parameterizing interatomic force fields for fixed chemical composition [6,7], and the discovery of novel ternary oxides for batteries [8]. Motivated by these, and other related efforts [9-12], we develop a non-linear regression ML model for computing molecular atomization energies in chemical compound space [3]. Our model is based on a measure of distance in compound space that accounts for both stoichiometry and configurational variation. After training, energies are predicted for new (out-of-sample) molecular systems, differing in composition and geometry, at negligible computational cost, i.e. milli seconds instead of hours on a conventional CPU. While the model is trained and tested using atomization energies calculated at the hybrid density-functional theory (DFT) level [2,13,14], any other training set or level of theory could be used as a starting point for subsequent ML training. Cross-validation on 7165 molecules yields a mean absolute error of 9.9 kcal/mol, which is an order of magnitude more accurate than counting bonds or semi-empirical quantum chemistry.We use the GDB data base, a library of nearly one billion organic molecules that ...

Communication: Understanding molecular representations in machine learning: The role of uniqueness and target similarity

“…2. (a) BAML and polarizability representation based ML learning curves for 9 molecular properties of 6k constitutional isomers of formula C 7 H 10 O 2 . 31 Results for Coulomb matrix (CM) 7 and bag-of-bonds (BoB) 23 are shown for comparison. (b) BAML learning curves for 134k QM9 molecules for the same 9 molecular properties.…”

“…1 Alternatively, Kernel-Ridge-Regression (KRR) based machine learning (ML) models 2 can also infer the observable in terms of a linear expansion in chemical compound space. [3][4][5][6] More specifically, any observable can be estimated using O inf (M) =  N i α i k(d(M, M i )), where k is the kernel function (e.g., Laplacian with training set dependent width), M is the molecular representation (typically in matrix or vector format), 7,8 and d is a metric (often the L 1 -norm). The sum runs over all reference molecules i used for training to obtain regression weights {α i }.…”

What makes Data Science different? A discussion involving Statistics2.0 and Computational Sciences

“…In many systems, pure computing power and algorithms are insufficient to obtain results within a reasonable time frame. At the same time, full mathematical models involving the full complexity of the system at hand are also computationally intractable, for example in quantum physics [65][66][67][68][69]. Building such hybrid strategies, we expect, will continue to be exciting research directions, at the interface between Statistics, Computer Science and application domains, see, e.g.…”

Discovering a Transferable Charge Assignment Model Using Machine Learning

“…[11] Trained to reference datasets, ML models can predict energies, forces, and other molecular properties. [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] They have been 3 used to discover materials [28][29][30][31][32][33][34][35][36][37] and study dynamical processes such as charge and exciton transfer. [38][39][40][41] Most related to this work are ML models of existing charge models, [9,[42][43][44] which are orders of magnitude faster than ab initio calculation.…”