Solving many-electron Schrödinger equation using deep neural networks

Han, Jiequn; Zhang, Linfeng; Weinan, E

doi:10.1016/j.jcp.2019.108929

Cited by 184 publications

(142 citation statements)

References 64 publications

Supporting

Mentioning

140

Contrasting

Unclassified

Order By: Relevance

“…A particularly important problem is the ground state of a quantum system [13,34,51]. Let H be the Hamiltonian operator of the quantum system, say on R d .…”

Section: Calculus Of Variationsmentioning

confidence: 99%

See 1 more Smart Citation

Machine learning from a continuous viewpoint, I

Weinan

2020

Sci. China Math.

Self Cite

View full text Add to dashboard Cite

We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, in the spirit of classical numerical analysis. We demonstrate that conventional machine learning models and algorithms, such as the random feature model, the two-layer neural network model and the residual neural network model, can all be recovered (in a scaled form) as particular discretizations of different continuous formulations. We also present examples of new models, such as the flow-based random feature model, and new algorithms, such as the smoothed particle method and spectral method, that arise naturally from this continuous formulation. We discuss how the issues of generalization error and implicit regularization can be studied under this framework.

show abstract

“…A particularly important problem is the ground state of a quantum system [13,34,51]. Let H be the Hamiltonian operator of the quantum system, say on R d .…”

Section: Calculus Of Variationsmentioning

confidence: 99%

“…An important application of machine learning is the numerical solution of high dimensional PDEs [13,22,27,[32][33][34]42,51,59]. Formulating these PDEs as variational problems is an important step in formulating machine learning based algorithms.…”

Section: Nonlinear Parabolic Pdesmentioning

confidence: 99%

Machine learning from a continuous viewpoint, I

Weinan

2020

Sci. China Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Applications of neural network Ansatz to chemical systems have been limited to date, presumably due to the complexity of Fermi-Dirac statistics. Existing work has been restricted to very small numbers of electrons [23], or has been of very low accuracy [24]. Unlike these other approaches, we use the Slater determinant as the starting point for our Ansatz, and then extend it by generalizing the single-electron orbitals to include generic exchangeable nonlinear interactions of all electrons.…”

Section: Introductionmentioning

confidence: 99%

Ab initio solution of the many-electron Schrödinger equation with deep neural networks

Pfau

Spencer

Matthews

et al. 2020

Phys. Rev. Research

389

386

View full text Add to dashboard Cite

Given access to accurate solutions of the many-electron Schrödinger equation, nearly all chemistry could be derived from first principles. Exact wave functions of interesting chemical systems are out of reach because they are NP-hard to compute in general, but approximations can be found using polynomially scaling algorithms. The key challenge for many of these algorithms is the choice of wave function approximation, or Ansatz, which must trade off between efficiency and accuracy. Neural networks have shown impressive power as accurate practical function approximators and promise as a compact wave-function Ansatz for spin systems, but problems in electronic structure require wave functions that obey Fermi-Dirac statistics. Here we introduce a novel deep learning architecture, the Fermionic neural network, as a powerful wave-function Ansatz for many-electron systems. The Fermionic neural network is able to achieve accuracy beyond other variational quantum Monte Carlo Ansatz on a variety of atoms and small molecules. Using no data other than atomic positions and charges, we predict the dissociation curves of the nitrogen molecule and hydrogen chain, two challenging strongly correlated systems, to significantly higher accuracy than the coupled cluster method, widely considered the most accurate scalable method for quantum chemistry at equilibrium geometry. This demonstrates that deep neural networks can improve the accuracy of variational quantum Monte Carlo to the point where it outperforms other ab initio quantum chemistry methods, opening the possibility of accurate direct optimization of wave functions for previously intractable many-electron systems.

show abstract

“…It is only recently that ML, specifically deep NNs, have started to be used to solve multi-electron SE. Han et al [83] computed ground-state energies of Be, B, LiH, and a chain of 10 hydrogen atoms. The antisymmetric nature of the wavefunction was imposed by a product representation whose components were represented with multi-layer NNs.…”

Section: Wavefunction Representationmentioning

confidence: 99%

Machine learning for the solution of the Schrödinger equation

Manzhos

2020

Mach. Learn.: Sci. Technol.

View full text Add to dashboard Cite

Machine learning (ML) methods have recently been increasingly widely used in quantum chemistry. While ML methods are now accepted as high accuracy approaches to construct interatomic potentials for applications, the use of ML to solve the Schrödinger equation, either vibrational or electronic, while not new, is only now making significant headway towards applications. We survey recent uses of ML techniques to solve the Schrödinger equation, including the vibrational Schrödinger equation, the electronic Schrödinger equation and the related problems of constructing functionals for density functional theory (DFT) as well as potentials which enter semi-empirical approximations to DFT. We highlight similarities and differences and specific difficulties that ML faces in these applications and possibilities for cross-fertilization of ideas.© 2020 The Author(s). Published by IOP Publishing Ltd Mach. Learn.: Sci. Technol. 1 (2020) 013002 S Manzhos approximator properties, with modifications specific to the problem of solving the SE or building functionals, including adapted architectures of ML methods and data selection schemes. It is therefore important to recognize similarities and differences in the requirements that can be addressed via ML among different applications.In this Perspective, we survey recent uses of ML techniques to solve the Schrödinger equation, including the vibrational Schrödinger equation and the electronic problem: the electronic Schrödinger equation and the related problems of constructing exchange-correlation and kinetic energy functionals for Kohn-Sham (KS-) and orbital-free (OF-) density functional theory (DFT) as well as use of ML for semi-empirical approximations used in density functional based approaches such as DFTB (density functional tight binding) and dispersion-corrected DFT (DFT-D). We only consider the use of ML for the solution of the vibrational and electronic SE or the KS equation or the Hohenberg-Kohn (HK) equation and not methods that aim to avoid such solutions (such as those directly mapping the molecular structure to the spectrum or energy or properties without solving for the density or wavefunction [29][30][31][32][33][34][35][36]); we also do not consider uses of ML for other types of quantum mechanical modelling or other types of differential or integral equations [37][38][39][40][41][42].We do not aim to present a comprehensive review but rather a survey allowing similarities and differences of ML uses in all these applications, as well as promising directions for future research, to transpire. This is not a review of ML methods; the key machine learning techniques that found use in quantum chemistry are well reviewed elsewhere [3-5, 43, 44], their description will not be repeated there; the reader is advised to consult the literature for their introduction, such as [45,46] for neural networks, [47,48] for Gaussian process regression (GPR), [49] for kernel ridge regression (KRR; note the similarity in the form of the function representation between GPR and KRR), [50] for...

show abstract

Solving many-electron Schrödinger equation using deep neural networks

Cited by 184 publications

References 64 publications

Machine learning from a continuous viewpoint, I

Machine learning from a continuous viewpoint, I

Ab initio solution of the many-electron Schrödinger equation with deep neural networks

Machine learning for the solution of the Schrödinger equation

Contact Info

Product

Resources

About