An accelerated linear method for optimizing non-linear wavefunctions in variational Monte Carlo

Sabzevari, Iliya; Mahajan, Ankit; Sharma, Sandeep

doi:10.1063/1.5125803

Cited by 22 publications

(29 citation statements)

References 76 publications

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“…To solve the eigenvalue equation with a memory efficient algorithm, we use the Davidson diagonalization scheme in which the lowest energy eigenvalues are computed without the explicit construction of the entire Hamiltonian and overlap matrices. 14 A similar procedure has recently been followed in ref ( 32 ).…”

Section: Methodsmentioning

confidence: 99%

Variational Principles in Quantum Monte Carlo: The Troubled Story of Variance Minimization

Cuzzocrea

Scemama

Briels

et al. 2020

J. Chem. Theory Comput.

124

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We investigate the use of different variational principles in quantum Monte Carlo, namely, energy and variance minimization, prompted by the interest in the robust and accurate estimation of electronic excited states. For two prototypical, challenging molecules, we readily reach the accuracy of the best available reference excitation energies using energy minimization in a state-specific or state-average fashion for states of different or equal symmetry, respectively. On the other hand, in variance minimization, where the use of suitable functionals is expected to target specific states regardless of the symmetry, we encounter severe problems for a variety of wave functions: as the variance converges, the energy drifts away from that of the selected state. This unexpected behavior is sometimes observed even when the target is the ground state and generally prevents the robust estimation of total and excitation energies. We analyze this problem using a very simple wave function and infer that the optimization finds little or no barrier to escape from a local minimum or local plateau, eventually converging to a lower-variance state instead of the target state. For the increasingly complex systems becoming in reach of quantum Monte Carlo simulations, variance minimization with current functionals appears to be an impractical route.

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Section: Methodsmentioning

confidence: 99%

Variational Principles in Quantum Monte Carlo: The Troubled Story of Variance Minimization

Cuzzocrea

Scemama

Briels

et al. 2020

J. Chem. Theory Comput.

124

View full text Add to dashboard Cite

show abstract

“…where E L is the local energy and we have dropped the dependence of ψ on θ for clarity. Recent developments [28,[35][36][37], including investigating first-order stochastic opitimization methods from the machine learning community [38,39], have enabled optimization of conventional wave functions with large parameter sets. We use a second-order method which can exploit the structure of the neural network.…”

Section: B Wave-function Optimizationmentioning

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

331

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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.

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“…The choice of efficient optimization algorithms for parameter updates in variational Monte Carlo has historically been a complex issue and is still under active debate (see e.g. [1,9,11,18,[40][41][42][43]). Among these works, Ref.…”

Section: B Optimizermentioning

confidence: 99%

Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation

Lin¹,

Goldshlager²,

Lin³

2021

Preprint

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The combination of neural networks and quantum Monte Carlo methods has arisen as a promising path forward for highly accurate electronic structure calculations. Previous proposals have combined equivariant neural network layers with a final antisymmetric layer in order to satisfy the antisymmetry requirements of the electronic wavefunction. However, to date it is unclear if one can represent antisymmetric functions of physical interest, and it is difficult to precisely measure the expressiveness of the antisymmetric layer. This work attempts to address this problem by introducing explicitly antisymmetrized universal neural network layers. This approach has a computational cost which increases factorially with respect to the system size, but we are nonetheless able to apply it to small systems to better understand how the structure of the antisymmetric layer affects its performance. We first introduce a generic antisymmetric (GA) neural network layer, which we use to replace the entire antisymmetric layer of the highly accurate ansatz known as the FermiNet. We demonstrate that the resulting FermiNet-GA architecture can yield effectively the exact ground state energy for small atoms and molecules. We then consider a factorized antisymmetric (FA) layer which more directly generalizes the FermiNet by replacing the products of determinants with products of antisymmetrized neural networks. We find, interestingly, that the resulting FermiNet-FA architecture does not outperform the FermiNet. This strongly suggests that the sum of products of antisymmetries is a key limiting aspect of the FermiNet architecture. To explore this further, we investigate a slight modification of the FermiNet, called the full determinant mode, which replaces each product of determinants with a single combined determinant. We find that the full single-determinant FermiNet closes a large part of the gap between the standard single-determinant FermiNet and FermiNet-GA on small atomic and molecular problems. Surprisingly, on the nitrogen molecule at a dissociating bond length of 4.0 Bohr, the full single-determinant FermiNet can significantly outperform the largest standard FermiNet calculation with 64 determinants, yielding an energy within 0.4 kcal/mol of the best available computational benchmark.

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An accelerated linear method for optimizing non-linear wavefunctions in variational Monte Carlo

Cited by 22 publications

References 76 publications

Variational Principles in Quantum Monte Carlo: The Troubled Story of Variance Minimization

Variational Principles in Quantum Monte Carlo: The Troubled Story of Variance Minimization

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

Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation

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