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arXiv:2405.07599 · Nature Computational Science

Transferable neural wavefunctions for solids

Leon Gerard, Michael Scherbela, Halvard Sutterud, W. M. C. Foulkes, Philipp Grohs

Published Oct 22, 2025• 4 citations
arXivHTMLPDF

Abstract

Abstract Deep-learning-based variational Monte Carlo has emerged as a highly accurate method for solving the many-electron Schrödinger equation. Despite favorable scaling with the number of electrons, $${\mathcal{O}}({{n}_{{\rm{el}}}}^{4})$$ O ( n el 4 ) , the practical value of deep-learning-based variational Monte Carlo is limited by the high cost of optimizing the neural network weights for every system studied. Recent research has proposed optimizing a single neural network across multiple systems, reducing the cost per system. Here we extend this approach to solids, which require numerous calculations across different geometries, boundary conditions and supercell sizes. We demonstrate that optimization of a single ansatz across these variations significantly reduces optimization steps. Furthermore, we successfully transfer a network trained on 2 × 2 × 2 supercells of LiH, to 3 × 3 × 3 supercells, reducing the number of optimization steps required to simulate the large system by a factor of 50 compared with previous work.

DOI
10.1038/s43588-025-00872-z

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Abstract Deep-learning-based variational Monte Carlo has emerged as a highly accurate method for solving the many-electron Schrödinger equation. Despite favorable scaling with the number of electrons, O(nel4){\mathcal{O}}({{n}_{{\rm{el}}}}^{4})O(nel​4) O ( n el 4 ) , the practical value of deep-learning-based variational Monte Carlo is limited by the high cost of optimizing the neural network weights for every system studied. Recent research has proposed optimizing a single neural network across multiple systems, reducing the cost per system. Here we extend this approach to solids, which require numerous calculations across different geometries, boundary conditions and supercell sizes. We demonstrate that optimization of a single ansatz across these variations significantly reduces optimization steps. Furthermore, we successfully transfer a network trained on 2 × 2 × 2 supercells of LiH, to 3 × 3 × 3 supercells, reducing the number of optimization steps required to simulate the large system by a factor of 50 compared with previous work.
Deep neural networks have become a highly accurate and powerful wavefunction ansatz in combination with variational Monte Carlo methods for solving the electronic Schrödinger equation. However, despite their success and favorable scaling, these methods are still computationally too costly for wide adoption. A significant obstacle is the requirement to optimize the wavefunction from scratch for each new system, thus requiring long optimization. In this work, we propose a neural network ansatz, which effectively maps uncorrelated, computationally cheap Hartree-Fock orbitals, to correlated, high-accuracy neural network orbitals. This ansatz is inherently capable of learning a single wavefunction across multiple compounds and geometries, as we demonstrate by successfully transferring a wavefunction model pre-trained on smaller fragments to larger compounds. Furthermore, we provide ample experimental evidence to support the idea that extensive pre-training of such a generalized wavefunction model across different compounds and geometries could lead to a foundation wavefunction model. Such a model could yield high-accuracy ab-initio energies using only minimal computational effort for fine-tuning and evaluation of observables.
Deep neural networks have become a highly accurate and powerful wavefunction ansatz in combination with variational Monte Carlo methods for solving the electronic Schrödinger equation. However, despite their success and favorable scaling, these methods are still computationally too costly for wide adoption. A significant obstacle is the requirement to optimize the wavefunction from scratch for each new system, thus requiring long optimization. In this work, we propose a novel neural network ansatz, which effectively maps uncorrelated, computationally cheap Hartree-Fock orbitals, to correlated, high-accuracy neural network orbitals. This ansatz is inherently capable of learning a single wavefunction across multiple compounds and geometries, as we demonstrate by successfully transferring a wavefunction model pre-trained on smaller fragments to larger compounds. Furthermore, we provide ample experimental evidence to support the idea that extensive pre-training of a such a generalized wavefunction model across different compounds and geometries could lead to a foundation wavefunction model. Such a model could yield high-accuracy ab-initio energies using only minimal computational effort for fine-tuning and evaluation of observables.
The solution of Schrödinger equation generates the quantisation properties of the system, which can completely describe the quantum behaviour of microscopic particles in a physical system. However, solving the Schrödinger equation is a non-deterministic polynomial time (NP)-hard problem. Numerical methods for solving the Schrödinger equation require careful selection of basis configurations, and the complete basis set has high accuracy but its complexity increases exponentially in the number of particles. Therefore, it is crucial to find an effective numerical method. To solve this problem, this paper present a deep learning architecture, VMCNet, using the powerful computational efficiency of neural networks to improve the speed of numerical computation. Moreover, this paper proposes a suitable wavefunction ansatz, witch determines the accuracy of neural networks, that achieves more accurate energy solutions of the many-electron Schrödinger equation compared to the conventional single-determinant variational molecular Monte Carlo. We introduce non-local and local blocks to represent quantum mechanical interactions, which contain more physical information about the particle system. The experimental results show that VMCNet can cover more than 93% of the correlation energy of the main group elements (Be–Ne) for the monoatomic system and more than 90% of the correlation energy for the diatomic system.
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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.
Deep neural networks have been shown as a potentially powerful ansatz in variational Monte Carlo for solving quantum many-body problems. We propose two improvements in this direction. The first is graph neural ansatz (GNA), which is a variational wavefunction universal to arbitrary geometry. GNA results in accurate ground-state energies on 2D Kagome lattices, triangular lattices, and randomly connected graphs. Secondly, we design a distributed workflow on multiple accelerators to scale up the computation. We compute Kagome lattices with sizes up to 432 sites on 128 TPU cores. The parameter sharing nature of the GNA also leads to transferability across different system sizes and geometries.