ProfessorMatthew Foulkes

Solving the Many-Electron Schrodinger Equation using Deep Neural Networks

Given access to accurate solutions of the many-electron Schrödinger equation, nearly all of 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 an approximate wave function, which must trade off between efficiency and accuracy.

Neural networks have shown impressive power as accurate practical function approximators and promise as a compact representation of wave functions for spin systems, but problems in electronic structure require wave functions that obey Fermi-Dirac statistics. In this work [1] we introduced a novel deep learning architecture, the Fermionic Neural Network, as a powerful way to approximate wave functions for many-electron systems.

The Fermionic Neural Network is able to achieve accuracy beyond other variational Monte Carlo trial wave functions on a variety of atoms and small molecules. Recent work has shown that neural network quantum Monte Carlo methods can discover quantum phase transition [2], predict superfluidity [3], and describe the binding of positrons to molecules [4]. The first review of this exciting new field of research was recently published [5].


Warm Dense Matter

Electrons are an elementary component of our world and determine many of the properties of solids and liquids, but despite their ubiquity, scientists have not yet been able to accurately describe the behaviour of large numbers of interacting electrons. This is especially true at extreme temperatures and densities, such as inside Jupiter and other gas giants, where the electrons form ‘warm dense matter’. Scientists have plenty of approximate models to choose from, but little idea of their accuracy or reliability. Warm dense matter can be created experimentally by illuminating a solid with a high-intensity conventional laser of a free-electron laser such as the new European XFEL in Hamburg. Warm dense matter is relevant for experiments with inertial confinement fusion, where fuel pellets are put under extreme pressure. This can cause chain reactions that might one day provide a virtually unlimited source of clean energy.

Earlier theories of warm dense matter behaviour used models based on approximations that are difficult to verify. However, by using sophisticated computer simulations, we are now able to precisely solve the complex quantum mechanical equations that describe matter in the warm dense regime. Working with groups from Kiel University, the Los Alamos National Laboratory, and the Lawrence Livermore National Laboratory, we used quantum Monte Carlo methods to construct the first accurate local density approximation for the exchange-correlation free energy of the warm dense electron gas [6]. This can be used, along with simpler simulation methods based on thermal density functional theory, to study more complicated warm dense objects. For this work, the American Physical Society presented our team with the 2023 John Dawson Award for Excellence in Plasma Physics Research.


References

[1] Ab-Initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks, by David Pfau, James S. Spencer, Alexander G. de G. Matthews, and W.M.C. Foulkes, Phys. Rev. Research 2, 033429 (2020)
[2] Discovering quantum phase transitions with neural-network wavefunctions, G. Cassella, H. Sutterud, S. Azadi, N.D. Drummond, D. Pfau, J.S. Spencer, and W.M.C. Foulkes, Phys. Rev. Lett. 130, 036401 (2023)
[3] Neural wave functions for superfuids, W.T. Lou, H. Sutterud, G. Cassella, W.M.C. Foulkes, J. Knolle, D. Pfau, and J.S. Spencer, accepted for publication in Phys. Rev. X, arXiv:2305.06989 (2024)
[4] G. Cassella, W.M.C. Foulkes, D. Pfau, and J.S. Spencer, Neural network variational Monte Carlo for positronic chemistry, accepted for publication in Nat. Commun, arXiv:2310.05607 (2024)
[5] Ab-initio quantum chemistry with neural-network wavefunctions, J. Hermann, J. Spencer, K. Choo, A. Mezzacapo, W.M.C. Foulkes, D. Pfau, G. Carleo, and Frank Noé, Nat. Rev. Chem. 7, 692 (2023)
[6] Ab initio Exchange-Correlation Free Energy of the Uniform Electron Gas at Warm Dense Matter Conditions, by Simon Groth, Tobias Dornheim, Travis Sjostrom, Fionn D. Malone, W. M. C. Foulkes, and Michael Bonitz, Phys. Rev. Lett. 119, 135001 (2017).