Conference Presentation (Invited)

PUBDB-2024-01766

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Learning basis sets in L$^2$ and application to computing excited states of molecules

Saleh, Y. (Corresponding author)CFEL*DESY*

2024

Abstract: Recently, there has been a significant research interest in using neural networks for solving partial differential equations (PDEs) in general, and Schrödinger equations in particular. The use of neural networks was shown to mitigate, or even break the curse of dimensionality encountered in standard numerical methods, such as finite-volume or spectral methods. In the context of quantum mechanics, neural networks were shown to accurately approximate ground-states of molecular systems, while scaling moderately with the dimension of the problem [4]. However, extensions to computing many excited states suffer from convergence issues and remain challenging.To extend the applicability of neural-network-based ansatzes to domains requiring computations of hundreds or even thousands of excited states, such as nuclear-motion problems we propose to learn problem-specific basis sets in $L^2$. In particular, rich families in $L^2$ are produced by pushing forward standard basis sets through differentiable mappings. I show that a bijectivity assumption on the mapping is a necessary condition for the resulting family to be dense in $L^2$. This allows us to model these mappings using normalizing flows, an important tool from generative machine learning. I present a nonlinear variational framework to approximate molecular wavefunctions in the linear span of these flow-induced families. The framework allowed to compute many eigenstates of various molecular vibrational and electronic systems with orders-of-magnitude improved accuracy over standard linear methods.The present approach can be seen as a nonlinear extension of spectral methods to a spectral learning framework, where basis sets are not predefined but learned in a manor tailored to the problem under consideration. The well-posedness of such a framework and convergence guarantees are discussed.

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Contributing Institute(s):

1. Universität Hamburg (UHH)
2. CFEL-CMI (FS-CFEL-CMI)
3. beauftragt von UNI (UNI/CUI)
4. Uni Hamburg / Experimentalphysik (UNI/EXP)

Research Program(s):

1. 631 - Matter – Dynamics, Mechanisms and Control (POF4-631) (POF4-631)
2. DFG project 390715994 - EXC 2056: CUI: Advanced Imaging of Matter (390715994) (390715994)

Experiment(s):

1. No specific instrument

Appears in the scientific report 2024

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