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@INPROCEEDINGS{Saleh:607067,
      author       = {Saleh, Yahya},
      title        = {{L}earning basis sets in {L}$^2$ and application to
                      computing excited states of molecules},
      reportid     = {PUBDB-2024-01766},
      year         = {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.},
      month         = {Apr},
      date          = {2024-04-21},
      organization  = {Workshop on Modern Methods for
                       Differential Equations of Quantum
                       Mechanics at Banff International
                       Research Station, Banff (Canada), 21
                       Apr 2024 - 26 Apr 2024},
      subtyp        = {Invited},
      cin          = {UHH / FS-CFEL-CMI / UNI/CUI / UNI/EXP},
      cid          = {I:(DE-H253)UHH-20231115 / I:(DE-H253)FS-CFEL-CMI-20220405 /
                      $I:(DE-H253)UNI_CUI-20121230$ /
                      $I:(DE-H253)UNI_EXP-20120731$},
      pnm          = {631 - Matter – Dynamics, Mechanisms and Control
                      (POF4-631) / DFG project 390715994 - EXC 2056: CUI: Advanced
                      Imaging of Matter (390715994)},
      pid          = {G:(DE-HGF)POF4-631 / G:(GEPRIS)390715994},
      experiment   = {EXP:(DE-MLZ)NOSPEC-20140101},
      typ          = {PUB:(DE-HGF)6},
      url          = {https://bib-pubdb1.desy.de/record/607067},
}