SAMOP 2023 – wissenschaftliches Programm
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MO: Fachverband Molekülphysik
MO 7: Machine Learning and Computational and Theoretical Molecular Physics
MO 7.1: Hauptvortrag
Mittwoch, 8. März 2023, 11:00–11:30, F142
Augmenting basis with normalizing flows for solving Schrödinger equations: theoretical analysis — •Yahya Saleh1,2, Armin Iske2, Andrey Yachmenev1,4, and Jochen Küpper1,3,4 — 1Center for Free-Electron Laser Science CFEL, Deutsches Elektronen-Synchrotron DESY, Hamburg — 2Department of Mathematics, Universität Hamburg — 3Department of Physics, Universität Hamburg — 4for Ultrafast Imaging, Universität Hamburg
Spectral methods are a popular class for solving time-independent Schrödinger equations. Here, one approximates the wavefunctions in the linear span of standard basis sets of L2. In spite of the well-posedness and the convergence guarantees of such methods, they suffer from the curse of dimenionality, as the computational expenses grow exponentially with the size of the quantum system.
Recently, nonlinear functions, e.g., neural networks have been proposed to model ground states and low-lying excited states of Schrödinger equations. Although they promise accurate results with smaller scaling than standard methods, extensions of such models to the simultaneous computation of many states are still lacking.
Here, we propose to model excited states of Schrödinger equations via augmented basis sets, where standard basis sets are composed with normalizing flows. We show that such a numerical scheme is well-posed and defines a richer approximation space than standard methods. Moreover, we provide convergence guarantees.