Research reports

Years: 2025 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Regularized dynamical parametric approximation

by M. Feischl and C. Lasser and C. Lubich and J. Nick

(Report number 2024-14)

Abstract
This paper studies the numerical approximation of evolution equations by nonlinear parametrizations $u(t)=\Phi(q(t))$ with time-dependent parameters $q(t)$, which are to be determined in the computation. The motivation comes from approximations in quantum dynamics by multiple Gaussians and approximations of various dynamical problems by tensor networks and neural networks. In all these cases, the parametrization is typically irregular: the derivative $\Phi'(q)$ can have arbitrarily small singular values and may have varying rank. We derive approximation results for a regularized approach in the time-continuous case as well as in time-discretized cases. With a suitable choice of the regularization parameter and the time stepsize, the approach can be successfully applied in irregular situations, even though it runs counter to the basic principle in numerical analysis to avoid solving ill-posed subproblems when aiming for a stable algorithm. Numerical experiments with sums of Gaussians for approximating quantum dynamics and with neural networks for approximating the flow map of a system of ordinary differential equations illustrate and complement the theoretical results.

Keywords: Time-dependent nonlinear parametric approximation, regularization, high-dimensional differential equation, Schr\"odinger equation, time integration, deep neural network, multi-Gaussian approximation

@Techreport{FLLN24_1096,
  author = {M. Feischl and C. Lasser and C. Lubich and J. Nick},
  title = {Regularized dynamical parametric approximation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-14},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-14.pdf },
  year = {2024}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).