Regularized dynamical parametric approximation
4 affiliations
- 1Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8–10, 1040, Wien, Austria
- 2Department of Mathematics, TU München, Boltzmannstraße 3, 85748, Garching bei München, Germany
- 3Mathematisches Institut, University Tübingen, Auf der Morgenstelle 10, 72076, Tubingen, Germany
- 4Seminar für Angewandte Mathematik, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland
Abstract
This paper studies the numerical approximation of evolution equations by nonlinear parametrizations u(t)=varPhi (θ (t)) u ( t ) = Φ ( θ ( t ) ) with time-dependent parameters θ (t) θ ( 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 varPhi '(θ ) Φ ′ ( θ ) 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. For the latter, there is a nontrivial interplay between the regularization parameter and the time stepsize, depending also on the defect size and local bounds of the second derivative of the parametrization map varPhi Φ . When this is appropriately taken into account, 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. The paper also includes two theoretical case studies: regularized parametric steepest descent for optimization and regularized parametric Strang splitting for the time-dependent Schrödinger equation. 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.
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BibTeX
@article{feischl2026regularized,
title = {{Regularized dynamical parametric approximation}},
author = {Michael Feischl and Caroline Lasser and Christian Lubich and Jörg Nick},
journal = {Numerische Mathematik},
year = {2026},
month = jul,
doi = {10.1007/s00211-026-01551-5},
url = {https://doi.org/10.1007/s00211-026-01551-5},
note = {CC BY},
}
RIS
TY - JOUR
TI - Regularized dynamical parametric approximation
AU - Michael Feischl
AU - Caroline Lasser
AU - Christian Lubich
AU - Jörg Nick
PY - 2026
DA - 2026/07/15
JO - Numerische Mathematik
DO - 10.1007/s00211-026-01551-5
UR - https://doi.org/10.1007/s00211-026-01551-5
AB - This paper studies the numerical approximation of evolution equations by nonlinear parametrizations u(t)=varPhi (θ (t)) u ( t ) = Φ ( θ ( t ) ) with time-dependent parameters θ (t) θ ( 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 varPhi '(θ ) Φ ′ ( θ ) 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. For the latter, there is a nontrivial interplay between the regularization parameter and the time stepsize, depending also on the defect size and local bounds of the second derivative of the parametrization map varPhi Φ . When this is appropriately taken into account, 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. The paper also includes two theoretical case studies: regularized parametric steepest descent for optimization and regularized parametric Strang splitting for the time-dependent Schrödinger equation. 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.
SN - 0029-599X
SN - 0945-3245
ER -