Regularized dynamical parametric approximation
CoRR(2024)
摘要
This paper studies the numerical approximation of evolution equations by
nonlinear parametrizations u(t)=Φ(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 Φ'(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.
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