Regularized dynamical parametric approximation

TL;DR

The paper develops a regularized dynamical parametric approximation framework to numerically solve evolution equations when the solution is represented by a nonlinear parametrization $u=\Phi(q)$. By solving a regularized least-squares problem for $\dot q$, it yields a well-defined tangent-space projection $\dot u = P_\varepsilon(q) f(u)$ that remains useful even when $\Phi'$ is ill-conditioned. It derives both a posteriori and a priori error bounds, analyzes time discretizations (explicit/implicit Euler, Runge–Kutta, Strang splitting), and introduces adaptive strategies for selecting the regularization parameter $\varepsilon$ and the time step $h$. The framework is illustrated through a Schrödinger equation case study and numerical experiments on neural-network-based flow-map approximation and sums of Gaussians for quantum dynamics, highlighting conservation aspects and practical performance. Overall, the work provides a rigorous pathway to stable approximate dynamics under irregular parametrizations and offers adaptive schemes to balance accuracy and stability in complex, high-dimensional settings.

Abstract

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.

Figures (4)

• Figure 1: Time convergence plot for the Lotka--Volterra system, computed with a fixed neural network architecture with three hidden layers and four neurons each, which is fully described by $q\in \mathbb R^{62}$. We fix the regularization parameter $\varepsilon$ and observe the error behaviour of the classical Runge--Kutta approximation to the regularized flow \ref{['reg-lsq']}. On the right-hand side, we plot the projection error term of the error bound described in Proposition \ref{['prop:glob-err-rk']}.
• Figure 2: The $\varepsilon-$ convergence of the same network architecture, with the same time discretization. We fix the number of time steps and vary the regularization parameter.
• Figure 3: Time convergence plot for the double-well system, computed with a sum of $M=36$ complex Gaussians, which is fully described by $q\in \mathbb C^{72}$. We fix the regularization parameter $\varepsilon$ and observe the error behaviour of the classical Runge--Kutta approximation to the regularized flow \ref{['reg-lsq']}. On the left-hand side, we plot the energy error $\left|\mathcal{E}(u(T))-\mathcal{E}(u(0)) \right|$, on the right-hand side the projection error term of the error bound described in Proposition \ref{['prop:glob-err-rk']}.
• Figure 4: The $\varepsilon-$convergence of the same sum of Gaussians, with the same time discretization. We fix the number of time steps and vary the regularization parameter. On the left-hand side, we plot the norm error $|\|u(T)\|^2_{\@fontswitch\mathcal{H}} -1|$, on the right-hand side the projection error.

Theorems & Definitions (41)

• remark thmcounterremark: Truncation vs. regularization
• remark thmcounterremark
• proposition thmcounterproposition
• proof
• lemma thmcounterlemma
• proof
• proposition thmcounterproposition
• proof
• proposition thmcounterproposition
• proof