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.

翻译：本文研究通过非线性参数化$u(t)=\Phi(q(t))$对演化方程进行数值逼近，其中时间依赖参数$q(t)$需在计算过程中确定。研究动机源于量子动力学中多高斯函数逼近、以及张量网络和神经网络对各类动力学问题的逼近。在上述所有情形中，参数化通常呈现不规则性：导数$\Phi'(q)$可能具有任意小的奇异值且秩可变。我们推导了连续时间情形及时间离散情形下正则化方法的逼近结果。通过合理选择正则化参数与时间步长，该方法可成功应用于不规则情形——尽管这有悖于数值分析中旨在获得稳定算法时避免求解病态子问题的基本原则。基于高斯函数和逼近量子动力学、以及神经网络逼近常微分方程组流映射的数值实验，验证并补充了理论结果。