Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well-posed and convergent, by provable approximation orders. On the down side, however, these methods often suffer from the curse of dimensionality, which limits their approximation behavior, especially in situations of highly oscillatory target functions. Nonlinear approximation methods, such as neural networks, were shown to be very efficient in approximating high-dimensional functions. We investigate nonlinear approximation methods that are constructed by composing standard basis sets with normalizing flows. Such models yield richer approximation spaces while maintaining the density properties of the initial basis set, as we show. Simulations to approximate eigenfunctions of a perturbed quantum harmonic oscillator indicate convergence with respect to the size of the basis set.

翻译：以线性宽度截断的基数组等值的近似函数是差别和整体方程数字解决方案的标准程序。 常用近似方法的概念通过可辨识的近似顺序被妥善保存和聚合。 但是,在下方,这些方法往往受到维度的诅咒,这限制了其近似行为,特别是在高度振动目标功能的情况下。 神经网络等非线性近似方法在近似高维函数时被证明非常高效。 我们调查非线性近似方法,这些方法是通过以正常流组合标准基数组来构建的。 这些模型产生更丰富的近似空间,同时保持初始基数组的密度特性,正如我们所显示的。 模拟相交错的量子感应振荡器的近似值表明与设定基数大小的趋同。