We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: first, we prove that continuous nonlinear functionals, functional derivatives and FDEs can be approximated uniformly on any compact subset of a real separable Hilbert space by high-dimensional multivariate functions and high-dimensional partial differential equations (PDEs), respectively. Second, we show that the convergence rate of such functional approximations can be exponential, depending on the regularity of the functional (in particular its Fr\'echet differentiability), and its domain. We also provide necessary and sufficient conditions for consistency, stability and convergence of functional approximations to linear FDEs. These results open the possibility to utilize numerical techniques for high-dimensional model representation such as deep neural networks and numerical tensor methods to approximate nonlinear functionals in terms of high-dimensional functions, and compute approximate solutions to FDEs by solving high-dimensional PDEs. Numerical demonstrations are presented and discussed for prototype nonlinear functionals in the space of square-integrable functions and for an initial value problem involving a linear FDE.

中文翻译：

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非线性泛函和泛函微分方程的谱方法

我们对非线性泛函、泛函导数和泛函微分方程 (FDE) 的圆柱近似进行了严格的收敛分析。这种分析的目的有两个：首先，我们证明了连续非线性泛函、泛函导数和 FDE 可以通过高维多元函数和高维偏微分方程 (PDE) 在实可分 Hilbert 空间的任何紧子集上均匀逼近）， 分别。其次，我们表明这种函数近似的收敛速度可以是指数级的，这取决于函数的规律性（特别是它的 Fr\'echet 可微性）及其域。我们还为线性 FDE 的函数逼近的一致性、稳定性和收敛性提供了充分必要条件。这些结果为利用深度神经网络和数值张量方法等高维模型表示的数值技术根据高维函数逼近非线性函数，并通过求解高维偏微分方程计算 FDE 的近似解提供了可能性。对平方可积函数空间中的原型非线性泛函和涉及线性 FDE 的初值问题进行了数值演示和讨论。