论文标题
非线性功能和功能微分方程的光谱方法
Spectral methods for nonlinear functionals and functional differential equations
论文作者
论文摘要
我们提出了严格的融合分析，用于非线性功能，功能衍生物和功能微分方程（FDES）的圆柱近似值。该分析的目的是双重的：首先，我们证明连续的非线性功能，功能衍生物和FDE可以在实际Banach空间的任何紧凑型子集上均匀地近似，从而分别通过高维多变量函数和高维度偏差差分（PDES）来承认基础。其次，我们表明，此类功能近似的收敛速率可以指数级，具体取决于功能的规律性（尤其是其fréchet的可不同性）及其域。我们还提供了必要和充分的条件，以使圆柱近似与线性FDE的一致性，稳定性和收敛性。这些结果为高维系统（例如深神经网络和数值张量方法）使用数值技术开辟了可能性，以高维函数近似非线性功能，并通过求解高维PDE来计算FDE的近似解决方案。为非线性函数以及涉及线性FDE的初始值问题提供了数值示例和讨论。
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 Banach space admitting a basis 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 Fréchet differentiability), and its domain. We also provide necessary and sufficient conditions for consistency, stability and convergence of cylindrical approximations to linear FDEs. These results open the possibility to utilize numerical techniques for high-dimensional systems 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 examples are presented and discussed for prototype nonlinear functionals and for an initial value problem involving a linear FDE.
