Spectral methods for nonlinear functionals and functional differential equations

Abstract

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.

Improving neural network predictions of material properties with limited data using transfer learning

“…where F [n] is some functional of n [24,23] while and the second term on the right had side represents the interaction of electrons with the external potential created from the nuclei. The Hohenberg-Kohn theorems showed that the functional F [n] exists.…”

“…The significant improvement in accuracy given by GGA functionals led to the wider adoption of density functional theory across chemistry and material science [13]. The exchange-correlation potential V XC in ( 7) is the first-order functional derivative [24] of…”

“…In (24) η ∈ [0, 1) is a predetermined parameter [6], and v must be initialized. The algorithm ( 23)-( 24) is also known as gradient descent with "momentum", in a physical analogy for the velocity of a ball rolling down a hill.…”

“…The ADAM algorithm updates individual learning rates based on previous learning rates in an exponential moving average [12]. Each parameter is updated similarly to ( 23)- (24). Specifically, in ADAM we have that ( 24) is replaced by…”

Adaptive integration of nonlinear evolution equations on tensor manifolds

“…Computing the solution of high-dimensional partial differential equations (PDEs) has become central to many new areas of application such as optimal mass transport [13,48], random dynamical systems [21,46,47], mean field optimal control [11,38], and functional-differential equations [45,44]. Classical numerical methods based on full tensor product discretizations are not viable in practice, due to the exponential growth of the degrees of freedom with the dimension.…”