Identifying hidden dynamics from observed data is a significant and challenging task in a wide range of applications. Recently, the combination of linear multistep methods (LMMs) and deep learning has been successfully employed to discover dynamics, whereas a complete convergence analysis of this approach is still under development.In this work, we consider the deep network-based LMMs for the discovery of dynamics. We put forward error estimates for these methods using the approximation property of deep networks. It indicates, for certain families of LMMs, that the ℓ 2 grid error is bounded by the sum of O(h p ) and the network approximation error, where h is the time step size and p is the local truncation error order. Numerical results of several physically relevant examples are provided to demonstrate our theory.
Some optical design problems arise from the study of photonic bandgap structure, including defect modes localization, that is, computing the optimal dielectric property to highly localize particular eigenfunctions of a Dirichlet model problem. The steepest descent method has been studied for this problem. In this paper, we present a new approach for the defect modes localization. Rather than focusing on the original objective and optimizing the structure along the gradient, a variant of the original problem is put forward with its corresponding method. Although the original problem and the variant presented in this work are not equivalent, our method is shown to solve both of them in numerical experiments. Furthermore, the algorithm in this paper can restart the optimization if the original gradient descent method gets stuck during the iteration.
Introduction.In recent years, there has been some research involving photonic crystals (first introduced in [8, 15]) and the bandgap phenomenon, in which the wave whose frequency is in some band cannot propagate in this medium. The phenomenon is related to loss mechanisms in optical and mechanical systems, thus leading to a series of optimal problems about the medium. For example, we can derive the eigenvalue problem from wave equations, then set up the objective whose variables are the eigenpairs and search for the optimal solution in an admissible set. Cox and Dobson [13,14] have considered the optimization of bandgap in two-dimensional periodic structures composed of two given dielectric materials, including the E-polarization and H-polarization cases. Another level set method can be used in this optimization [5]; Lipton, Shipman, and Venakides [12] have optimized the electromagnetic resonant properties in periodic photonic crystal slabs, from the relation between resonance and transmission; Kao and Santosa [6] have derived the maximization of quality factor from wave equations in an inhomogeneous medium, whose index of refraction is the design variable. This optimization is related to a nonlinear eigenvalue problem, and the quality factor is defined as the ratio between the real part of the complex eigenfrequency and the imaginary part. Numerical results demonstrate that the optimal solution is a piecewise constant function, which is analytically proved by Karabash [9]; Heider et al. [10] have optimized scattering resonances in micro-and nano-scale components, thus decreasing the radiative loss, which is the magnitude of the imaginary part of scattering resonances; Osting [3] considered the optimization of two spectral functions from the Dirichlet-Laplacian eigenvalue problem. The variable domains are
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.