Related papers: Solving partial differential equations with wavegu…
In classical continuum physics, a wave is a mechanical disturbance. Whether the disturbance is stationary or traveling and whether it is caused by the motion of atoms and molecules or the vibration of a lattice structure, a wave can be…
Neural networks have proven effective for solving many difficult computational problems. Implementing complex neural networks in software is very computationally expensive. To explore the limits of information processing, it will be…
The ability of mechanical systems to perform basic computations has gained traction over recent years, providing an unconventional alternative to digital computing in off grid, low power, and severe environments which render the majority of…
We develop a fast, accurate, and robust technique to model the electromagnetic response of a two-dimensional (2D) waveguide-fed metasurface aperture. The geometry under consideration consists of a parallel-plate waveguide with an array of…
We adapt boundary deformation techniques to solve a Neumann problem for the Helmholtz equation with rough electric potentials in bounded domains. In particular, we study the dependance of Neumann eigenvalues of the perturbed Laplacian with…
We consider the wave equation $\varepsilon^2(-\partial_t^2 + \Delta)u + f(u) = 0$ for $0<\varepsilon\ll 1$, where $f$ is the derivative of a balanced, double-well potential, the model case being $f(u) = u-u^3$. For equations of this form,…
We review the recent advances reported in the field of integrated photonic waveguide meshes, both from the theoretical as well as from the experimental point of view. We show how these devices can be programmed to implement both traditional…
This article proposes a novel approach for determining exact solutions to nonlinear ordinary differential equations. The recommended iterative method provides the solution via a rapidly converging series that readily approaches a closed…
In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded physical domain. With the help of tensor product structure, we…
We investigate the Helmholtz equation in a two dimensional open waveguide with a thin and high contrast core layer. We develop an asymptotic analysis of the Green function of the problem, and through it we identify and characterize the…
We introduce and analyze a virtual element method (VEM) for the Helmholtz problem with approximating spaces made of products of low order VEM functions and plane waves. We restrict ourselves to the 2D Helmholtz equation with impedance…
This paper establishes a comprehensive mathematical framework connecting optical physics equations to generative models, demonstrating how light propagation dynamics inspire powerful artificial intelligence approaches. We analyze six…
In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for…
Learning the full family of solutions to parameterized partial differential equations (PDEs) is a central challenge to our ability to model the behavior of heterogeneous systems, with a variety of fundamental and application-oriented…
Photonic topological insulators (PTIs) have been widely studied due to the robustness of energy transport via supported edge modes immune to structural disorder. In this work, a topological gap waveguide is constructed by introducing line…
The multigrid algorithm is a multilevel approach to accelerate the numerical solution of discretized differential equations in physical problems involving long-range interactions. Multiresolution analysis of wavelet theory provides an…
We consider solutions to the linear wave equation on non-compact Riemannian manifolds without boundary when the geodesic flow admits a filamentary hyperbolic trapped set. We obtain a polynomial rate of local energy decay with exponent…
This article is devoted to the proof of the well-posedness of a model describing waves propagating in shallow water in horizontal dimension $d=2$ and in the presence of a fixed partially immersed object. We first show that this…
In this paper, we propose a new numerical method for scattering problems in periodic waveguide, based on the newly established contour integral representation of solutions in a previous paper by the author (see [Zhadf]). For this kind of…
We introduce a simple initial working system in which relations (such as part-whole) are directly represented via an architecture with operating and learning rules fundamentally distinct from standard artificial neural network methods.…