Showing papers on "Bilinear interpolation published in 2022"

Novel trial functions and rogue waves of generalized breaking soliton equation via bilinear neural network method

Abstract: In this work, some new test functions are constructed by setting generalized activation functions in different artificial network models. Bilinear neural network method is introduced to solve the explicit solution of a generalized breaking soliton equation. Rogue waves of generalized breaking soliton equation are obtained by symbolic computing technology and displayed intuitively with the help of Maple software.

Fate of Measurement-Induced Phase Transition in Long-Range Interactions

Abstract: We consider quantum many-body dynamics under quantum measurements, where the measurement-induced phase transitions (MIPs) occur when changing the frequency of the measurement. In this work, we consider the robustness of the MIP for long-range interaction that decays as r^{-α} with distance r. The effects of long-range interactions are classified into two regimes: (i) the MIP is observed (α>α_{c}), and (ii) the MIP is absent even for arbitrarily strong measurements (α<α_{c}). Using fermion models, we demonstrate both regimes in integrable and nonintegrable cases. We identify the underlying mechanism and propose sufficient conditions to observe the MIP, that is, α>d/2+1 for general bilinear systems and α>d+1 for general nonintegrable systems (d: spatial dimension). Numerical calculation indicates that these conditions are optimal.

Soliton solutions by means of Hirota bilinear forms

Abstract: The paper aims to provide a brief overview of soliton solutions obtained through the Hirota direct method. A bilinear formulation of soliton solutions in both (1+1)-dimensions and (2+1)-dimensions is discussed, together with applications to various integrable equations. The Hirota conditions for N-soliton solutions are analyzed and a few open questions regarding higher-dimensional cases and generalized bilinear equations are presented.

ReLU Fields: The Little Non-linearity That Could

Abstract: In many recent works, multi-layer perceptions (MLPs) have been shown to be suitable for modeling complex spatially-varying functions including images and 3D scenes. Although the MLPs are able to represent complex scenes with unprecedented quality and memory footprint, this expressive power of the MLPs, however, comes at the cost of long training and inference times. On the other hand, bilinear/trilinear interpolation on regular grid-based representations can give fast training and inference times, but cannot match the quality of MLPs without requiring significant additional memory. Hence, in this work, we investigate what is the smallest change to grid-based representations that allows for retaining the high fidelity result of MLPs while enabling fast reconstruction and rendering times. We introduce a surprisingly simple change that achieves this task – simply allowing a fixed non-linearity (ReLU) on interpolated grid values. When combined with coarse-to-fine optimization, we show that such an approach becomes competitive with the state-of-the-art. We report results on radiance fields, and occupancy fields, and compare against multiple existing alternatives. Code and data for the paper are available at https://geometry.cs.ucl.ac.uk/projects/2022/relu_fields.

Regarding the shallow water in an ocean via a Whitham-Broer-Kaup-like system: hetero-Bäcklund transformations, bilinear forms and M solitons

Abstract: Considering the water waves, people have investigated many systems. In this paper, what we study is a Whitham-Broer-Kaup-like system for the dispersive long waves in the shallow water in an ocean. With respect to the water-wave horizontal velocity and deviation height from the equilibrium of the water, we construct (A) two branches of the hetero-Bäcklund transformations, from that system to a known constant-coefficient nonlinear dispersive-wave system, (B) two branches of the bilinear forms and (C) two branches of the M -soliton solutions, with M as a positive integer. Results rely upon the oceanic shallow-water coefficients in that system. • Whitham-Broer-Kaup-like system for the dispersive long waves in oceanic shallow water. • For the velocity of water wave and deviation from the equilibrium position of water. • As to the oceanic shallow water, building up (1) two branches of the hetero-Bäcklund transformations, • (2) two branches of the bilinear forms and (3) two branches of the M -soliton solutions. • Results relying on different dispersion/diffusion powers from the oceanic shallow water.

A novel and efficient method for obtaining Hirota’s bilinear form for the nonlinear evolution equation in (n+1) dimensions

Abstract: Bilinearization of nonlinear partial differential equations (PDEs) is essential in the Hirota method, which is a widely used and robust mathematical tool for finding soliton solutions of nonlinear PDEs in a variety of fields, including nonlinear dynamics, mathematical physics, and engineering sciences. We present a novel systematic computational approach for determining the bilinear form of a class of nonlinear PDEs in this article. It can be easily implemented in symbolic system software like Mathematica, Matlab, and Maple because of its simplicity. The proven results are obtained by using a developed method in Mathematica and applying a logarithmic transformation to the dependent variable. Finally, the findings validate the implemented technique’s competence, productivity, and dependability. The approach is a useful, authentic, and simple mathematical tool for calculating multiple soliton solutions to nonlinear evolution equations encountered in nonlinear sciences, plasma physics, ocean engineering, applied mathematics, and fluid dynamics.

Overall recursive least squares and overall stochastic gradient algorithms and their convergence for feedback nonlinear controlled autoregressive systems

Abstract: This article deals with the problems of the parameter estimation for feedback nonlinear controlled autoregressive systems (i.e., feedback nonlinear equation‐error systems). The bilinear‐in‐parameter identification model is formulated to describe the feedback nonlinear system. An overall recursive least squares algorithm is developed to handle the difficulty of the bilinear‐in‐parameter. For the purpose of avoiding the heavy computational burden, an overall stochastic gradient algorithm is deduced and the forgetting factor is introduced to improve the convergence rate. Furthermore, the convergence analysis of the proposed algorithms are established by means of the stochastic process theory. The effectiveness of the proposed algorithms are illustrated by the simulation example.

Abundant soliton wave solutions and the linear superposition principle for generalized (3+1)-D nonlinear wave equation in liquid with gas bubbles by bilinear analysis

Abstract: In this article, we study the generalized (3+1)-dimensional nonlinear wave equation where is investigated in soliton theory and by employing the Hirota’s bilinear method the bilinear form is obtained, and the N-soliton solutions are constructed. In addition, some Lump solutions, Lump-kink solutions, Lump-two kink solutions, Lump-periodic solutions, its Interaction solutions, Cross-kink wave, Breather-type, Multi wave, Periodic wave solutions, and Solitary wave solutions of the addressed equation with specific coefficients are shown. Finally, under certain conditions the asymptotic behavior of the two-soliton solution is analyzed to prove that the collision of the two-soliton is elastic. Moreover, we employ the linear superposition principle to determine N-soliton wave solutions for the generalized (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles.

Three types of periodic solutions of new (3 + 1)‐dimensional Boiti–Leon–Manna–Pempinelli equation via bilinear neural network method

Abstract: In the paper, we search for some analytical solutions for a new (3 + 1)‐dimension Boiti–Leon–Manna–Pempinelli (BLMP) equation. Based on the Hirota bilinear method, bilinear neural network framework expands to more than one hidden layer to construct test functions. By using the symbolic computation software Maple, periodic‐type I, II, and III solutions of the new (3 + 1)‐dimension BLMP equation are obtained. Besides, the evolution and dynamical characteristics of these solutions derived via the appropriate real values are also exhibited.

Resonance Y-shape solitons and mixed solutions for a (2+1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics

Abstract: Under the well-known bilinear method of Hirota, the specific expression for N-soliton solutions of (2+1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada(gCDGKS) equation in fluid mechanics is given. By defining a novel restrictive condition on N-soliton solutions, resonant Y-type and X-type soliton solutions are generated. Under the new proposed constraint, combined with the velocity resonance method and module resonant method, the mixed solutions of resonant Y-type solitons and line waves and breather solutions are found. Finally, with the support of long-wave limit method, the interaction between resonant Y-type solitons and higher-order lumps is shown, and the motion trajectory equation before and after the interaction between lumps and resonant Y-type solitons is derived. These new results greatly extend the exact solution of (2+1)-dimensional gCDGKS equation already available in the literature and provide new ideas for studying the dynamical behaviors of fluid mechanic, soliton and shallow water wave and so on.

Using Low-Rank Representation of Abundance Maps and Nonnegative Tensor Factorization for Hyperspectral Nonlinear Unmixing

Abstract: Tensor-based methods have been widely studied to attack inverse problems in hyperspectral imaging since a hyperspectral image (HSI) cube can be naturally represented as a third-order tensor, which can perfectly retain the spatial information in the image. In this article, we extend the linear tensor method to the nonlinear tensor method and propose a nonlinear low-rank tensor unmixing algorithm to solve the generalized bilinear model (GBM). Specifically, the linear and nonlinear parts of the GBM can both be expressed as tensors. Furthermore, the low-rank structures of abundance maps and nonlinear interaction abundance maps are exploited by minimizing their nuclear norm, thus taking full advantage of the high spatial correlation in HSIs. Synthetic and real-data experiments show that the low rank of abundance maps and nonlinear interaction abundance maps exploited in our method can improve the performance of the nonlinear unmixing. A MATLAB demo of this work will be available at https://github.com/LinaZhuang for the sake of reproducibility.