Related papers: Direct linearization method for nonlinear PDE's an…
We present a systematic weight pruning framework of deep neural networks (DNNs) using the alternating direction method of multipliers (ADMM). We first formulate the weight pruning problem of DNNs as a constrained nonconvex optimization…
Partial differential equations (PDEs) involving high contrast and oscillating coefficients are common in scientific and industrial applications. Numerical approximation of these PDEs is a challenging task that can be addressed, for example,…
We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation property of this type of networks, both methods extend the…
This paper aims at reviewing nonlinear methods for model order reduction of structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear based…
Solutions of partial differential equations (PDEs) on manifolds have provided important applications in different fields in science and engineering. Existing methods are majorly based on discretization of manifolds as implicit functions,…
Using the rudiments of pde jets theory in a nonstandard setting, we first deepen and extend previous nonstandard existence results for generalized solutions of linear differential equations and second extend the previous results for linear…
We consider a biochemical model that consists of a system of partial differential equations based on reaction terms and subject to non--homogeneous Dirichlet boundary conditions. The model is discretised using the gradient discretisation…
Motion planning under uncertainty is essential for reliable robot operation. Despite substantial advances over the past decade, the problem remains difficult for systems with complex dynamics. Most state-of-the-art methods perform search…
Linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data, due to their simple geometric interpretations and typically attractive computational properties. These methods capture many data features of…
We present a novel method for using Neural Networks (NNs) for finding solutions to a class of Partial Differential Equations (PDEs). Our method builds on recent advances in Neural Radiance Field research (NeRFs) and allows for a NN to…
We investigate the resolution of parabolic PDEs via Extreme Learning Machine (ELMs) Neural Networks, which have a single hidden layer and can be trained at a modest computational cost as compared with Deep Learning Neural Networks. Our…
We propose a new method for computing Dynamic Mode Decomposition (DMD) evolution matrices, which we use to analyze dynamical systems. Unlike the majority of existing methods, our approach is based on a variational formulation consisting of…
Equation discovery is aimed at directly extracting physical laws from data and has emerged as a pivotal research domain. Previous methods based on symbolic mathematics have achieved substantial advancements, but often require the design of…
Atkinson developed a strategy which splits solution of a PDE system into homogeneous and particular solutions, where the former have to satisfy the boundary and governing equation, while the latter only need to satisfy the governing…
We present a method to linearize, without approximation, a specific class of eigenvalue problems with eigenvector nonlinearities (NEPv), where the nonlinearities are expressed by scalar functions that are defined by a quotient of linear…
A novel approach is introduced for deriving exact solutions to nonlinear systems of ordinary differential equations. This method consists of four parts. In the initial part, the examined nonlinear differential equation system is transformed…
One of the oldest and most studied subject in scientific computing is algorithms for solving partial differential equations (PDEs). A long list of numerical methods have been proposed and successfully used for various applications. In…
This paper is concerned with a few novel RBF-based numerical schemes discretizing partial differential equations. For boundary-type methods, we derive the indirect and direct symmetric boundary knot methods (BKM). The resulting…
A very simple and efficient local variational iteration method for solving problems of nonlinear science is proposed in this paper. The analytical iteration formula of this method is derived first using a general form of first order…
We present a method of deriving linearizing transformations for a class of second order nonlinear ordinary differential equations. We construct a general form of a nonlinear ordinary differential equation that admits Bernoulli equation as…