Related papers: Two-layers neural networks for Schr{\"o}dinger eig…
Physics-informed neural networks have been widely applied to learn general parametric solutions of differential equations. Here, we propose a neural network to discover parametric eigenvalue and eigenfunction surfaces of quantum systems. We…
The energy landscape of high-dimensional non-convex optimization problems is crucial to understanding the effectiveness of modern deep neural network architectures. Recent works have experimentally shown that two different solutions found…
In this paper, we study the estimation of partially linear models for spatial data distributed over complex domains. We use bivariate splines over triangulations to represent the nonparametric component on an irregular two-dimensional…
A recent strategy to circumvent the exploding and vanishing gradient problem in RNNs, and to allow the stable propagation of signals over long time scales, is to constrain recurrent connectivity matrices to be orthogonal or unitary. This…
Two aspects of neural networks that have been extensively studied in the recent literature are their function approximation properties and their training by gradient descent methods. The approximation problem seeks accurate approximations…
We present an algorithm for the minimization of a nonconvex quadratic function subject to linear inequality constraints and a two-sided bound on the 2-norm of its solution. The algorithm minimizes the objective using an active-set method by…
We explore the block nature of the matrix representation of multiplex networks, introducing a new formalism to deal with its spectral properties as a function of the inter-layer coupling parameter. This approach allows us to derive…
We develop a corrective mechanism for neural network approximation: the total available non-linear units are divided into multiple groups and the first group approximates the function under consideration, the second group approximates the…
The two-dimensional cubic nonlinear Schr\"{o}dinger equation is used to describe the propagation of an intense laser beam through a medium with Kerr nonlinearity. The coupled two-dimensional cubic nonlinear Schr\"{o}dinger equations are…
Let $A$ be a set and $V$ a real Hilbert space. Let $H$ be a real Hilbert space of functions $f:A\to V$ and assume $H$ is continuously embedded in the Banach space of bounded functions. For $i=1,\cdots,n$, let $(x_i,y_i)\in A\times V$…
The goal of this paper is to survey the properties of the eigenvalue relaxation for least squares binary problems. This relaxation is a convex program which is obtained as the Lagrangian dual of the original problem with an implicit compact…
We provide a characterization of the spectral minimum for a random Schr\"odinger operator of the form $H=-\Delta + \sum_{i \in \Z^d}q(x-i-\omega_i)$ in $L^2(\R^d)$, where the single site potential $q$ is reflection symmetric, compactly…
In this paper, we apply the optimized Schwarz method to the two dimensional nonlinear Schr{\"o}dinger equation and extend this method to the simulation of Bose-Einstein condensates (Gross-Pitaevskii equation). We propose an extended version…
In this article we are interested for the numerical study of nonlinear eigenvalue problems. We begin with a review of theoretical results obtained by functional analysis methods, especially for the Schrodinger pencils. Some recall are given…
One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model. This is the set of functions endowed with a quantity which can control the approximation and estimation…
A new method to solve computationally challenging (random) parametric obstacle problems is developed and analyzed, where the parameters can influence the related partial differential equation (PDE) and determine the position and surface…
This paper develops expansive gradient dynamics in deep neural network-induced mapping spaces. Specifically, we generate tools and concepts for minimizing a class of energy functionals in an abstract Hilbert space setting covering a wide…
We propose automatic optimisation methods considering the geometry of matrix manifold for the normalised parameters of neural networks. Layerwise weight normalisation with respect to Frobenius norm is utilised to bound the Lipschitz…
In this paper, we introduce a new functional point of view on bilevel optimization problems for machine learning, where the inner objective is minimized over a function space. These types of problems are most often solved by using methods…
Deep learning has received considerable empirical successes in recent years. However, while many ad hoc tricks have been discovered by practitioners, until recently, there has been a lack of theoretical understanding for tricks invented in…