Related papers: Gradient-preserving hyper-reduction of nonlinear d…
We propose and study an algorithm for computing a nearest passive system to a given non-passive linear time-invariant system (with much freedom in the choice of the metric defining `nearest', which may be restricted to structured…
We propose a linearly implicit structure-preserving numerical method for semilinear Hamiltonian systems with polynomial nonlinearities, combining Kahan's method and exponential integrator. This approach efficiently balances computational…
Gradient normalization and soft clipping are two popular techniques for tackling instability issues and improving convergence of stochastic gradient descent (SGD) with momentum. In this article, we study these types of methods through the…
In the last two decades, significant effort has been put in understanding and designing so-called structure-preserving numerical methods for the simulation of mechanical systems. Geometric integrators attempt to preserve the geometry…
In this paper, we analyze the preservation of asymptotic properties of partially dissipative hyperbolic systems when switching to a discrete setting. We prove that one of the simplest consistent and unconditionally stable numerical methods…
We discuss structure-preserving model order reduction for port-Hamiltonian systems based on an approximation of the full-order state by a linear combination of ansatz functions which depend themselves on the state of the reduced-order…
As deep learning applications continue to deploy increasingly large artificial neural networks, the associated high energy demands are creating a need for alternative neuromorphic approaches. Optics and photonics are particularly compelling…
We introduce a method for the nonlinear dimension reduction of a high-dimensional function $u:\mathbb{R}^d\rightarrow\mathbb{R}$, $d\gg1$. Our objective is to identify a nonlinear feature map $g:\mathbb{R}^d\rightarrow\mathbb{R}^m$, with a…
Though ubiquitous as first-principles models for conservative phenomena, Hamiltonian systems present numerous challenges for model reduction even in relatively simple, linear cases. Here, we present a method for the projection-based model…
Locally exact integrators preserve linearization of the original system at every point. We construct energy-preserving locally exact discrete gradient schemes for arbitrary multidimensional canonical Hamiltonian systems by modifying…
We present a new Subset Simulation approach using Hamiltonian neural network-based Monte Carlo sampling for reliability analysis. The proposed strategy combines the superior sampling of the Hamiltonian Monte Carlo method with…
We obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of a sublinear and a superlinear part in the sense of Barles and Souganidis (2001). We…
Gradient descent is one of the most widely used iterative algorithms in modern statistical learning. However, its precise algorithmic dynamics in high-dimensional settings remain only partially understood, which has limited its broader…
A great number of works is devoted to qualitative investigation of Hamiltonian systems. One of tools of such investigation is the method of skew-symmetric differential forms. In present work, under investigation Hamiltonian systems in…
We consider the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel general…
In this paper an approach is proposed to represent a class of dissipative mechanical systems by corresponding infinite-dimensional Hamiltonian systems. This approach is based upon the following structure: for any non-conservative classical…
In this article, we develop a reduced basis method for efficiently solving the coupled Stokes/Darcy equations with parametric internal geometry. To accommodate possible changes in topology, we define the Stokes and Darcy domains implicitly…
Modern supervised learning techniques, particularly those using deep nets, involve fitting high dimensional labelled data sets with functions containing very large numbers of parameters. Much of this work is empirical. Interesting phenomena…
The reduction of dynamical systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an exact manner - as is the case with…
We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves…