Results 21 to 30 of about 2,529,562 (187)
Geometric integration on spheres and some interesting applications [PDF]
J. Computational and Applied Mathematics, v.151(2003) n.1,
pp.141-170, 2011 Geometric integration theory can be employed when numerically solving ODEs or PDEs with constraints. In this paper, we present several one-step algorithms of various orders for ODEs on a collection of spheres. To demonstrate the versatility of these algorithms, we present representative calculations for reduced free rigid body motion (a conservative ...
arxiv +1 more source
Correcting auto-differentiation in neural-ODE training [PDF]
arXiv, 2023 Does the use of auto-differentiation yield reasonable updates to deep neural networks that represent neural ODEs? Through mathematical analysis and numerical evidence, we find that when the neural network employs high-order forms to approximate the underlying ODE flows (such as the Linear Multistep Method (LMM)), brute-force computation using auto ...
arxiv
Rosenbrock-Krylov Methods for Large Systems of Differential Equations
, 2014 This paper develops a new class of Rosenbrock-type integrators based on a Krylov space solution of the linear systems. The new family, called Rosenbrock-Krylov (Rosenbrock-K), is well suited for solving large scale systems of ODEs or semi-discrete PDEs ...
Sandu, Adrian, Tranquilli, Paul
core +1 more source
Robust learning with implicit residual networks
, 2020 In this effort, we propose a new deep architecture utilizing residual blocks inspired by implicit discretization schemes. As opposed to the standard feed-forward networks, the outputs of the proposed implicit residual blocks are defined as the fixed ...
Reshniak, Viktor, Webster, Clayton
core +1 more source
Embedding Capabilities of Neural ODEs [PDF]
arXiv, 2023 A class of neural networks that gained particular interest in the last years are neural ordinary differential equations (neural ODEs). We study input-output relations of neural ODEs using dynamical systems theory and prove several results about the exact embedding of maps in different neural ODE architectures in low and high dimension.
arxiv
, 2018 In this paper, Multirate Partial Differential Equations (MPDEs) are used for the efficient simulation of problems with 2-level pulsed excitations as they often occur in power electronics, e.g., DC-DC switch-mode converters.
Gyselinck, Johan +3 more
core +1 more source
Classical-quantum correspondence for shape-invariant systems
, 2015 A quantization procedure, which has recently been introduced for the analysis of Painlev\'e equations, is applied to a general time-independent potential of a Newton equation.
Grundland, A. M., Riglioni, D.
core +1 more source
Globally nilpotent differential operators and the square Ising model
, 2008 We recall various multiple integrals related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility, to the (lattice) form factors, to the two-point correlation functions and ...
Bostan, A. +5 more
core +3 more sources
Neural Ordinary Differential Equation based Recurrent Neural Network Model [PDF]
arXiv, 2020 Neural differential equations are a promising new member in the neural network family. They show the potential of differential equations for time series data analysis. In this paper, the strength of the ordinary differential equation (ODE) is explored with a new extension.
arxiv
Edge of Chaos Theory Unveils the First and Simplest Ever Reported Hodgkin–Huxley Neuristor
Advanced Electronic Materials, EarlyView.This manuscript presents the first and simplest ever‐reported electrical cell, which leverages one memristor on Edge of Chaos to reproduce the three‐bifurcation cascade, marking the entire life cycle from birth to extinction via All‐to‐None effect of an electrical spike, also referred to as Action Potential, across axon membranes under monotonic ...
Alon Ascoli +12 more
wiley +1 more source