Error estimation in the neural network solution of ordinary differential equations

2010 ◽  
Vol 23 (5) ◽  
pp. 614-617 ◽  
Author(s):  
Cristian Filici
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Error Estimation ◽  
Network Solution ◽  
The Neural Network

A HORSESHOE IN A CELLULAR NEURAL NETWORK OF FOUR-DIMENSIONAL AUTONOMOUS ORDINARY DIFFERENTIAL EQUATIONS

2007 ◽  
Vol 17 (09) ◽  
pp. 3211-3218 ◽  
Author(s):  
XIAO-SONG YANG ◽  
QINGDU LI
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Poincaré Map ◽  
Cellular Neural Network ◽  
Poincare Map ◽  
Invariant Subset ◽  
The Neural Network

We obtain numerically a horseshoe in a Poincaré map derived from a cellular neural network described by four-dimensional autonomous ordinary differential equations. Contrary to the horseshoe numerically found in the Hodgkin–Huxley model, which showed evidence that the Poincaré map derived from the Hodgkin–Huxley model has just one expanding direction on some invariant subset, the horseshoe obtained in this paper proves that the Poincaré map derived from the neural network have two expanding directions on some invariant subset.

scholarly journals Discretization and machine learning approximation of BSDEs with a constraint on the Gains-process

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Idris Kharroubi ◽  
Thomas Lim ◽  
Xavier Warin
Keyword(s):  
Neural Network ◽  
Machine Learning ◽  
Neural Networks ◽  
Differential Equations ◽  
Numerical Experiments ◽  
Optimization Problem ◽  
Learning Approach ◽  
The Neural Network ◽  
Machine Learning Approach ◽  
Mesh Grid

AbstractWe study the approximation of backward stochastic differential equations (BSDEs for short) with a constraint on the gains process. We first discretize the constraint by applying a so-called facelift operator at times of a grid. We show that this discretely constrained BSDE converges to the continuously constrained one as the mesh grid converges to zero. We then focus on the approximation of the discretely constrained BSDE. For that we adopt a machine learning approach. We show that the facelift can be approximated by an optimization problem over a class of neural networks under constraints on the neural network and its derivative. We then derive an algorithm converging to the discretely constrained BSDE as the number of neurons goes to infinity. We end by numerical experiments.

scholarly journals Data Driven Solutions and Discoveries in Mechanics Using Physics Informed Neural Network

2020 ◽  
Author(s):  
Qi Zhang ◽  
Yilin Chen ◽  
Ziyi Yang
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
Automatic Differentiation ◽  
Numerical Solutions ◽  
Data Driven ◽  
Attractive Alternative ◽  
The Finite Element Method ◽  
The Neural Network ◽  
Engineering Problems ◽  
Free Parameters

Deep learning has achieved remarkable success in diverse computer science applications, however, its use in other traditional engineering fields has emerged only recently. In this project, we solved several mechanics problems governed by differential equations, using physics informed neural networks (PINN). The PINN embeds the differential equations into the loss of the neural network using automatic differentiation. We present our developments in the context of solving two main classes of problems: data-driven solutions and data-driven discoveries, and we compare the results with either analytical solutions or numerical solutions using the finite element method. The remarkable achievements of the PINN model shown in this report suggest the bright prospect of the physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters. More broadly, this study shows that PINN provides an attractive alternative to solve traditional engineering problems.

scholarly journals Nonindependent Session Recommendation Based on Ordinary Differential Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Zhenyu Yang ◽  
Mingge Zhang ◽  
Guojing Liu ◽  
Mingyu Li
Keyword(s):  
Neural Network ◽  
Differential Equation ◽  
Neural Networks ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Deep Neural Networks ◽  
Semantic Information ◽  
User Behavior ◽  
User Behaviors

The recommendation method based on user sessions is mainly to model sessions as sequences in the assumption that user behaviors are independent and identically distributed, and then to use deep semantic information mining through Deep Neural Networks. Nevertheless, user behaviors may be a nonindependent intention at irregular points in time. For example, users may buy painkillers, books, or clothes for different reasons at different times. However, this has not been taken seriously in previous studies. Therefore, we propose a session recommendation method based on Neural Differential Equations in an attempt to predict user behavior forward or backward from any point in time. We used Ordinary Differential Equations to train the Graph Neural Network and could predict forward or backward at any point in time to model the user's nonindependent sessions. We tested for four real datasets and found that our model achieved the expected results and was superior to the existing session-based recommendations.

scholarly journals Solving parametric PDE problems with artificial neural networks

2020 ◽  
pp. 1-15
Author(s):  
YUEHAW KHOO ◽  
JIANFENG LU ◽  
LEXING YING
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Artificial Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Time Evolution ◽  
Random Coefficients ◽  
Numerical Partial Differential Equations ◽  
The Neural Network ◽  
Physical Quantities

The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modelled into the equations as random coefficients. However, very often the variability of physical quantities derived from PDE can be captured by a few features on the space of the coefficient fields. Based on such observation, we propose using neural network to parameterise the physical quantity of interest as a function of input coefficients. The representability of such quantity using a neural network can be justified by viewing the neural network as performing time evolution to find the solutions to the PDE. We further demonstrate the simplicity and accuracy of the approach through notable examples of PDEs in engineering and physics.

Sign in / Sign up

Export Citation Format

Share Document