• Corpus ID: 118674660

Solving Differential Equation with Constrained Multilayer Feedforward Network

@article{Liu2019SolvingDE,
  title={Solving Differential Equation with Constrained Multilayer Feedforward Network},
  author={Zeyu Liu and Yantao Yang and Qingdong Cai},
  journal={arXiv: Numerical Analysis},
  year={2019}
}
In this paper, we present a novel framework to solve differential equations based on multilayer feedforward network. Previous works indicate that solvers based on neural network have low accuracy due to that the boundary conditions are not satisfied accurately. The boundary condition is now inserted directly into the model as boundary term, and the model is a combination of a boundary term and a multilayer feedforward network with its weight function. As the boundary condition becomes… 
View PDF on arXiv
9 Citations
Highly Influential Citations
1
Background Citations
2
Methods Citations
3

Figures from this paper

9 Citations

ODEN: A Framework to Solve Ordinary Differential Equations using Artificial Neural Networks

A specific loss function is proved to be a suitable standard metric to evaluate neural networks' performance and its ability to outperform traditional standard numerical techniques is illustrated.

PFNN-2: A Domain Decomposed Penalty-Free Neural Network Method for Solving Partial Differential Equations

Experiments results demonstrate that PFNN-2 can outperform state-of-the-art neural network methods in various aspects such as numerical accuracy, convergence speed, and parallel scalability.

Solving Huxley equation using an improved PINN method

An improved deep learning method to recover the new soliton solution of Huxley equation and results show that the improved algorithm can use fewer sample points to reconstruct the exact solution of the Huxleys equation with faster convergence speed and better simulation effect.

Do Physics-informed Neural Networks Satisfy Local and Global Mass Balance?

Physics-informed neural networks (PINN) have recently become attractive to solve partial differential equations (PDEs) that describe physics laws. By including PDE-based loss functions, physics laws…

Deep learning and American options via free boundary framework

The proposed deep learning method presents an efficient and alternative way of pricing options with early exercise features and establishes equations that approximate the early exercise boundary and its derivative directly from the DNN output based on some linear relationships at the left boundary.

Fixed-budget online adaptive mesh learning for physics-informed neural networks. Towards parameterized problem inference

A Fixed-Budget Online Adaptive Learning (FBOAL) method is proposed, which decomposes the domain into sub-domains, for training collocation points based on local maxima and local minima of the PDEs residuals and is shown to be able to identify the high-gradient locations and give better predictions for some physical fields than the classical PINNs with collocations points sampled on a pre-adapted finite element mesh.

Physics Informed by Deep Learning: Numerical Solutions of Modified Korteweg-de Vries Equation

The method used in this paper had demonstrated the powerful mathematical and physical ability of deep learning to flexibly simulate the physical dynamic state represented by differential equations and also opens the way for us to understand more physical phenomena later.

Data-driven Discovery of Modified Kortewegde Vries Equation, Kdv–Burger Equation and Huxley Equation by Deep Learning

The method has demonstrated the powerful mathematical and physical ability of deep learning to flexibly simulate the physical dynamic state represented by differential equations, which opens the way for us to understand more physical phenomena later.

A deep energy method for functionally graded porous beams

The proposed method completely eliminates the need of a discretization technique, such as the finite element method, and optimizes the potential energy of the beam to train the neural network.

31 References

Artificial neural networks for solving ordinary and partial differential equations

This article illustrates the method by solving a variety of model problems and presents comparisons with solutions obtained using the Galekrkin finite element method for several cases of partial differential equations.

Neural network as a function approximator and its application in solving differential equations

This work explores the use of NN for the function approximation and proposes a universal solver for ODEs and PDEs, and exhibits high accuracy for not only the unknown functions but also their derivatives.

A unified deep artificial neural network approach to partial differential equations in complex geometries

Numerical solution of differential equations using multiquadric radial basis function networks

Machine-learning solver for modified diffusion equations

This work presents a universal, machine-learning based solver for multi-variable partial differential equations that approximates the target functions by neural networks and adjusts the network parameters to approach the desirable solutions.

Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations

This two part treatise introduces physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations and demonstrates how these networks can be used to infer solutions topartial differential equations, and obtain physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters.

Chebyshev Neural Network based model for solving Lane-Emden type equations

Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations

We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial…

Approximation by superpositions of a sigmoidal function

  • G. Cybenko
  • Computer Science
    Math. Control. Signals Syst.
  • 1989
In this paper we demonstrate that finite linear combinations of compositions of a fixed, univariate function and a set of affine functionals can uniformly approximate any continuous function ofn real…

Automatic differentiation in machine learning: a survey

By precisely defining the main differentiation techniques and their interrelationships, this work aims to bring clarity to the usage of the terms “autodiff’, “automatic differentiation”, and “symbolic differentiation" as these are encountered more and more in machine learning settings.