Numerical solution of differential equations by radial basis function neural networks

Li, Jianyu; Luo, Siwei; Qi, Yingjian; Huang, Yaping

doi:10.1109/ijcnn.2002.1005571

Cited by 7 publications

(2 citation statements)

References 4 publications

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“…Other related work. Early works on approximating the ODE solutions without numerical solvers used splines or radial basis functions [55,50], or functions similar to modern ResNets [45]. More recently, [66] approximate the solution by minimizing the error of the solution points and of the boundary condition.…”

Section: Discussionmentioning

confidence: 99%

Neural Flows: Efficient Alternative to Neural ODEs

Biloš¹,

Sommer²,

Rangapuram³

et al. 2021

Preprint

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Neural ordinary differential equations describe how values change in time. This is the reason why they gained importance in modeling sequential data, especially when the observations are made at irregular intervals. In this paper we propose an alternative by directly modeling the solution curves -the flow of an ODE -with a neural network. This immediately eliminates the need for expensive numerical solvers while still maintaining the modeling capability of neural ODEs. We propose several flow architectures suitable for different applications by establishing precise conditions on when a function defines a valid flow. Apart from computational efficiency, we also provide empirical evidence of favorable generalization performance via applications in time series modeling, forecasting, and density estimation.

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Section: Discussionmentioning

confidence: 99%

Neural Flows: Efficient Alternative to Neural ODEs

Biloš¹,

Sommer²,

Rangapuram³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…S. He et al used feedforward neural networks to solve a special class of linear first-order partial differential equations [8]. The radial basis function (RBF) network architecture [10] has also been used for the solution of differential equations in [9], where Jianyu et al demonstrated a method for solving linear ordinary differential equations based on multiquadric RBF networks. Ramuhalli et al proposed an artificial neural network with an embedded finite-element model, for the solution of partial differential equations [11].…”

Section: Introductionmentioning

confidence: 99%

Constructing Runge–Kutta methods with the use of artificial neural networks

Anastassi

2013

Neural Comput & Applic

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A methodology that can generate the optimal coefficients of a numerical method with the use of an artificial neural network is presented in this work. The network can be designed to produce a finite difference algorithm that solves a specific system of ordinary differential equations numerically. The case we are examining here concerns an explicit two-stage Runge-Kutta method for the numerical solution of the two-body problem. Following the implementation of the network, the latter is trained to obtain the optimal values for the coefficients of the Runge-Kutta method. The comparison of the new method to others that are well known in the literature proves its efficiency and demonstrates the capability of the network to provide efficient algorithms for specific problems.

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