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

Anastassi, Angelos A.

doi:10.1007/s00521-013-1476-x

Cited by 12 publications

(8 citation statements)

References 15 publications

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“…As a special case, the explicit Euler scheme leads to a one-block architecture. Such NN representation of numerical schemes have been investigated in previous works [9,10] to estimate coefficients {α i } i and {β i } i for a given ODE. The objective is here to learn all the parameters of the NN representation of the unkown ODE governing observed time series.…”

Section: Runge-kutta Methods As Residual Neural Netsmentioning

confidence: 99%

Bilinear Residual Neural Network for the Identification and Forecasting of Geophysical Dynamics

Fablet¹,

Ouala²,

Herzet³

2018

2018 26th European Signal Processing Conference (EUSIPCO)

107

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Due to the increasing availability of large-scale observation and simulation datasets, data-driven representations arise as efficient and relevant computation representations of dynamical systems for a wide range of applications, where modeldriven models based on ordinary differential equation remain the state-of-the-art approaches. In this work, we investigate neural networks (NN) as physically-sound data-driven representations of such systems. Reinterpreting Runge-Kutta methods as graphical models, we consider a residual NN architecture and introduce bilinear layers to embed non-linearities which are intrinsic features of dynamical systems. From numerical experiments for classic dynamical systems, we demonstrate the relevance of the proposed NN-based architecture both in terms of forecasting performance and model identification.

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Section: Runge-kutta Methods As Residual Neural Netsmentioning

confidence: 99%

Bilinear Residual Neural Network for the Identification and Forecasting of Geophysical Dynamics

Fablet¹,

Ouala²,

Herzet³

2018

2018 26th European Signal Processing Conference (EUSIPCO)

107

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“…If f (x, u(t, x)) is the same discretization as in (7), then the equation ( 8) leads to the system of 2000 ODEs…”

Section: Lie Transform-based Neural Networkmentioning

confidence: 99%

“…Other approaches rely on the implementation of a traditional step-by-step integrating method in a neural network basis [7,8]. In the article [8], the author proposes such an architecture.…”

Section: Introductionmentioning

confidence: 99%

Matrix Lie Maps and Neural Networks for Solving Differential Equations

Ivanov,

Andrianov

2019

Preprint

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The coincidence between polynomial neural networks and matrix Lie maps is discussed in the article. The matrix form of Lie transform is an approximation of the general solution of the nonlinear system of ordinary differential equations. It can be used for solving systems of differential equations more efficiently than traditional step-by-step numerical methods. Implementation of the Lie map as a polynomial neural network provides a tool for both simulation and data-driven identification of dynamical systems. If the differential equation is provided, training a neural network is unnecessary. The weights of the network can be directly calculated from the equation. On the other hand, for data-driven system learning, the weights can be fitted without any assumptions in view of differential equations. The proposed technique is discussed in the examples of both ordinary and partial differential equations. The building of a polynomial neural network that simulates the Van der Pol oscillator is discussed. For this example, we consider learning the dynamics from a single solution of the system. We also demonstrate the building of the neural network that describes the solution of Burgers' equation that is a fundamental partial differential equation.

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“…To alleviate the computational cost, Ramuhalli, et al [12] proposed a parallelly formed finite element NN or FENN method with directly calculated weights. Anastassi [13] proposed a method by combining the NN and Runge-Kutta method for solving the two-body problems. The efficiency of the method was proven by comparing it with those of the classical methods.…”

Section: Introductionmentioning

confidence: 99%

A Novel Neural Network Cell Method for Solving Nonlinear Electromagnetic Problems

Zhu

et al. 2021

Advcd Theory and Sims

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Effective analysis of nonlinear electromagnetic fields is essential for the accurate modeling of electromagnetic devices, such as transformers, generators, and motors. This paper proposes a novel approach of coupled neural network (NN) and cell method (CM) or NNCM for solving nonlinear electromagnetic problems with ferromagnetic domains. While the topologically linear relations of the cell complexes are mathematically assembled through a transformation in the Tonti diagram by the CM, and the constitutive nonlinear magnetic relations are dealt with by partially connected NN for the fast prediction of the permeability distribution inside the ferromagnetic domain. Since the construction of NN is directly related to the grid connections, a partially connected NN structure with a small number of neurons can reduce the computational cost of the training process. By using a compact NN, the proposed NNCM can effectively eliminate the time consuming iterations for determining the nonlinear permeability distribution, and improve the computational efficiency significantly. The NNCM is employed to analyze the transient electromagnetic field distribution inside a cylindrical ferromagnetic core. The results are compared with those obtained by the traditional iterative CM, which determines the nonlinear permeability distribution by lengthy numerical iterations, to verify the feasibility and effectiveness of the proposed NNCM.

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

Cited by 12 publications

References 15 publications

Bilinear Residual Neural Network for the Identification and Forecasting of Geophysical Dynamics

Bilinear Residual Neural Network for the Identification and Forecasting of Geophysical Dynamics

Matrix Lie Maps and Neural Networks for Solving Differential Equations

A Novel Neural Network Cell Method for Solving Nonlinear Electromagnetic Problems

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