Feedforward Neural Network for Solving Partial Differential Equations

Hayati, Mohsen; Karami, Behnam

doi:10.3923/jas.2007.2812.2817

Cited by 34 publications

(13 citation statements)

References 6 publications

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“…For example, one of the problems proposed in the Airbus Quantum Computing Challenge [19] is building a neural network that solves Burgers' equation with at least the same level of accuracy and higher computational performance as the traditional numerical methods. In the article [20], a feedforward neural network is trained to satisfy Burgers' equation and certain initial conditions, but the computational performance of the approach is not estimated. In this section, we demonstrate how to build a Lie transform-based neural network that solves Burgers' equation.…”

Section: Partial Differential Equationsmentioning

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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Section: Partial Differential Equationsmentioning

confidence: 99%

Matrix Lie Maps and Neural Networks for Solving Differential Equations

Ivanov,

Andrianov

2019

Preprint

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“…However, their work is not easily extensible to non-rectangular domains as it requires construction of auxiliary functions, which is only feasible for rectangular domains. Other researchers extended the approach of Dissanayake and Phan-Thien to the solution of nonlinear Schrodinger equation [35], chemical reactor problems [36], self-gravitating systems of N bodies [37], and the solution of one-dimensional Burgers equation [38].…”

Section: Introductionmentioning

confidence: 99%

Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion

Basir,

Senocak

2021

Preprint

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Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE). In PINNs, the residual form of the PDE of interest and its boundary conditions are lumped into a composite objective function as an unconstrained optimization problem, which is then used to train a deep feed-forward neural network.Here, we show that this specific way of formulating the objective function is the source of severe limitations in the PINN approach when applied to different kinds of PDEs. To address these limitations, we propose a versatile framework that can tackle both inverse and forward problems. The framework is adept at multi-fidelity data fusion and can seamlessly constrain the governing physics equations with proper initial and boundary conditions. The backbone of the proposed framework is a nonlinear, equality-constrained optimization problem formulation aimed at minimizing a loss functional, where an augmented Lagrangian method (ALM) is used to formally convert a constrained-optimization problem into an unconstrainedoptimization problem. We implement the ALM within a stochastic, gradient-descent type training algorithm in a way that scrupulously focuses on meeting the constraints without sacrificing other loss terms. Additionally, as a modification of the original residual layers, we propose lean residual layers in our neural network architecture to address the so-called vanishing-gradient problem. We demonstrate the efficacy and versatility of our physics-and equality-constrained deep-learning framework by applying it to learn the solutions of various multi-dimensional PDEs, including a nonlinear inverse problem from the hydrology field with multi-fidelity data fusion. The results produced with our proposed model match exact solutions very closely for all the cases considered.

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“…A number of methods for this approach include Runge-Kutta, finite difference, etc. are available for approximate the solution accurately and efficiently [5][6][7]. However, while these methods are efficient and well-studied, these traditional methods are require much memory space and time.…”

Section: Introductionmentioning

confidence: 99%

“…However, while these methods are efficient and well-studied, these traditional methods are require much memory space and time. Thus made the approximation computational process costly [7]. As alternative approach, we can replace traditional numerical discretization method with Artificial Neural Networks (ANNs) to approximates the PDE solution [8].…”

Section: Introductionmentioning

confidence: 99%

ANNs-Based Method for Solving Partial Differential Equations : A Survey

Pratama¹,

Bakar²,

Mohan³

et al. 2021

Preprint

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Conventionally, partial differential equations (PDE) problems are solved numerically through discretization process by using finite difference approximations. The algebraic systems generated by this process are then finalized by using an iterative method. Recently, scientists invented a short cut approach, without discretization process, to solve the PDE problems, namely by using machine learning (ML). This is potential to make scientific machine learning as a new sub-field of research. Thus, given the interest in developing ML for solving PDEs, it makes an abundance of an easy-to-use methods that allows researchers to quickly set up and solve problems. In this review paper, we discussed at least three methods for solving high dimensional of PDEs, namely PyDEns, NeuroDiffEq, and Nangs, which are all based on artificial neural networks (ANNs). ANN is one of the methods under ML which proven to be a universal estimator function. Comparison of numerical results presented in solving the classical PDEs such as heat, wave, and Poisson equations, to look at the accuracy and efficiency of the methods. The results showed that the NeuroDiffEq and Nangs algorithms performed better to solve higher dimensional of PDEs than the PyDEns.

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Feedforward Neural Network for Solving Partial Differential Equations

Cited by 34 publications

References 6 publications

Matrix Lie Maps and Neural Networks for Solving Differential Equations

Matrix Lie Maps and Neural Networks for Solving Differential Equations

Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion

ANNs-Based Method for Solving Partial Differential Equations : A Survey

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