A Neural Network Study of Blasius Equation

Abstract

In this work we applied a feed forward neural network to solve Blasius equation which is a thirdorder nonlinear differential equation. Blasius equation is a kind of boundary layer flow. We solved Blasius equation without reducing it into a system of first order equation. Numerical results are presented and a comparison according to some studies is made in the form of their results. Obtained results are found to be in good agreement with the given studies. I. * Electronic address: halilmutuk@gmail.com arXiv:1811.08936v1 [cs.LG] 8 Nov 2018 * The neural network based solution of a differential equation is differentiable and is in closed analytic form that can be used in any subsequent calculation. On the other hand most other techniques offer a discrete solution or a solution of limited differentiability. * Trial solutions of ANN include a single independent variable regardless of the dimension of the problem. * The neural network based method to solve a differential equation provides a solution with very good generalization properties. * The solutions are continuous over all the domain of the integration. Traditional numerical methods provide solutions over discrete points and the solution among these points must be interpolated. * The required number of model parameters is far less than any other solution technique and therefore, compact solution models are obtained, with very low demand on memory space. * The method is general and can be applied to ordinary differential equations (ODEs), systems of ODEs and to partial differential equations (PDEs) as well. * The method is general and can be applied to the systems defined on either orthogonal box boundaries or on irregular arbitrary shaped boundaries. * The method can be realized in hardware, using neuroprocessors, and hence offer the opportunity to tackle in real time difficult differential equation problems arising in many engineering applications. * The method can also be implemented on parallel architectures.

Solving Schrödinger Wave Equation for the Charmonium Spectrum Using Artificial Neural Networks

“…Artificial neural networks are being used to solve different problems relevant to the optimization of high-energy physics processes [12][13][14][15][16][17]. In theoretical HEP, artificial neural networks are being used to calculate the mass spectra of particles by solving the Schrödinger wave equation [16][17][18][19][20], while in experimental high-energy physics, artificial neural networks are being used in event classification [21,22], object reconstruction [23,24], triggering [25,26], and track fitting [27,28]. Apart from these, artificial neural networks are also being used to solve quantum many-body problems [29] through ordinary and partial differential equations in different domains [30][31][32].…”

“…He suggested that neural networks can automatically discover high-level features from the data [47]. In 2018, Mutuk also used the trial function method based on neural networks to solve the Blasius differential equation for fluid mechanics and compared the results with the existing numerical techniques [16]. Baldi et al [14] used a parameterized neural network for the analysis and prediction of new particles in the field of high-energy physics.…”

“…We followed the formalism used by Lagaris et al [30] and Mutuk [16][17][18][19][20] for the implementation of ANN to solve the Schrodinger wave equation for the charmonium system. First, we provide the steps for solving any differential equation by employing ANNs under this technique [16][17][18][19][20]30].…”

Physics-informed Neural Networks approach to solve the Blasius function

“…Still, specific methods were needed to handle the unbounded boundary conditions. The Blasius benchmark problem has been resolved [8] using the trial function method put out by Lagaris [9] or a hybrid approach [10]. The solution function is seen to have a singularity on the negative real axis at approximately -5.69.…”

Wavelets based physics informed neural networks to solve non-linear differential equations

“…Khandelwal et al 50 introduced the Adomian Mohand transform method which gave numerical values as well as power series close-form solutions. Mutuk 51 used a feed-forward neural network to solve the Blasius problem using the method proposed by Lagaris et al 2 . Recently Bararnia 52 et al carried out the first study of PINN in solving the problem by exploring the application of PINN in an unbounded domain.…”