Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation

Abstract

In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) approximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable large time-steps and a relatively small number of nodes with satisfactory accuracy. The numerical examples show that the proposed method is very attractive for simulating the high-dimensional SGEs.Keywords Sine-Gordon equations · Meshless methods · High dimensional model representation 1 IntroductionThe sine-Gordon equation (SGE) is a nonlinear hyperbolic partial differential equation (PDE) involving the d'Alembert operator and the sine of the unknown function, and the SGE plays an important role in many mathematical physics applications. It was originally introduced by Bour [6] and rediscovered

Exploring Physics‐Informed Neural Networks for the Generalized Nonlinear Sine‐Gordon Equation

“…Function. Similarly to the above example, the loss function is expressed as the summation of the square of the diference corresponding to each of the equations in (31). Te loss function used to train the PINN with the parameter P is given by (15) where…”

“…where Te train and test loss of this model is shown in Figure 16. Since we have previously demonstrated that, for example one, the combined Adam and L-BFGS-B optimization algorithm the optimal optimization for our model, we employed this mixed optimization technique to minimize the loss function of the problem (31). Similarly, we used the tanh as an activation function to predict the solution of the suggested instances given by (31) by PINN.…”

“…Figures 17 and 18 show the exact solution and the resulting PINN solution with the corresponding absolute error to the problem (31). Te graphs of the 2D and 3D solution plots for the combined model optimization Adam and L-BFGS-B allow a comparison of the two solutions.…”

“…Te clique polynomial is regarded as a fundamental function for operational matrices in this method. For more details, refer to the following references: [30][31][32][33][34][35][36].…”

A technical pipeline for screening microbial communities as a function of substrate specificity through single cell fluorescent imaging

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