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

Using a wavelet, one can represent a given function in several scale components36. The wavelet function (ψ) and the scaling function (ϕ) are the two functions that define wavelets. Because of the excellent properties of wavelets, many researchers have shown a strong interest in numerical analysis using wavelet theory. Wavelets can represent a given function with a lower number of coefficients and a faster algorithm. A few other properties include space and frequency localization.

In this study, PINN using wavelet as an activation function is applied to solve five problems: firstly the Blasius equation (a nonlinear differential equation defined on an unbounded domain), a linear and non-linear coupled equation, and then the Burger’s equation for two different cases (nonlinear partial differential equation).

Hornik53 established neural networks as universal function approximators. Hence, we can approximate any function using an appropriate neural network.

We demonstrate the versatility of the suggested method by extending its application to solve both linear coupled equations, non-linear coupled equations and a partial differential equation in addition to the Blasius equation.

In this work, the Blasius equation was solved using PINN with wavelet as an activation function, and the method was then extended to solve a linear coupled equation, a non-linear coupled equation and partial differential equations. Three distinct wavelet functions were applied, and the results were compared. The findings of all the problems that were taken into consideration showed that utilizing wavelet as the activation function in PINN produces results with improved accuracy.