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

Physics informed neural networks (PINNs), a type of machine learning approach, can be used to find the solution of differential equations by including all of the physics into the loss function and building a neural network that approximates the solution. In PINN, the neural network is optimized in such a way that the loss function is taken as a residual of the governing differential equation, boundary conditions, and initial conditions. The fundamental idea of PINN is that the neural network approximates the solution of a differential equation and satisfies any given constraints such that the loss function is minimized. A few of the earliest examples of using artificial neural networks to determine the solution of differential equations are in the work of Dissanayake et al.1 and I.E. The differential equation is presumed to be satisfied by a trial solution in the approach suggested by Lagaris et al..