Physics-informed neural network solution of 2nd order ODE:s

A PINN employed to solve c(x)y''+c'(x)y'-f = 0, y(0)=y(1)=0, using symbolic differentiation and the gradient decent method.

This rutine presents the design of a physics-informed neural networks applicable to solve initial- and boundary value problems described by linear ODE:s. The objective not to develop a numerical solution procedure which is more accurate and efficient than standard finite element or finite difference based methods, but to present the concept of the construction of a PINN, in the context of hydrodynamic lubrication. It is, however, worth to notice that the present PINN, contrary to FEM and FDM, is a meshless method and that it is not a datadriven machine learning program. This concept may, of course, be generalised, and perhaps it turns out to be more accurate and efficient than existing routines in solving related but nonlinear problems. This is, however, scope of future research in this direction.

Almqvist, A. (2021). Fundamentals of Physics-Informed Neural Networks Applied to Solve the Reynolds Boundary Value Problem. Lubricants, 9(8). https://doi.org/10.3390/lubricants9080082