*Technical Note* **Fundamentals of Physics-Informed Neural Networks Applied to Solve the Reynolds Boundary Value Problem**

**Andreas Almqvist**

Division of Machine Elements, Luleå University of Technology, SE-971 87 Luleå, Sweden; andreas.almqvist@ltu.se

**Abstract:** This paper presents a complete derivation and design of a physics-informed neural network (PINN) applicable to solve initial and boundary value problems described by linear ordinary differential equations. The objective with this technical note is not to develop a numerical solution procedure which is more accurate and efficient than standard finite element- or finite difference-based methods, but to give a fully explicit mathematical description of a PINN and to present an application example in the context of hydrodynamic lubrication. It is, however, worth noticing that the PINN developed herein, contrary to FEM and FDM, is a meshless method and that training does not require big data which is typical in machine learning.

**Keywords:** PINN; machine learning; reynolds equation
