**1. Introduction**

There are various categories of artificial neural networks (ANN) and a physicsinformed neural network (PINN), see [1] for a recent review on the matter, is a neural network trained to solve both supervised and unsupervised learning tasks while satisfying some given laws of physics, which may be described in terms of nonlinear partial differential equations (PDE). For example, the balance of momentum and conservation laws in solid- and fluid mechanics and various types of initial value problems (IVP) and boundary value problems (BVP), see e.g., [2,3]. The application of a PINN (of this type) to solve differential equations, renders meshless numerical solution procedures [4], and an important feature from a machine learning perspective, is that it is not a data-driven approach requiring a large set of training data to learn the solution.

In fluid mechanics, under certain assumptions, i.e., that the fluid is incompressible, iso-viscous, the balance of linear momentum and the continuity equation, for flows in narrow interfaces reduces to the classical Reynolds equation [5]. For more recent work establishing lower-dimensional models in a similar manner, see e.g., [6–8]. The present work describes how a PINN can be adapted and trained to solve both initial and boundary value problems, described by ordinary differential equations, numerically. The theoretical description starts by presenting the neural network's architecture and it is first applied to solve an initial value problem, which is described by a first order ODE, which can be solved analytically so that the validity of the solution can be thoroughly assessed. Thereafter, it is used to obtain a PINN for the classical one-dimensional Reynolds equation, which is a boundary value problem governing, e.g., the flow of lubricant between the runner and the stator in a 1D slider bearing. The novelty and originality of the present work lays in the explicit mathematical description of the cost function, which constitutes the *physics-informed* feature of the ANN, and the associated gradient with respect to the networks weights and bias. Important features of this particular numerical solution procedure, that is publicly available here: [9], are that it is not data driven, i.e., no training data need to be provided and that it is a meshless method [4].

**Citation:** Almqvist, A. Fundamentals of Physics-Informed Neural Networks Applied to Solve the Reynolds Boundary Value Problem. *Lubricants* **2021**, *9*, 82. https://doi.org/10.3390/ lubricants9080082

Received: 31 July 2021 Accepted: 14 August 2021 Published: 19 August 2021

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