Fundamentals of Physics-Informed Neural Networks Applied to Solve the Reynolds Boundary Value Problem
Andreas Almqvist
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.
Topics & Concepts
Finite element methodContext (archaeology)Artificial neural networkBoundary value problemComputer scienceApplied mathematicsPartial differential equationBoundary (topology)Ordinary differential equationFinite difference methodMathematical optimizationCalculus (dental)MathematicsDifferential equationMathematical analysisArtificial intelligenceEngineeringMedicineStructural engineeringPaleontologyDentistryBiologyModel Reduction and Neural NetworksHydraulic and Pneumatic SystemsHeat Transfer Mechanisms