Adapting PINN Models of Physical Entities to Dynamical Data

Abstract

This article examines the possibilities of adapting approximate solutions of boundary value problems for differential equations using physics-informed neural networks (PINNs) to changes in data about the physical entity being modelled. Two types of models are considered: PINN and parametric PINN (PPINN). The former is constructed for a fixed parameter of the problem, while the latter includes the parameter for the number of input variables. The models are tested on three problems. The first problem involves modelling the bending of a cantilever rod under varying loads. The second task is a non-stationary problem of a thermal explosion in the plane-parallel case. The initial model is constructed based on an ordinary differential equation, while the modelling object satisfies a partial differential equation. The third task is to solve a partial differential equation of mixed type depending on time. In all cases, the initial models are adapted to the corresponding pseudo-measurements generated based on changing equations. A series of experiments are carried out for each problem with different functions of a parameter that reflects the character of changes in the object. A comparative analysis of the quality of the PINN and PPINN models and their resistance to data changes has been conducted for the first time in this study.

Keywords

• PINN
• parametric PINN
• dynamical system modelling
• data-driven adaptation
• multi-fidelity
• mixed type equation