Inverse Problem of Identification of Thermophysical Parameters of the Green-Nagdi Type III Model for an Elastic Rod Based on a Physically Informed Neural Network

Abstract

In this paper, we study the inverse problem of identifying the dimensionless thermal conductivity coefficient for the Green–Naghdi equation of type III, which describes the propagation of thermal disturbances with a finite velocity and takes into account the inertial effects of heat flux. For the inverse problem, the stability requirement (Hadamard criteria) is violated, as a result of which even minimal data distortions lead to significant errors in parameter identification. As a solution method, we use an approach based on physically informed neural networks (PINN), which combines the capabilities of deep learning with a priori knowledge of the structure of the differential equation. The parameter is included among the trained variables, and the loss function is formed based on the deviation from the differential equation, boundary conditions, initial conditions, and noisy experimental data from a point sensor. The results of computational experiments are presented, demonstrating high accuracy of parameter recovery (error less than 0.03%) and the stability of the method with respect to the presence of additive Gaussian noise in the data. The PINN method has proven itself to be an effective tool for solving ill-posed inverse problems of mathematical physics.