Application of machine learning technologies to the study of thermoelastic wave processes


Аuthors

Son P. T.

Le Quy Don Technical University, 236 Hoang Quoc Viet, Ha Noi, Viet Nam

e-mail: sonphantungk49@gmail.com

Abstract

In recent years, there has been a rapid development of artificial intelligence technologies and, in particular, deep learning methods, which opens up new horizons for solving applied problems in mathematical physics. One of the most promising areas in this field is the use of Physically Informed Neural Networks (PINNs), which allow prior information about physical and mathematical models to be embedded directly into the training process. This enables the construction of solutions to complex differential equations describing various processes in continuum mechanics, without the need for meshing, which is typical of traditional numerical methods.
This paper is devoted to the application of deep learning methods to the modeling and analysis of transient processes in structural elements, such as elastic and thermoelastic layers. The study focuses on solving direct problems in the mechanics of deformable solids: the problem of nonstationary oscillations of an elastic layer, the problem of heat conduction in a thermally conductive layer, and the coupled thermoelasticity problem with mutual influence of mechanical and thermal fields. These problems have significant practical relevance for analyzing structures under varying loads and thermal effects, especially in aerospace and mechanical engineering.
The proposed approach is based on the use of PINNs, which approximate the desired solutions by parametric neural network models. Each problem formulation includes a corresponding loss function that incorporates residuals of the governing differential equations, initial and boundary conditions, and model parameters. Collocation points used for evaluating the loss function are randomly distributed in the space-time domain, which ensures flexibility and generality of the method.
For each problem, the results obtained via PINNs are compared with exact analytical solutions (constructed using the method of separation of variables) and numerical results obtained by the finite difference method. It is shown that with proper selection of the neural network architecture and hyperparameters (such as number of hidden layers, neurons per layer, activation functions, optimization algorithms, etc.), high approximation accuracy can be achieved, comparable to that of classical numerical techniques. Special attention is paid to the analysis of convergence and error estimation. The numerical experiments demonstrate that increasing the network complexity improves accuracy, although at the cost of higher computational expenses. The hyperbolic tangent activation function proved to be the most effective in this study.
A key advantage of the proposed approach is its universality: the same code, with minor modifications, can be used for both forward and inverse problem settings. This is especially important in the presence of experimental data with uncertainties, where traditional methods become unstable. Furthermore, the method can be easily adapted to problems with variable physical parameters, complex geometries, or high dimensionality.
In conclusion, it is emphasized that the PINN-based approach demonstrates high accuracy, robustness, and flexibility in solving problems of continuum mechanics. It can be effectively used as either an auxiliary or standalone tool for the analysis of transient thermoelastic processes, acoustics, structural dynamics, as well as in solving inverse design and system diagnostics problems. The results presented in this work confirm the strong potential of further research in this area, including noisy and contact problems, as well as problems involving nonlinear material behavior.

Keywords:

thermoelasticity, wave processes, physically informed neural networks (PINN), deep learning, machine learning, transient processes, mechanics of deformable solids, inverse problems, heat conduction

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