TY - JOUR
TI - Uniqueness of the solution for the conjugation problem of third-order partial differential equations and its application in neural network regularization
AU - Arkabaev Nurkasym K
AU - Abdelsalam Hanadi
AU - Daqqab Ibtisam
AU - Alkhalafc Salem
AU - Ibnoufd Hawa
AU - Khaleke Sayed Abdel
JN - Thermal Science
PY - 2025
VL - 29
IS - 1
SP - 371
EP - 382
PT - Article
AB - This article investigates the uniqueness of the solution for the conjugation  problem of third-order PDE with a characteristic line and its application in  neural network regularization. A theorem on the uniqueness of the solution  for the considered class of equations is proven. Based on the obtained  theoretical results, a novel neural network regularization method is  developed that accounts for the physical constraints of the problem. A  comparison is made between the classical finite difference method and an  innovative approach based on physics-informed neural networks. Numerical  experiments demonstrated that the proposed method provides higher accuracy  and better adherence to the physical constraints of the problem. The  regularized neural networks exhibited lower mean square error, better  compliance with conjugation conditions, and higher resilience to input data  variations compared to classical methods and standard neural networks. The  research opens new perspectives for integrating classical mathematical  methods with modern machine learning technologies.