PHYSICS-INFORMED NEURAL NETWORK (PINN) METHOD BASED ON SELF-SIMILAR SOLUTIONS

In the numerical solution of partial differential equations that describe complex physical processes such as heat conduction and gas dynamics, substantial computational resources are often required. To address these challenges, Physics-Informed Neural Networks (PINNs) have gained increasing attention in recent years within the fields of science and engineering. This paper investigates the application of the PINN methodology to obtain solutions to the heat conduction and gas dynamics equations. Unlike traditional numerical approaches, the physics-informed neural network framework incorporates governing physical laws directly into the neural network architecture. Consequently, the solution is constrained not only by data but also by the underlying differential equations. The paper presents the architecture of the PINN framework and details the structure of loss functions, demonstrating their relationship with the heat equation and the Euler equations using specific examples. Furthermore, the implementation of initial and boundary conditions is discussed, along with an analysis of factors influencing the stability and accuracy of the obtained solutions. The results highlight the efficiency of PINNs and demonstrate their potential for solving complex multiphase and high-dimensional problems in the future. Additionally, current research directions aimed at accelerating the computational process and enhancing the robustness of PINNs are outlined.

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