Development and optimization of physics-informed neural networks for solving partial differential equations
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Date
2025
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SDU University
Abstract
This thesis talks about using physics-informed neural networks (PINNs) to solve Poisson equations in both one-dimensional and two-dimensional areas. These equations are common in many physical problems, like heat transfer and electrostatics. The results from PINNs are compared to the finite difference method (FDM), which is a classical numerical method often used to solve these kinds of equations. The study shows that PINNs can give results that are close to those from FDM, with the added benefit of being more flexible for different types of problems. Another part of this work focuses on using multi-task learning with PINNs. In this part, the neural network does more than one job. It not only finds the solution of the differential equation, but it also learns unknown values or parameters that are part of the equation. For example, in one test problem, the equation had a source term and a coefficient that changes depending on the position. The PINN was able to learn both of them correctly while still solving the equation with a low training error. The results show that PINNs can work well even when the equation is more complex or has unknown parts. The model showed good performance on new or unseen data and was able to find the correct hidden values in the system. Because of this, PINNs may be very useful in future applications for solving advanced problems in science and engineering, especially where traditional methods might be harder to use.
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Keywords
physics-informed neural networks, differential equations, optimization
Citation
Sharimbayev B / Development and optimization of physics-informed neural networks for solving partial differential equations / SDU University / Faculty of Engineering and Natural Sciences