Joint Research on Partial Differential Equation Solving and Computer Simulation Technology In Electromagnetic Field Numerical Simulation

Many in the mathematical community study partial differential equations (PDEs), which are essential yet often difficult to solve analytically. Due to the lack of general analytical tools, numerical methods are widely used, though researchers continue exploring new approaches. Deep learning has recently achieved remarkable success in areas like image classification and natural language processing, and its application to PDEs is gaining attention. Deep neural networks have proven highly effective at function approximation, making them promising tools for solving PDEs. A growing interdisciplinary field is using deep learning to tackle PDEs in engineering and physics. Physics-Informed Neural Networks (PINNs) have emerged as a standard method by embedding physical laws directly into the learning process. This study presents a PINN-based framework enhanced by analytical insights and electromagnetic domain knowledge. It applies the method to electromagnetic field problems using Maxwell, wave, diffusion, and Laplace equations. PINNs accurately reproduce physical behaviors, aligning well with theoretical solutions. By incorporating PDE-based physical information as a regularizing term, PINNs improve neural network performance. This study also evaluates the wave, KdV-Burgers, and KdV equations. Experimental results confirm that PINNs are highly effective for solving PDEs and warrant further research in scientific computing.