Helmholtz Equation

The Helmholtz equation, a partial differential equation describing wave propagation, is a cornerstone of numerous scientific fields. Current research focuses on improving its numerical solution, particularly for high-frequency waves and complex geometries, using techniques like physics-informed neural networks (PINNs), multigrid methods, and novel loss functions such as the wave loss function, often incorporated into architectures like U-Nets or RVFL networks. These advancements enhance the accuracy, efficiency, and robustness of solving the Helmholtz equation, impacting diverse applications from acoustic modeling and fluid dynamics to medical imaging and machine learning.