Elliptic Partial Differential Equation

Elliptic partial differential equations (PDEs) describe a wide range of steady-state physical phenomena, and solving them efficiently is crucial across many scientific and engineering disciplines. Current research focuses on developing and analyzing novel numerical methods, particularly those leveraging neural networks, including architectures like operator networks and physics-informed neural networks, to overcome computational challenges associated with high dimensionality and complex geometries. These advanced techniques aim to improve accuracy, speed, and scalability compared to traditional methods, ultimately impacting fields like fluid dynamics, materials science, and optimal control.